Stress Testing Credit Risk Portfolios

Michael Jacobs PhD CFA

Senior Financial Economist

Credit Risk Analysis Division

US Office of the Comptroller of the Currency

Risk Incisive Media Training March 2012

The views expressed herein are those of the author and do not necessarily represent the views of the Office of the Comptroller of the Currency or the Department of the Treasury

>

Outlinebull Introduction bull The Function of Stress Testingbull Supervisory Requirements and Expectationsbull The Credit Risk Parameters for Stress Testingbull Interpretation of Stress Test Resultsbull A Typology of Stress Tests

ndash Uniform Testingndash Risk Factor Sensitivitiesndash Scenario Analysis

bull Historical Scenariosbull Statistical Scenariosbull Hypothetical Scenarios

bull Procedures for Conducting Stress Testsbull A Simple Stress Testing Example

Introduction Overview

bull Modern credit risk modeling (eg Merton 1974) increasingly relies on advanced mathematical statistical and numerical techniques to measure and manage risk in credit portfolios

bull This gives rise to model risk (OCC 2011-16) and the possibility of understating inherent dangers stemming from very rare yet plausible occurrencs perhaps not in our reference data-sets

bull International supervisors have recognized the importance of stress testing credit risk in the Basel framework (BCBS 2009)

bull It can and has been argued that the art and science of stress testing has lagged in the domain of credit vs other types of risk (eg market) and our objective is to help fill this vacuum

bull We aim to present classifications amp established techniques that will help practitioners formulate robust credit risk stress tests

Introduction Motivation in the Financial Crisis

Reproduced from Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

Figure 3 Average Ratio of Total Charge-offs to Total Value of Loans for Top 50 Banks as of 4Q09

(Call Report Data 1984-2009)

0

0005

001

0015

002

0025

003

0035

19840331

19841231

19850930

19860630

19870331

19871231

19880930

19890630

19900331

19901231

19910930

19920630

19930331

19931231

19940930

19950630

19960331

19961231

19970930

19980630

19990331

19991231

20000930

20010630

20020331

20021231

20030930

20040630

20050331

20051231

20060930

20070630

20080331

20081231

20090930

bull Bank losses in the recent financial crisis exceed levels observed in recent history

bull This illustrates the inherent limitations of backward looking models ndash we must anticipate risk

Introduction Motivation in the Imprecision of Value-at-Risk

Gaussian Copula Bootstrapped (Margins) Distribution of 9997 Percentile VaR

VaR997=764e+8 q25=626e+8 q975=894e+8 CV=35379997 Percentile Value-at-Risk for 5 Risk Types(CrMktOpsLiquampIntRt) Top 200 Banks (1984-2008)

Den

sity

5e+08 6e+08 7e+08 8e+08 9e+08 1e+09

0e+0

01e

-09

2e-0

93e

-09

4e-0

95e

-09

6e-0

9

bull Sampling variation in VaR inputs leads to huge confidence bounds for risk estimates (coefficient of variation =354)

bull This is even assuming we have the correct model

Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

Conceptual Issues in Stress Testing Risk vs Uncertainty

bull Knight (1921) uncertainty is when a probability distribution is unmeasurable or unknown arguably a realistic scenario

bull Rely upon empirical data to estimate loss distributions but this is complicated because of changing economic conditions

bull Popper (1945) situations of uncertainty closely associated amp inherent to changes in knowledge amp behavior (no historicism)

bull Shackle (1990) predictions reliable only for immediate future as impact othersrsquo choices after time has an appreciable effect

bull This role of human behavior in economic theory was a key impetus behind rational expectations amp behavioral finance

bull Implication is that risk managers must be aware of model limitations amp how an EC regime itself changes behavior

bull Although we face uncertainty valuable to estimate loss distributions in that helps make explicit sources of uncertainty

The Function of Stress Testingbull A possible definition of stress testing (ST) is the investigation of

unexpected loss (UL) under conditions outside our ordinary realm of experience (eg extreme events not in our data-sets)

bull Many reasons for conducting periodic ST are largely due to the relationship between UL and economic capital (EC)

bull EC is generally thought of as the difference between Value-at-Risk (VaR) or extreme loss at some confidence level (eg a high quantile of a loss distribution) and expected loss (EL)

bull This purpose for ST hinges on our definition of UL ndash while it is commonly thought that EC should cover this in that UL may not only be unexpected but not credible as it is a statistical concept

bull Therefore some argue that results of an ST should be used for EC vs UL but this is rare as we usually do not have probability distributions associated with stress events

Function of Stress Testing Expected vs Unexpected Loss

001 002 003 004 005

20

40

60

80

Unexpected Losses

Expected Losses

VaR 9995ldquoBody of the Distributionrdquo

ldquoTail of the Distributionrdquo

Pro

babi

lity

Losses

EL

Economic Capital

Vasicek distribution (theta = 001 rho = 006)

Figure 1

The Function of Stress Testing (continued)

bull ST can and commonly have been used to challenge the adequacy of regulatory (RC) or EC amp derive a buffer for losses exceeding the VaR especially for new products or portfolios

bull Another advantage to ST to determine capital is that it can easily aggregate different risk types (eg credit market amp operational) problematic under standard EC methodologiesndash Eg different horizons and confidence levels for market vs credit riskndash Powerful dependencies between risk types in periods of stress

bull Quantification of ST appear and can be deployed several aspects of risk management with respect to extreme lossesndash Risk buffers determined or testedndash Risk capacity of a financial institutionndash Setting sub-portfolio limits especially if low-default situationndash Risk policy tolerance and appetite

Function of Stress Testing The Risk Aggregation Problem

-2 0 2

x 108

-5 0 5

x 108

-2 0 2

x 107

0 2 4

x 107

0 2 4

x 107

-2

0

2

x 108

-5

0

5

x 108

-2

0

2

x 107

0

2

4

x 107

Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)

0

2

4

x 107

Credit

Liqu

Operat

Market

IntRt

corr(crops)= 06517

corr(mktliqu)= 01127

corr(intliqu)= 01897

corr(crmkt)= 02241

corr(opsliqu)= 01533

corr(mktint)= 02478

corr(crliqu)= 05343

corr(opsint)= -01174

corr(opsmkt)= 01989

corr(crint)= -01328

bull Correlations amongst different risk types are in many cases large and cannot be ignored

bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure

Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

The Function of Stress Testing (continued)

bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal

market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or

extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and

hence economic capital and mitigate the vulnerability to important risk relevant effects

ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk

Supervisory Requirements and Expectations

bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management

bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie

risk management strategies to respond to the outcome of ST)

bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress

bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such

Supervisory Requirements and Expectations (continued)

bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required

bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage

bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating

migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the

impact of smaller deterioration in the credit environment

Supervisory Requirements and Expectations Regulatory Capital

000 005 010 015

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions

Credit Loss

Pro

ba

bili

ty D

en

sity

EL-norm=040

EL-stress=090

CVaR-norm=678

CVaR-stress=1579

NormalPD=1LGD=40Rho=01

StressedPD=15LGD=60Rho=015

Stressed Capital

Regulatory Capital

bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate

Supervisory Requirements and Expectations (continued)

bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy

bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately

bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk

bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)

bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)

Supervisory Requirements and Expectations (continued)

bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets

bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc

bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits

bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and

quantitative way

The Credit Risk Parameters for Stress Testing (continued)

bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters

bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation

bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited

bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate

bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be

key amp incorporate sufficient conservatism naturally but that varies

The Credit Risk Parameters for Stress Testing LGD

bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)

bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure

bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher

bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution

bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo

bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice

bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)

1 RecoveryRate

Discounted RecoveriesLGD=1- EAD

Discounted Direct amp Indirect Workout Costs

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Contractual features more senior and secured instruments do better

bull Absolute Priority Rule some violations (but usually small)

bull More senior instruments tend to be better secured

bull Debt cushion as distinct from position in the capital structure

bull High LGD for senior debt with little sub-debt

bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value

19

SENIORITY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees Trade Creditors Lawyers

Banks

Bondholders

Shareholders

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Outlinebull Introduction bull The Function of Stress Testingbull Supervisory Requirements and Expectationsbull The Credit Risk Parameters for Stress Testingbull Interpretation of Stress Test Resultsbull A Typology of Stress Tests

ndash Uniform Testingndash Risk Factor Sensitivitiesndash Scenario Analysis

bull Historical Scenariosbull Statistical Scenariosbull Hypothetical Scenarios

bull Procedures for Conducting Stress Testsbull A Simple Stress Testing Example

Introduction Overview

bull Modern credit risk modeling (eg Merton 1974) increasingly relies on advanced mathematical statistical and numerical techniques to measure and manage risk in credit portfolios

bull This gives rise to model risk (OCC 2011-16) and the possibility of understating inherent dangers stemming from very rare yet plausible occurrencs perhaps not in our reference data-sets

bull International supervisors have recognized the importance of stress testing credit risk in the Basel framework (BCBS 2009)

bull It can and has been argued that the art and science of stress testing has lagged in the domain of credit vs other types of risk (eg market) and our objective is to help fill this vacuum

bull We aim to present classifications amp established techniques that will help practitioners formulate robust credit risk stress tests

Introduction Motivation in the Financial Crisis

Reproduced from Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

Figure 3 Average Ratio of Total Charge-offs to Total Value of Loans for Top 50 Banks as of 4Q09

(Call Report Data 1984-2009)

0

0005

001

0015

002

0025

003

0035

19840331

19841231

19850930

19860630

19870331

19871231

19880930

19890630

19900331

19901231

19910930

19920630

19930331

19931231

19940930

19950630

19960331

19961231

19970930

19980630

19990331

19991231

20000930

20010630

20020331

20021231

20030930

20040630

20050331

20051231

20060930

20070630

20080331

20081231

20090930

bull Bank losses in the recent financial crisis exceed levels observed in recent history

bull This illustrates the inherent limitations of backward looking models ndash we must anticipate risk

Introduction Motivation in the Imprecision of Value-at-Risk

Gaussian Copula Bootstrapped (Margins) Distribution of 9997 Percentile VaR

VaR997=764e+8 q25=626e+8 q975=894e+8 CV=35379997 Percentile Value-at-Risk for 5 Risk Types(CrMktOpsLiquampIntRt) Top 200 Banks (1984-2008)

Den

sity

5e+08 6e+08 7e+08 8e+08 9e+08 1e+09

0e+0

01e

-09

2e-0

93e

-09

4e-0

95e

-09

6e-0

9

bull Sampling variation in VaR inputs leads to huge confidence bounds for risk estimates (coefficient of variation =354)

bull This is even assuming we have the correct model

Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

Conceptual Issues in Stress Testing Risk vs Uncertainty

bull Knight (1921) uncertainty is when a probability distribution is unmeasurable or unknown arguably a realistic scenario

bull Rely upon empirical data to estimate loss distributions but this is complicated because of changing economic conditions

bull Popper (1945) situations of uncertainty closely associated amp inherent to changes in knowledge amp behavior (no historicism)

bull Shackle (1990) predictions reliable only for immediate future as impact othersrsquo choices after time has an appreciable effect

bull This role of human behavior in economic theory was a key impetus behind rational expectations amp behavioral finance

bull Implication is that risk managers must be aware of model limitations amp how an EC regime itself changes behavior

bull Although we face uncertainty valuable to estimate loss distributions in that helps make explicit sources of uncertainty

The Function of Stress Testingbull A possible definition of stress testing (ST) is the investigation of

unexpected loss (UL) under conditions outside our ordinary realm of experience (eg extreme events not in our data-sets)

bull Many reasons for conducting periodic ST are largely due to the relationship between UL and economic capital (EC)

bull EC is generally thought of as the difference between Value-at-Risk (VaR) or extreme loss at some confidence level (eg a high quantile of a loss distribution) and expected loss (EL)

bull This purpose for ST hinges on our definition of UL ndash while it is commonly thought that EC should cover this in that UL may not only be unexpected but not credible as it is a statistical concept

bull Therefore some argue that results of an ST should be used for EC vs UL but this is rare as we usually do not have probability distributions associated with stress events

Function of Stress Testing Expected vs Unexpected Loss

001 002 003 004 005

20

40

60

80

Unexpected Losses

Expected Losses

VaR 9995ldquoBody of the Distributionrdquo

ldquoTail of the Distributionrdquo

Pro

babi

lity

Losses

EL

Economic Capital

Vasicek distribution (theta = 001 rho = 006)

Figure 1

The Function of Stress Testing (continued)

bull ST can and commonly have been used to challenge the adequacy of regulatory (RC) or EC amp derive a buffer for losses exceeding the VaR especially for new products or portfolios

bull Another advantage to ST to determine capital is that it can easily aggregate different risk types (eg credit market amp operational) problematic under standard EC methodologiesndash Eg different horizons and confidence levels for market vs credit riskndash Powerful dependencies between risk types in periods of stress

bull Quantification of ST appear and can be deployed several aspects of risk management with respect to extreme lossesndash Risk buffers determined or testedndash Risk capacity of a financial institutionndash Setting sub-portfolio limits especially if low-default situationndash Risk policy tolerance and appetite

Function of Stress Testing The Risk Aggregation Problem

-2 0 2

x 108

-5 0 5

x 108

-2 0 2

x 107

0 2 4

x 107

0 2 4

x 107

-2

0

2

x 108

-5

0

5

x 108

-2

0

2

x 107

0

2

4

x 107

Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)

0

2

4

x 107

Credit

Liqu

Operat

Market

IntRt

corr(crops)= 06517

corr(mktliqu)= 01127

corr(intliqu)= 01897

corr(crmkt)= 02241

corr(opsliqu)= 01533

corr(mktint)= 02478

corr(crliqu)= 05343

corr(opsint)= -01174

corr(opsmkt)= 01989

corr(crint)= -01328

bull Correlations amongst different risk types are in many cases large and cannot be ignored

bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure

Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

The Function of Stress Testing (continued)

bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal

market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or

extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and

hence economic capital and mitigate the vulnerability to important risk relevant effects

ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk

Supervisory Requirements and Expectations

bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management

bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie

risk management strategies to respond to the outcome of ST)

bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress

bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such

Supervisory Requirements and Expectations (continued)

bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required

bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage

bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating

migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the

impact of smaller deterioration in the credit environment

Supervisory Requirements and Expectations Regulatory Capital

000 005 010 015

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions

Credit Loss

Pro

ba

bili

ty D

en

sity

EL-norm=040

EL-stress=090

CVaR-norm=678

CVaR-stress=1579

NormalPD=1LGD=40Rho=01

StressedPD=15LGD=60Rho=015

Stressed Capital

Regulatory Capital

bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate

Supervisory Requirements and Expectations (continued)

bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy

bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately

bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk

bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)

bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)

Supervisory Requirements and Expectations (continued)

bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets

bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc

bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits

bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and

quantitative way

The Credit Risk Parameters for Stress Testing (continued)

bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters

bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation

bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited

bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate

bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be

key amp incorporate sufficient conservatism naturally but that varies

The Credit Risk Parameters for Stress Testing LGD

bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)

bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure

bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher

bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution

bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo

bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice

bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)

1 RecoveryRate

Discounted RecoveriesLGD=1- EAD

Discounted Direct amp Indirect Workout Costs

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Contractual features more senior and secured instruments do better

bull Absolute Priority Rule some violations (but usually small)

bull More senior instruments tend to be better secured

bull Debt cushion as distinct from position in the capital structure

bull High LGD for senior debt with little sub-debt

bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value

19

SENIORITY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees Trade Creditors Lawyers

Banks

Bondholders

Shareholders

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Introduction Overview

bull Modern credit risk modeling (eg Merton 1974) increasingly relies on advanced mathematical statistical and numerical techniques to measure and manage risk in credit portfolios

bull This gives rise to model risk (OCC 2011-16) and the possibility of understating inherent dangers stemming from very rare yet plausible occurrencs perhaps not in our reference data-sets

bull International supervisors have recognized the importance of stress testing credit risk in the Basel framework (BCBS 2009)

bull It can and has been argued that the art and science of stress testing has lagged in the domain of credit vs other types of risk (eg market) and our objective is to help fill this vacuum

bull We aim to present classifications amp established techniques that will help practitioners formulate robust credit risk stress tests

Introduction Motivation in the Financial Crisis

Reproduced from Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

Figure 3 Average Ratio of Total Charge-offs to Total Value of Loans for Top 50 Banks as of 4Q09

(Call Report Data 1984-2009)

0

0005

001

0015

002

0025

003

0035

19840331

19841231

19850930

19860630

19870331

19871231

19880930

19890630

19900331

19901231

19910930

19920630

19930331

19931231

19940930

19950630

19960331

19961231

19970930

19980630

19990331

19991231

20000930

20010630

20020331

20021231

20030930

20040630

20050331

20051231

20060930

20070630

20080331

20081231

20090930

bull Bank losses in the recent financial crisis exceed levels observed in recent history

bull This illustrates the inherent limitations of backward looking models ndash we must anticipate risk

Introduction Motivation in the Imprecision of Value-at-Risk

Gaussian Copula Bootstrapped (Margins) Distribution of 9997 Percentile VaR

VaR997=764e+8 q25=626e+8 q975=894e+8 CV=35379997 Percentile Value-at-Risk for 5 Risk Types(CrMktOpsLiquampIntRt) Top 200 Banks (1984-2008)

Den

sity

5e+08 6e+08 7e+08 8e+08 9e+08 1e+09

0e+0

01e

-09

2e-0

93e

-09

4e-0

95e

-09

6e-0

9

bull Sampling variation in VaR inputs leads to huge confidence bounds for risk estimates (coefficient of variation =354)

bull This is even assuming we have the correct model

Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

Conceptual Issues in Stress Testing Risk vs Uncertainty

bull Knight (1921) uncertainty is when a probability distribution is unmeasurable or unknown arguably a realistic scenario

bull Rely upon empirical data to estimate loss distributions but this is complicated because of changing economic conditions

bull Popper (1945) situations of uncertainty closely associated amp inherent to changes in knowledge amp behavior (no historicism)

bull Shackle (1990) predictions reliable only for immediate future as impact othersrsquo choices after time has an appreciable effect

bull This role of human behavior in economic theory was a key impetus behind rational expectations amp behavioral finance

bull Implication is that risk managers must be aware of model limitations amp how an EC regime itself changes behavior

bull Although we face uncertainty valuable to estimate loss distributions in that helps make explicit sources of uncertainty

The Function of Stress Testingbull A possible definition of stress testing (ST) is the investigation of

unexpected loss (UL) under conditions outside our ordinary realm of experience (eg extreme events not in our data-sets)

bull Many reasons for conducting periodic ST are largely due to the relationship between UL and economic capital (EC)

bull EC is generally thought of as the difference between Value-at-Risk (VaR) or extreme loss at some confidence level (eg a high quantile of a loss distribution) and expected loss (EL)

bull This purpose for ST hinges on our definition of UL ndash while it is commonly thought that EC should cover this in that UL may not only be unexpected but not credible as it is a statistical concept

bull Therefore some argue that results of an ST should be used for EC vs UL but this is rare as we usually do not have probability distributions associated with stress events

Function of Stress Testing Expected vs Unexpected Loss

001 002 003 004 005

20

40

60

80

Unexpected Losses

Expected Losses

VaR 9995ldquoBody of the Distributionrdquo

ldquoTail of the Distributionrdquo

Pro

babi

lity

Losses

EL

Economic Capital

Vasicek distribution (theta = 001 rho = 006)

Figure 1

The Function of Stress Testing (continued)

bull ST can and commonly have been used to challenge the adequacy of regulatory (RC) or EC amp derive a buffer for losses exceeding the VaR especially for new products or portfolios

bull Another advantage to ST to determine capital is that it can easily aggregate different risk types (eg credit market amp operational) problematic under standard EC methodologiesndash Eg different horizons and confidence levels for market vs credit riskndash Powerful dependencies between risk types in periods of stress

bull Quantification of ST appear and can be deployed several aspects of risk management with respect to extreme lossesndash Risk buffers determined or testedndash Risk capacity of a financial institutionndash Setting sub-portfolio limits especially if low-default situationndash Risk policy tolerance and appetite

Function of Stress Testing The Risk Aggregation Problem

-2 0 2

x 108

-5 0 5

x 108

-2 0 2

x 107

0 2 4

x 107

0 2 4

x 107

-2

0

2

x 108

-5

0

5

x 108

-2

0

2

x 107

0

2

4

x 107

Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)

0

2

4

x 107

Credit

Liqu

Operat

Market

IntRt

corr(crops)= 06517

corr(mktliqu)= 01127

corr(intliqu)= 01897

corr(crmkt)= 02241

corr(opsliqu)= 01533

corr(mktint)= 02478

corr(crliqu)= 05343

corr(opsint)= -01174

corr(opsmkt)= 01989

corr(crint)= -01328

bull Correlations amongst different risk types are in many cases large and cannot be ignored

bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure

Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

The Function of Stress Testing (continued)

bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal

market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or

extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and

hence economic capital and mitigate the vulnerability to important risk relevant effects

ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk

Supervisory Requirements and Expectations

bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management

bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie

risk management strategies to respond to the outcome of ST)

bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress

bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such

Supervisory Requirements and Expectations (continued)

bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required

bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage

bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating

migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the

impact of smaller deterioration in the credit environment

Supervisory Requirements and Expectations Regulatory Capital

000 005 010 015

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions

Credit Loss

Pro

ba

bili

ty D

en

sity

EL-norm=040

EL-stress=090

CVaR-norm=678

CVaR-stress=1579

NormalPD=1LGD=40Rho=01

StressedPD=15LGD=60Rho=015

Stressed Capital

Regulatory Capital

bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate

Supervisory Requirements and Expectations (continued)

bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy

bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately

bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk

bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)

bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)

Supervisory Requirements and Expectations (continued)

bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets

bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc

bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits

bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and

quantitative way

The Credit Risk Parameters for Stress Testing (continued)

bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters

bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation

bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited

bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate

bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be

key amp incorporate sufficient conservatism naturally but that varies

The Credit Risk Parameters for Stress Testing LGD

bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)

bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure

bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher

bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution

bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo

bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice

bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)

1 RecoveryRate

Discounted RecoveriesLGD=1- EAD

Discounted Direct amp Indirect Workout Costs

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Contractual features more senior and secured instruments do better

bull Absolute Priority Rule some violations (but usually small)

bull More senior instruments tend to be better secured

bull Debt cushion as distinct from position in the capital structure

bull High LGD for senior debt with little sub-debt

bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value

19

SENIORITY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees Trade Creditors Lawyers

Banks

Bondholders

Shareholders

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Introduction Motivation in the Financial Crisis

Reproduced from Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

Figure 3 Average Ratio of Total Charge-offs to Total Value of Loans for Top 50 Banks as of 4Q09

(Call Report Data 1984-2009)

0

0005

001

0015

002

0025

003

0035

19840331

19841231

19850930

19860630

19870331

19871231

19880930

19890630

19900331

19901231

19910930

19920630

19930331

19931231

19940930

19950630

19960331

19961231

19970930

19980630

19990331

19991231

20000930

20010630

20020331

20021231

20030930

20040630

20050331

20051231

20060930

20070630

20080331

20081231

20090930

bull Bank losses in the recent financial crisis exceed levels observed in recent history

bull This illustrates the inherent limitations of backward looking models ndash we must anticipate risk

Introduction Motivation in the Imprecision of Value-at-Risk

Gaussian Copula Bootstrapped (Margins) Distribution of 9997 Percentile VaR

VaR997=764e+8 q25=626e+8 q975=894e+8 CV=35379997 Percentile Value-at-Risk for 5 Risk Types(CrMktOpsLiquampIntRt) Top 200 Banks (1984-2008)

Den

sity

5e+08 6e+08 7e+08 8e+08 9e+08 1e+09

0e+0

01e

-09

2e-0

93e

-09

4e-0

95e

-09

6e-0

9

bull Sampling variation in VaR inputs leads to huge confidence bounds for risk estimates (coefficient of variation =354)

bull This is even assuming we have the correct model

Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

Conceptual Issues in Stress Testing Risk vs Uncertainty

bull Knight (1921) uncertainty is when a probability distribution is unmeasurable or unknown arguably a realistic scenario

bull Rely upon empirical data to estimate loss distributions but this is complicated because of changing economic conditions

bull Popper (1945) situations of uncertainty closely associated amp inherent to changes in knowledge amp behavior (no historicism)

bull Shackle (1990) predictions reliable only for immediate future as impact othersrsquo choices after time has an appreciable effect

bull This role of human behavior in economic theory was a key impetus behind rational expectations amp behavioral finance

bull Implication is that risk managers must be aware of model limitations amp how an EC regime itself changes behavior

bull Although we face uncertainty valuable to estimate loss distributions in that helps make explicit sources of uncertainty

The Function of Stress Testingbull A possible definition of stress testing (ST) is the investigation of

unexpected loss (UL) under conditions outside our ordinary realm of experience (eg extreme events not in our data-sets)

bull Many reasons for conducting periodic ST are largely due to the relationship between UL and economic capital (EC)

bull EC is generally thought of as the difference between Value-at-Risk (VaR) or extreme loss at some confidence level (eg a high quantile of a loss distribution) and expected loss (EL)

bull This purpose for ST hinges on our definition of UL ndash while it is commonly thought that EC should cover this in that UL may not only be unexpected but not credible as it is a statistical concept

bull Therefore some argue that results of an ST should be used for EC vs UL but this is rare as we usually do not have probability distributions associated with stress events

Function of Stress Testing Expected vs Unexpected Loss

001 002 003 004 005

20

40

60

80

Unexpected Losses

Expected Losses

VaR 9995ldquoBody of the Distributionrdquo

ldquoTail of the Distributionrdquo

Pro

babi

lity

Losses

EL

Economic Capital

Vasicek distribution (theta = 001 rho = 006)

Figure 1

The Function of Stress Testing (continued)

bull ST can and commonly have been used to challenge the adequacy of regulatory (RC) or EC amp derive a buffer for losses exceeding the VaR especially for new products or portfolios

bull Another advantage to ST to determine capital is that it can easily aggregate different risk types (eg credit market amp operational) problematic under standard EC methodologiesndash Eg different horizons and confidence levels for market vs credit riskndash Powerful dependencies between risk types in periods of stress

bull Quantification of ST appear and can be deployed several aspects of risk management with respect to extreme lossesndash Risk buffers determined or testedndash Risk capacity of a financial institutionndash Setting sub-portfolio limits especially if low-default situationndash Risk policy tolerance and appetite

Function of Stress Testing The Risk Aggregation Problem

-2 0 2

x 108

-5 0 5

x 108

-2 0 2

x 107

0 2 4

x 107

0 2 4

x 107

-2

0

2

x 108

-5

0

5

x 108

-2

0

2

x 107

0

2

4

x 107

Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)

0

2

4

x 107

Credit

Liqu

Operat

Market

IntRt

corr(crops)= 06517

corr(mktliqu)= 01127

corr(intliqu)= 01897

corr(crmkt)= 02241

corr(opsliqu)= 01533

corr(mktint)= 02478

corr(crliqu)= 05343

corr(opsint)= -01174

corr(opsmkt)= 01989

corr(crint)= -01328

bull Correlations amongst different risk types are in many cases large and cannot be ignored

bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure

Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

The Function of Stress Testing (continued)

bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal

market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or

extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and

hence economic capital and mitigate the vulnerability to important risk relevant effects

ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk

Supervisory Requirements and Expectations

bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management

bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie

risk management strategies to respond to the outcome of ST)

bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress

bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such

Supervisory Requirements and Expectations (continued)

bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required

bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage

bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating

migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the

impact of smaller deterioration in the credit environment

Supervisory Requirements and Expectations Regulatory Capital

000 005 010 015

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions

Credit Loss

Pro

ba

bili

ty D

en

sity

EL-norm=040

EL-stress=090

CVaR-norm=678

CVaR-stress=1579

NormalPD=1LGD=40Rho=01

StressedPD=15LGD=60Rho=015

Stressed Capital

Regulatory Capital

bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate

Supervisory Requirements and Expectations (continued)

bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy

bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately

bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk

bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)

bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)

Supervisory Requirements and Expectations (continued)

bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets

bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc

bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits

bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and

quantitative way

The Credit Risk Parameters for Stress Testing (continued)

bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters

bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation

bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited

bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate

bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be

key amp incorporate sufficient conservatism naturally but that varies

The Credit Risk Parameters for Stress Testing LGD

bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)

bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure

bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher

bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution

bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo

bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice

bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)

1 RecoveryRate

Discounted RecoveriesLGD=1- EAD

Discounted Direct amp Indirect Workout Costs

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Contractual features more senior and secured instruments do better

bull Absolute Priority Rule some violations (but usually small)

bull More senior instruments tend to be better secured

bull Debt cushion as distinct from position in the capital structure

bull High LGD for senior debt with little sub-debt

bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value

19

SENIORITY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees Trade Creditors Lawyers

Banks

Bondholders

Shareholders

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Introduction Motivation in the Imprecision of Value-at-Risk

Gaussian Copula Bootstrapped (Margins) Distribution of 9997 Percentile VaR

VaR997=764e+8 q25=626e+8 q975=894e+8 CV=35379997 Percentile Value-at-Risk for 5 Risk Types(CrMktOpsLiquampIntRt) Top 200 Banks (1984-2008)

Den

sity

5e+08 6e+08 7e+08 8e+08 9e+08 1e+09

0e+0

01e

-09

2e-0

93e

-09

4e-0

95e

-09

6e-0

9

bull Sampling variation in VaR inputs leads to huge confidence bounds for risk estimates (coefficient of variation =354)

bull This is even assuming we have the correct model

Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

Conceptual Issues in Stress Testing Risk vs Uncertainty

bull Knight (1921) uncertainty is when a probability distribution is unmeasurable or unknown arguably a realistic scenario

bull Rely upon empirical data to estimate loss distributions but this is complicated because of changing economic conditions

bull Popper (1945) situations of uncertainty closely associated amp inherent to changes in knowledge amp behavior (no historicism)

bull Shackle (1990) predictions reliable only for immediate future as impact othersrsquo choices after time has an appreciable effect

bull This role of human behavior in economic theory was a key impetus behind rational expectations amp behavioral finance

bull Implication is that risk managers must be aware of model limitations amp how an EC regime itself changes behavior

bull Although we face uncertainty valuable to estimate loss distributions in that helps make explicit sources of uncertainty

The Function of Stress Testingbull A possible definition of stress testing (ST) is the investigation of

unexpected loss (UL) under conditions outside our ordinary realm of experience (eg extreme events not in our data-sets)

bull Many reasons for conducting periodic ST are largely due to the relationship between UL and economic capital (EC)

bull EC is generally thought of as the difference between Value-at-Risk (VaR) or extreme loss at some confidence level (eg a high quantile of a loss distribution) and expected loss (EL)

bull This purpose for ST hinges on our definition of UL ndash while it is commonly thought that EC should cover this in that UL may not only be unexpected but not credible as it is a statistical concept

bull Therefore some argue that results of an ST should be used for EC vs UL but this is rare as we usually do not have probability distributions associated with stress events

Function of Stress Testing Expected vs Unexpected Loss

001 002 003 004 005

20

40

60

80

Unexpected Losses

Expected Losses

VaR 9995ldquoBody of the Distributionrdquo

ldquoTail of the Distributionrdquo

Pro

babi

lity

Losses

EL

Economic Capital

Vasicek distribution (theta = 001 rho = 006)

Figure 1

The Function of Stress Testing (continued)

bull ST can and commonly have been used to challenge the adequacy of regulatory (RC) or EC amp derive a buffer for losses exceeding the VaR especially for new products or portfolios

bull Another advantage to ST to determine capital is that it can easily aggregate different risk types (eg credit market amp operational) problematic under standard EC methodologiesndash Eg different horizons and confidence levels for market vs credit riskndash Powerful dependencies between risk types in periods of stress

bull Quantification of ST appear and can be deployed several aspects of risk management with respect to extreme lossesndash Risk buffers determined or testedndash Risk capacity of a financial institutionndash Setting sub-portfolio limits especially if low-default situationndash Risk policy tolerance and appetite

Function of Stress Testing The Risk Aggregation Problem

-2 0 2

x 108

-5 0 5

x 108

-2 0 2

x 107

0 2 4

x 107

0 2 4

x 107

-2

0

2

x 108

-5

0

5

x 108

-2

0

2

x 107

0

2

4

x 107

Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)

0

2

4

x 107

Credit

Liqu

Operat

Market

IntRt

corr(crops)= 06517

corr(mktliqu)= 01127

corr(intliqu)= 01897

corr(crmkt)= 02241

corr(opsliqu)= 01533

corr(mktint)= 02478

corr(crliqu)= 05343

corr(opsint)= -01174

corr(opsmkt)= 01989

corr(crint)= -01328

bull Correlations amongst different risk types are in many cases large and cannot be ignored

bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure

Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

The Function of Stress Testing (continued)

bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal

market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or

extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and

hence economic capital and mitigate the vulnerability to important risk relevant effects

ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk

Supervisory Requirements and Expectations

bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management

bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie

risk management strategies to respond to the outcome of ST)

bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress

bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such

Supervisory Requirements and Expectations (continued)

bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required

bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage

bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating

migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the

impact of smaller deterioration in the credit environment

Supervisory Requirements and Expectations Regulatory Capital

000 005 010 015

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions

Credit Loss

Pro

ba

bili

ty D

en

sity

EL-norm=040

EL-stress=090

CVaR-norm=678

CVaR-stress=1579

NormalPD=1LGD=40Rho=01

StressedPD=15LGD=60Rho=015

Stressed Capital

Regulatory Capital

bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate

Supervisory Requirements and Expectations (continued)

bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy

bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately

bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk

bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)

bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)

Supervisory Requirements and Expectations (continued)

bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets

bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc

bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits

bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and

quantitative way

The Credit Risk Parameters for Stress Testing (continued)

bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters

bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation

bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited

bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate

bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be

key amp incorporate sufficient conservatism naturally but that varies

The Credit Risk Parameters for Stress Testing LGD

bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)

bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure

bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher

bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution

bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo

bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice

bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)

1 RecoveryRate

Discounted RecoveriesLGD=1- EAD

Discounted Direct amp Indirect Workout Costs

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Contractual features more senior and secured instruments do better

bull Absolute Priority Rule some violations (but usually small)

bull More senior instruments tend to be better secured

bull Debt cushion as distinct from position in the capital structure

bull High LGD for senior debt with little sub-debt

bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value

19

SENIORITY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees Trade Creditors Lawyers

Banks

Bondholders

Shareholders

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Conceptual Issues in Stress Testing Risk vs Uncertainty

bull Knight (1921) uncertainty is when a probability distribution is unmeasurable or unknown arguably a realistic scenario

bull Rely upon empirical data to estimate loss distributions but this is complicated because of changing economic conditions

bull Popper (1945) situations of uncertainty closely associated amp inherent to changes in knowledge amp behavior (no historicism)

bull Shackle (1990) predictions reliable only for immediate future as impact othersrsquo choices after time has an appreciable effect

bull This role of human behavior in economic theory was a key impetus behind rational expectations amp behavioral finance

bull Implication is that risk managers must be aware of model limitations amp how an EC regime itself changes behavior

bull Although we face uncertainty valuable to estimate loss distributions in that helps make explicit sources of uncertainty

The Function of Stress Testingbull A possible definition of stress testing (ST) is the investigation of

unexpected loss (UL) under conditions outside our ordinary realm of experience (eg extreme events not in our data-sets)

bull Many reasons for conducting periodic ST are largely due to the relationship between UL and economic capital (EC)

bull EC is generally thought of as the difference between Value-at-Risk (VaR) or extreme loss at some confidence level (eg a high quantile of a loss distribution) and expected loss (EL)

bull This purpose for ST hinges on our definition of UL ndash while it is commonly thought that EC should cover this in that UL may not only be unexpected but not credible as it is a statistical concept

bull Therefore some argue that results of an ST should be used for EC vs UL but this is rare as we usually do not have probability distributions associated with stress events

Function of Stress Testing Expected vs Unexpected Loss

001 002 003 004 005

20

40

60

80

Unexpected Losses

Expected Losses

VaR 9995ldquoBody of the Distributionrdquo

ldquoTail of the Distributionrdquo

Pro

babi

lity

Losses

EL

Economic Capital

Vasicek distribution (theta = 001 rho = 006)

Figure 1

The Function of Stress Testing (continued)

bull ST can and commonly have been used to challenge the adequacy of regulatory (RC) or EC amp derive a buffer for losses exceeding the VaR especially for new products or portfolios

bull Another advantage to ST to determine capital is that it can easily aggregate different risk types (eg credit market amp operational) problematic under standard EC methodologiesndash Eg different horizons and confidence levels for market vs credit riskndash Powerful dependencies between risk types in periods of stress

bull Quantification of ST appear and can be deployed several aspects of risk management with respect to extreme lossesndash Risk buffers determined or testedndash Risk capacity of a financial institutionndash Setting sub-portfolio limits especially if low-default situationndash Risk policy tolerance and appetite

Function of Stress Testing The Risk Aggregation Problem

-2 0 2

x 108

-5 0 5

x 108

-2 0 2

x 107

0 2 4

x 107

0 2 4

x 107

-2

0

2

x 108

-5

0

5

x 108

-2

0

2

x 107

0

2

4

x 107

Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)

0

2

4

x 107

Credit

Liqu

Operat

Market

IntRt

corr(crops)= 06517

corr(mktliqu)= 01127

corr(intliqu)= 01897

corr(crmkt)= 02241

corr(opsliqu)= 01533

corr(mktint)= 02478

corr(crliqu)= 05343

corr(opsint)= -01174

corr(opsmkt)= 01989

corr(crint)= -01328

bull Correlations amongst different risk types are in many cases large and cannot be ignored

bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure

Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

The Function of Stress Testing (continued)

bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal

market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or

extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and

hence economic capital and mitigate the vulnerability to important risk relevant effects

ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk

Supervisory Requirements and Expectations

bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management

bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie

risk management strategies to respond to the outcome of ST)

bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress

bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such

Supervisory Requirements and Expectations (continued)

bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required

bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage

bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating

migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the

impact of smaller deterioration in the credit environment

Supervisory Requirements and Expectations Regulatory Capital

000 005 010 015

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions

Credit Loss

Pro

ba

bili

ty D

en

sity

EL-norm=040

EL-stress=090

CVaR-norm=678

CVaR-stress=1579

NormalPD=1LGD=40Rho=01

StressedPD=15LGD=60Rho=015

Stressed Capital

Regulatory Capital

bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate

Supervisory Requirements and Expectations (continued)

bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy

bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately

bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk

bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)

bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)

Supervisory Requirements and Expectations (continued)

bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets

bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc

bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits

bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and

quantitative way

The Credit Risk Parameters for Stress Testing (continued)

bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters

bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation

bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited

bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate

bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be

key amp incorporate sufficient conservatism naturally but that varies

The Credit Risk Parameters for Stress Testing LGD

bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)

bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure

bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher

bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution

bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo

bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice

bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)

1 RecoveryRate

Discounted RecoveriesLGD=1- EAD

Discounted Direct amp Indirect Workout Costs

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Contractual features more senior and secured instruments do better

bull Absolute Priority Rule some violations (but usually small)

bull More senior instruments tend to be better secured

bull Debt cushion as distinct from position in the capital structure

bull High LGD for senior debt with little sub-debt

bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value

19

SENIORITY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees Trade Creditors Lawyers

Banks

Bondholders

Shareholders

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

The Function of Stress Testingbull A possible definition of stress testing (ST) is the investigation of

unexpected loss (UL) under conditions outside our ordinary realm of experience (eg extreme events not in our data-sets)

bull Many reasons for conducting periodic ST are largely due to the relationship between UL and economic capital (EC)

bull EC is generally thought of as the difference between Value-at-Risk (VaR) or extreme loss at some confidence level (eg a high quantile of a loss distribution) and expected loss (EL)

bull This purpose for ST hinges on our definition of UL ndash while it is commonly thought that EC should cover this in that UL may not only be unexpected but not credible as it is a statistical concept

bull Therefore some argue that results of an ST should be used for EC vs UL but this is rare as we usually do not have probability distributions associated with stress events

Function of Stress Testing Expected vs Unexpected Loss

001 002 003 004 005

20

40

60

80

Unexpected Losses

Expected Losses

VaR 9995ldquoBody of the Distributionrdquo

ldquoTail of the Distributionrdquo

Pro

babi

lity

Losses

EL

Economic Capital

Vasicek distribution (theta = 001 rho = 006)

Figure 1

The Function of Stress Testing (continued)

bull ST can and commonly have been used to challenge the adequacy of regulatory (RC) or EC amp derive a buffer for losses exceeding the VaR especially for new products or portfolios

bull Another advantage to ST to determine capital is that it can easily aggregate different risk types (eg credit market amp operational) problematic under standard EC methodologiesndash Eg different horizons and confidence levels for market vs credit riskndash Powerful dependencies between risk types in periods of stress

bull Quantification of ST appear and can be deployed several aspects of risk management with respect to extreme lossesndash Risk buffers determined or testedndash Risk capacity of a financial institutionndash Setting sub-portfolio limits especially if low-default situationndash Risk policy tolerance and appetite

Function of Stress Testing The Risk Aggregation Problem

-2 0 2

x 108

-5 0 5

x 108

-2 0 2

x 107

0 2 4

x 107

0 2 4

x 107

-2

0

2

x 108

-5

0

5

x 108

-2

0

2

x 107

0

2

4

x 107

Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)

0

2

4

x 107

Credit

Liqu

Operat

Market

IntRt

corr(crops)= 06517

corr(mktliqu)= 01127

corr(intliqu)= 01897

corr(crmkt)= 02241

corr(opsliqu)= 01533

corr(mktint)= 02478

corr(crliqu)= 05343

corr(opsint)= -01174

corr(opsmkt)= 01989

corr(crint)= -01328

bull Correlations amongst different risk types are in many cases large and cannot be ignored

bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure

Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

The Function of Stress Testing (continued)

bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal

market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or

extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and

hence economic capital and mitigate the vulnerability to important risk relevant effects

ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk

Supervisory Requirements and Expectations

bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management

bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie

risk management strategies to respond to the outcome of ST)

bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress

bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such

Supervisory Requirements and Expectations (continued)

bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required

bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage

bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating

migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the

impact of smaller deterioration in the credit environment

Supervisory Requirements and Expectations Regulatory Capital

000 005 010 015

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions

Credit Loss

Pro

ba

bili

ty D

en

sity

EL-norm=040

EL-stress=090

CVaR-norm=678

CVaR-stress=1579

NormalPD=1LGD=40Rho=01

StressedPD=15LGD=60Rho=015

Stressed Capital

Regulatory Capital

bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate

Supervisory Requirements and Expectations (continued)

bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy

bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately

bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk

bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)

bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)

Supervisory Requirements and Expectations (continued)

bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets

bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc

bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits

bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and

quantitative way

The Credit Risk Parameters for Stress Testing (continued)

bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters

bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation

bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited

bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate

bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be

key amp incorporate sufficient conservatism naturally but that varies

The Credit Risk Parameters for Stress Testing LGD

bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)

bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure

bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher

bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution

bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo

bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice

bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)

1 RecoveryRate

Discounted RecoveriesLGD=1- EAD

Discounted Direct amp Indirect Workout Costs

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Contractual features more senior and secured instruments do better

bull Absolute Priority Rule some violations (but usually small)

bull More senior instruments tend to be better secured

bull Debt cushion as distinct from position in the capital structure

bull High LGD for senior debt with little sub-debt

bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value

19

SENIORITY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees Trade Creditors Lawyers

Banks

Bondholders

Shareholders

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Function of Stress Testing Expected vs Unexpected Loss

001 002 003 004 005

20

40

60

80

Unexpected Losses

Expected Losses

VaR 9995ldquoBody of the Distributionrdquo

ldquoTail of the Distributionrdquo

Pro

babi

lity

Losses

EL

Economic Capital

Vasicek distribution (theta = 001 rho = 006)

Figure 1

The Function of Stress Testing (continued)

bull ST can and commonly have been used to challenge the adequacy of regulatory (RC) or EC amp derive a buffer for losses exceeding the VaR especially for new products or portfolios

bull Another advantage to ST to determine capital is that it can easily aggregate different risk types (eg credit market amp operational) problematic under standard EC methodologiesndash Eg different horizons and confidence levels for market vs credit riskndash Powerful dependencies between risk types in periods of stress

bull Quantification of ST appear and can be deployed several aspects of risk management with respect to extreme lossesndash Risk buffers determined or testedndash Risk capacity of a financial institutionndash Setting sub-portfolio limits especially if low-default situationndash Risk policy tolerance and appetite

Function of Stress Testing The Risk Aggregation Problem

-2 0 2

x 108

-5 0 5

x 108

-2 0 2

x 107

0 2 4

x 107

0 2 4

x 107

-2

0

2

x 108

-5

0

5

x 108

-2

0

2

x 107

0

2

4

x 107

Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)

0

2

4

x 107

Credit

Liqu

Operat

Market

IntRt

corr(crops)= 06517

corr(mktliqu)= 01127

corr(intliqu)= 01897

corr(crmkt)= 02241

corr(opsliqu)= 01533

corr(mktint)= 02478

corr(crliqu)= 05343

corr(opsint)= -01174

corr(opsmkt)= 01989

corr(crint)= -01328

bull Correlations amongst different risk types are in many cases large and cannot be ignored

bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure

Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

The Function of Stress Testing (continued)

bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal

market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or

extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and

hence economic capital and mitigate the vulnerability to important risk relevant effects

ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk

Supervisory Requirements and Expectations

bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management

bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie

risk management strategies to respond to the outcome of ST)

bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress

bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such

Supervisory Requirements and Expectations (continued)

bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required

bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage

bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating

migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the

impact of smaller deterioration in the credit environment

Supervisory Requirements and Expectations Regulatory Capital

000 005 010 015

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions

Credit Loss

Pro

ba

bili

ty D

en

sity

EL-norm=040

EL-stress=090

CVaR-norm=678

CVaR-stress=1579

NormalPD=1LGD=40Rho=01

StressedPD=15LGD=60Rho=015

Stressed Capital

Regulatory Capital

bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate

Supervisory Requirements and Expectations (continued)

bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy

bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately

bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk

bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)

bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)

Supervisory Requirements and Expectations (continued)

bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets

bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc

bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits

bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and

quantitative way

The Credit Risk Parameters for Stress Testing (continued)

bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters

bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation

bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited

bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate

bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be

key amp incorporate sufficient conservatism naturally but that varies

The Credit Risk Parameters for Stress Testing LGD

bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)

bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure

bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher

bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution

bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo

bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice

bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)

1 RecoveryRate

Discounted RecoveriesLGD=1- EAD

Discounted Direct amp Indirect Workout Costs

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Contractual features more senior and secured instruments do better

bull Absolute Priority Rule some violations (but usually small)

bull More senior instruments tend to be better secured

bull Debt cushion as distinct from position in the capital structure

bull High LGD for senior debt with little sub-debt

bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value

19

SENIORITY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees Trade Creditors Lawyers

Banks

Bondholders

Shareholders

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

The Function of Stress Testing (continued)

bull ST can and commonly have been used to challenge the adequacy of regulatory (RC) or EC amp derive a buffer for losses exceeding the VaR especially for new products or portfolios

bull Another advantage to ST to determine capital is that it can easily aggregate different risk types (eg credit market amp operational) problematic under standard EC methodologiesndash Eg different horizons and confidence levels for market vs credit riskndash Powerful dependencies between risk types in periods of stress

bull Quantification of ST appear and can be deployed several aspects of risk management with respect to extreme lossesndash Risk buffers determined or testedndash Risk capacity of a financial institutionndash Setting sub-portfolio limits especially if low-default situationndash Risk policy tolerance and appetite

Function of Stress Testing The Risk Aggregation Problem

-2 0 2

x 108

-5 0 5

x 108

-2 0 2

x 107

0 2 4

x 107

0 2 4

x 107

-2

0

2

x 108

-5

0

5

x 108

-2

0

2

x 107

0

2

4

x 107

Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)

0

2

4

x 107

Credit

Liqu

Operat

Market

IntRt

corr(crops)= 06517

corr(mktliqu)= 01127

corr(intliqu)= 01897

corr(crmkt)= 02241

corr(opsliqu)= 01533

corr(mktint)= 02478

corr(crliqu)= 05343

corr(opsint)= -01174

corr(opsmkt)= 01989

corr(crint)= -01328

bull Correlations amongst different risk types are in many cases large and cannot be ignored

bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure

Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

The Function of Stress Testing (continued)

bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal

market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or

extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and

hence economic capital and mitigate the vulnerability to important risk relevant effects

ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk

Supervisory Requirements and Expectations

bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management

bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie

risk management strategies to respond to the outcome of ST)

bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress

bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such

Supervisory Requirements and Expectations (continued)

bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required

bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage

bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating

migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the

impact of smaller deterioration in the credit environment

Supervisory Requirements and Expectations Regulatory Capital

000 005 010 015

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions

Credit Loss

Pro

ba

bili

ty D

en

sity

EL-norm=040

EL-stress=090

CVaR-norm=678

CVaR-stress=1579

NormalPD=1LGD=40Rho=01

StressedPD=15LGD=60Rho=015

Stressed Capital

Regulatory Capital

bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate

Supervisory Requirements and Expectations (continued)

bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy

bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately

bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk

bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)

bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)

Supervisory Requirements and Expectations (continued)

bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets

bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc

bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits

bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and

quantitative way

The Credit Risk Parameters for Stress Testing (continued)

bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters

bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation

bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited

bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate

bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be

key amp incorporate sufficient conservatism naturally but that varies

The Credit Risk Parameters for Stress Testing LGD

bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)

bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure

bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher

bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution

bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo

bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice

bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)

1 RecoveryRate

Discounted RecoveriesLGD=1- EAD

Discounted Direct amp Indirect Workout Costs

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Contractual features more senior and secured instruments do better

bull Absolute Priority Rule some violations (but usually small)

bull More senior instruments tend to be better secured

bull Debt cushion as distinct from position in the capital structure

bull High LGD for senior debt with little sub-debt

bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value

19

SENIORITY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees Trade Creditors Lawyers

Banks

Bondholders

Shareholders

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Function of Stress Testing The Risk Aggregation Problem

-2 0 2

x 108

-5 0 5

x 108

-2 0 2

x 107

0 2 4

x 107

0 2 4

x 107

-2

0

2

x 108

-5

0

5

x 108

-2

0

2

x 107

0

2

4

x 107

Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)

0

2

4

x 107

Credit

Liqu

Operat

Market

IntRt

corr(crops)= 06517

corr(mktliqu)= 01127

corr(intliqu)= 01897

corr(crmkt)= 02241

corr(opsliqu)= 01533

corr(mktint)= 02478

corr(crliqu)= 05343

corr(opsint)= -01174

corr(opsmkt)= 01989

corr(crint)= -01328

bull Correlations amongst different risk types are in many cases large and cannot be ignored

bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure

Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

The Function of Stress Testing (continued)

bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal

market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or

extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and

hence economic capital and mitigate the vulnerability to important risk relevant effects

ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk

Supervisory Requirements and Expectations

bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management

bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie

risk management strategies to respond to the outcome of ST)

bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress

bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such

Supervisory Requirements and Expectations (continued)

bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required

bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage

bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating

migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the

impact of smaller deterioration in the credit environment

Supervisory Requirements and Expectations Regulatory Capital

000 005 010 015

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions

Credit Loss

Pro

ba

bili

ty D

en

sity

EL-norm=040

EL-stress=090

CVaR-norm=678

CVaR-stress=1579

NormalPD=1LGD=40Rho=01

StressedPD=15LGD=60Rho=015

Stressed Capital

Regulatory Capital

bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate

Supervisory Requirements and Expectations (continued)

bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy

bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately

bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk

bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)

bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)

Supervisory Requirements and Expectations (continued)

bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets

bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc

bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits

bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and

quantitative way

The Credit Risk Parameters for Stress Testing (continued)

bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters

bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation

bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited

bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate

bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be

key amp incorporate sufficient conservatism naturally but that varies

The Credit Risk Parameters for Stress Testing LGD

bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)

bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure

bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher

bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution

bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo

bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice

bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)

1 RecoveryRate

Discounted RecoveriesLGD=1- EAD

Discounted Direct amp Indirect Workout Costs

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Contractual features more senior and secured instruments do better

bull Absolute Priority Rule some violations (but usually small)

bull More senior instruments tend to be better secured

bull Debt cushion as distinct from position in the capital structure

bull High LGD for senior debt with little sub-debt

bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value

19

SENIORITY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees Trade Creditors Lawyers

Banks

Bondholders

Shareholders

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

The Function of Stress Testing (continued)

bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal

market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or

extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and

hence economic capital and mitigate the vulnerability to important risk relevant effects

ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk

Supervisory Requirements and Expectations

bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management

bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie

risk management strategies to respond to the outcome of ST)

bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress

bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such

Supervisory Requirements and Expectations (continued)

bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required

bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage

bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating

migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the

impact of smaller deterioration in the credit environment

Supervisory Requirements and Expectations Regulatory Capital

000 005 010 015

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions

Credit Loss

Pro

ba

bili

ty D

en

sity

EL-norm=040

EL-stress=090

CVaR-norm=678

CVaR-stress=1579

NormalPD=1LGD=40Rho=01

StressedPD=15LGD=60Rho=015

Stressed Capital

Regulatory Capital

bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate

Supervisory Requirements and Expectations (continued)

bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy

bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately

bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk

bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)

bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)

Supervisory Requirements and Expectations (continued)

bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets

bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc

bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits

bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and

quantitative way

The Credit Risk Parameters for Stress Testing (continued)

bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters

bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation

bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited

bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate

bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be

key amp incorporate sufficient conservatism naturally but that varies

The Credit Risk Parameters for Stress Testing LGD

bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)

bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure

bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher

bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution

bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo

bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice

bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)

1 RecoveryRate

Discounted RecoveriesLGD=1- EAD

Discounted Direct amp Indirect Workout Costs

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Contractual features more senior and secured instruments do better

bull Absolute Priority Rule some violations (but usually small)

bull More senior instruments tend to be better secured

bull Debt cushion as distinct from position in the capital structure

bull High LGD for senior debt with little sub-debt

bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value

19

SENIORITY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees Trade Creditors Lawyers

Banks

Bondholders

Shareholders

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Supervisory Requirements and Expectations

bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management

bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie

risk management strategies to respond to the outcome of ST)

bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress

bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such

Supervisory Requirements and Expectations (continued)

bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required

bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage

bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating

migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the

impact of smaller deterioration in the credit environment

Supervisory Requirements and Expectations Regulatory Capital

000 005 010 015

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions

Credit Loss

Pro

ba

bili

ty D

en

sity

EL-norm=040

EL-stress=090

CVaR-norm=678

CVaR-stress=1579

NormalPD=1LGD=40Rho=01

StressedPD=15LGD=60Rho=015

Stressed Capital

Regulatory Capital

bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate

Supervisory Requirements and Expectations (continued)

bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy

bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately

bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk

bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)

bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)

Supervisory Requirements and Expectations (continued)

bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets

bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc

bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits

bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and

quantitative way

The Credit Risk Parameters for Stress Testing (continued)

bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters

bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation

bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited

bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate

bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be

key amp incorporate sufficient conservatism naturally but that varies

The Credit Risk Parameters for Stress Testing LGD

bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)

bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure

bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher

bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution

bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo

bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice

bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)

1 RecoveryRate

Discounted RecoveriesLGD=1- EAD

Discounted Direct amp Indirect Workout Costs

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Contractual features more senior and secured instruments do better

bull Absolute Priority Rule some violations (but usually small)

bull More senior instruments tend to be better secured

bull Debt cushion as distinct from position in the capital structure

bull High LGD for senior debt with little sub-debt

bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value

19

SENIORITY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees Trade Creditors Lawyers

Banks

Bondholders

Shareholders

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Supervisory Requirements and Expectations (continued)

bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required

bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage

bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating

migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the

impact of smaller deterioration in the credit environment

Supervisory Requirements and Expectations Regulatory Capital

000 005 010 015

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions

Credit Loss

Pro

ba

bili

ty D

en

sity

EL-norm=040

EL-stress=090

CVaR-norm=678

CVaR-stress=1579

NormalPD=1LGD=40Rho=01

StressedPD=15LGD=60Rho=015

Stressed Capital

Regulatory Capital

bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate

Supervisory Requirements and Expectations (continued)

bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy

bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately

bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk

bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)

bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)

Supervisory Requirements and Expectations (continued)

bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets

bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc

bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits

bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and

quantitative way

The Credit Risk Parameters for Stress Testing (continued)

bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters

bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation

bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited

bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate

bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be

key amp incorporate sufficient conservatism naturally but that varies

The Credit Risk Parameters for Stress Testing LGD

bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)

bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure

bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher

bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution

bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo

bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice

bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)

1 RecoveryRate

Discounted RecoveriesLGD=1- EAD

Discounted Direct amp Indirect Workout Costs

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Contractual features more senior and secured instruments do better

bull Absolute Priority Rule some violations (but usually small)

bull More senior instruments tend to be better secured

bull Debt cushion as distinct from position in the capital structure

bull High LGD for senior debt with little sub-debt

bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value

19

SENIORITY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees Trade Creditors Lawyers

Banks

Bondholders

Shareholders

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Supervisory Requirements and Expectations Regulatory Capital

000 005 010 015

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions

Credit Loss

Pro

ba

bili

ty D

en

sity

EL-norm=040

EL-stress=090

CVaR-norm=678

CVaR-stress=1579

NormalPD=1LGD=40Rho=01

StressedPD=15LGD=60Rho=015

Stressed Capital

Regulatory Capital

bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate

Supervisory Requirements and Expectations (continued)

bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy

bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately

bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk

bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)

bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)

Supervisory Requirements and Expectations (continued)

bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets

bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc

bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits

bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and

quantitative way

The Credit Risk Parameters for Stress Testing (continued)

bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters

bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation

bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited

bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate

bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be

key amp incorporate sufficient conservatism naturally but that varies

The Credit Risk Parameters for Stress Testing LGD

bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)

bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure

bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher

bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution

bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo

bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice

bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)

1 RecoveryRate

Discounted RecoveriesLGD=1- EAD

Discounted Direct amp Indirect Workout Costs

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Contractual features more senior and secured instruments do better

bull Absolute Priority Rule some violations (but usually small)

bull More senior instruments tend to be better secured

bull Debt cushion as distinct from position in the capital structure

bull High LGD for senior debt with little sub-debt

bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value

19

SENIORITY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees Trade Creditors Lawyers

Banks

Bondholders

Shareholders

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Supervisory Requirements and Expectations (continued)

bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy

bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately

bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk

bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)

bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)

Supervisory Requirements and Expectations (continued)

bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets

bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc

bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits

bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and

quantitative way

The Credit Risk Parameters for Stress Testing (continued)

bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters

bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation

bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited

bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate

bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be

key amp incorporate sufficient conservatism naturally but that varies

The Credit Risk Parameters for Stress Testing LGD

bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)

bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure

bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher

bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution

bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo

bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice

bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)

1 RecoveryRate

Discounted RecoveriesLGD=1- EAD

Discounted Direct amp Indirect Workout Costs

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Contractual features more senior and secured instruments do better

bull Absolute Priority Rule some violations (but usually small)

bull More senior instruments tend to be better secured

bull Debt cushion as distinct from position in the capital structure

bull High LGD for senior debt with little sub-debt

bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value

19

SENIORITY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees Trade Creditors Lawyers

Banks

Bondholders

Shareholders

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Supervisory Requirements and Expectations (continued)

bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets

bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc

bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits

bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and

quantitative way

The Credit Risk Parameters for Stress Testing (continued)

bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters

bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation

bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited

bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate

bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be

key amp incorporate sufficient conservatism naturally but that varies

The Credit Risk Parameters for Stress Testing LGD

bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)

bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure

bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher

bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution

bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo

bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice

bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)

1 RecoveryRate

Discounted RecoveriesLGD=1- EAD

Discounted Direct amp Indirect Workout Costs

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Contractual features more senior and secured instruments do better

bull Absolute Priority Rule some violations (but usually small)

bull More senior instruments tend to be better secured

bull Debt cushion as distinct from position in the capital structure

bull High LGD for senior debt with little sub-debt

bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value

19

SENIORITY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees Trade Creditors Lawyers

Banks

Bondholders

Shareholders

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

The Credit Risk Parameters for Stress Testing (continued)

bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters

bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation

bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited

bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate

bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be

key amp incorporate sufficient conservatism naturally but that varies

The Credit Risk Parameters for Stress Testing LGD

bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)

bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure

bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher

bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution

bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo

bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice

bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)

1 RecoveryRate

Discounted RecoveriesLGD=1- EAD

Discounted Direct amp Indirect Workout Costs

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Contractual features more senior and secured instruments do better

bull Absolute Priority Rule some violations (but usually small)

bull More senior instruments tend to be better secured

bull Debt cushion as distinct from position in the capital structure

bull High LGD for senior debt with little sub-debt

bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value

19

SENIORITY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees Trade Creditors Lawyers

Banks

Bondholders

Shareholders

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

The Credit Risk Parameters for Stress Testing LGD

bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)

bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure

bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher

bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution

bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo

bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice

bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)

1 RecoveryRate

Discounted RecoveriesLGD=1- EAD

Discounted Direct amp Indirect Workout Costs

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Contractual features more senior and secured instruments do better

bull Absolute Priority Rule some violations (but usually small)

bull More senior instruments tend to be better secured

bull Debt cushion as distinct from position in the capital structure

bull High LGD for senior debt with little sub-debt

bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value

19

SENIORITY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees Trade Creditors Lawyers

Banks

Bondholders

Shareholders

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Contractual features more senior and secured instruments do better

bull Absolute Priority Rule some violations (but usually small)

bull More senior instruments tend to be better secured

bull Debt cushion as distinct from position in the capital structure

bull High LGD for senior debt with little sub-debt

bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value

19

SENIORITY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees Trade Creditors Lawyers

Banks

Bondholders

Shareholders

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)

bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)

Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

The Credit Risk Parameters for Stress Testing LGD (continued)

00 02 04 06 08 10

00

05

10

15

Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)

LGD

Den

sity

-02 00 02 04 06 08 10

00

05

10

15

20

25

Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)

LGD

Den

sity

-02 00 02 04 06 08 10 120

00

51

01

52

0

Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)

LGD

Den

sity

-02 00 02 04 06 08 10 12

00

05

10

15

20

25

Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12

Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38

All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198

Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416

2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602

1537 174 581 368 0 NA 8 412 7 449 1 600 2134 229129 431 0 NA 1147 514 451 708 358 717 64 808 2149 592

1666 194 581 368 1147 514 459 703 365 712 65 805 4283 411

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moodys Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

bull Distributions of Moodyrsquos Defaulted Bonds amp Loan LGD (DRS Database 1970-2010)

bull Lower the quality of collateral the higher the LGD

bull Lower ranking of the creditor class the higher the LGD

bull And higher seniority debt tends to have better collateral

Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10

Reproduced with permission Moodyrsquos URD Release 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during

economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise

Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

The Credit Risk Parameters for Stress Testing LGD (continued)

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

The Credit Risk Parameters for Stress Testing LGD (continued)

bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously

bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery

bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)

bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -00903 255E-03

Book Value -00814 00174

Tobins Q 00729 873E-03

Intangibles Ratio 00978 702E-03

Working Capital Total Assets -01347 454E-03

Operating Cash Flow -831E-03 00193

Profit Margin - Industry -00917 120E-03

Industry - Utility -01506 818E-03

Industry - Technology 00608 203E-03Senior Secured 00432 00482Senior Unsecured 00725 311E-03Senior Subordinated 02266 121E-03Junior Subordinated 01088 00303Collateral Rank 01504 426E-12Percent Debt Above 01241 384E-03Percent Debt Below -02930 765E-06

Time Between Defaults -01853 740E-04

Time-to-Maturity 00255 00084

Number of Creditor Classes 00975 120E-03

Percent Secured Debt -01403 756E-03

Percent Bank Debt -02382 745E-03Investment Grade at Origination -00720 481E-03Principal at Default 899E-03 114E-03Cumulative Abnormal Returns -02753 176E-04Ultimate LGD - Obligor 05643 782E-06LGD at Default - Obligor 01906 405E-04LGD at Default - Instrument 02146 118E-14

Prepackaged Bankruptcy -00406 538E-03

Bankruptcy Filing 01429 500E-031989-1991 Recession 00678 004742000-2002 Recession 01074 00103Moodys Speculative Default Rate 00726 172E-04SampP 500 Return -01392 288E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 172E-10 960E-08 172E-10 960E-08Pseudo R-Squared 06997 06119 05822 04744Hoshmer-Lemeshow 04115 03345 05204 03907Area under ROC Curve 08936 07653 08983 07860Kolmogorov-Smirnov 112E-07 489E-06 142E-07 687E-06

Table 3 of Jacobs amp Karagozoglu (2010)Simultaneous Equation Modeling of Discounted Instrument amp Oligor LGD Full

Information Maximum Likelihood Estimation (Moodys URD 1985ndash2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y

Mar

ket

Lega

lM

acro

Jacobs Jr M and Karagozoglu A 2011 Modeling ultimate loss given default on corporate debt The Journal of Fixed Income 211 (Summer) 6-20

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

The Credit Risk Parameters for Stress Testing EAD

bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon

bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk

bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to

default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is

deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the

case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk

may actually become lower as banks cut lines

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

The Credit Risk Parameters for Stress Testing EAD (continued)

bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability

t tE t tTt

f ttT t t t t t t

t t

O - OEAD = O + LEQ times L - O O + | T times L - O

L - O

XX X

bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation

bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing

bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

EAD Example for Credit Models Jacobs (2010) Study

bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt

bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion

1 2 3 4 5 gt5

AAA-BBB 6456 6526 8493 9286 8458 000 6906

BB 3890 4213 4591 4391 4235 000 4079

B 4151 4392 4260 5277 4994 1400 4266

CCC-CC 3297 4738 5480 5505 5530 000 3685

C 2821 971 4764 2567 000 000 2022

Total 4081 4489 4779 5400 5205 1400 4221

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff P-Value

Utilization Used Amount Limit () -03508 253E-06

Total Commitment Line Limit ($) 364E-05 00723

Undrawn Headroom on line ($) 327E-05 742E-03

Time-to-Default (years) 00516 172E-05

Rating 1 BB (base = AAA-BBB) -01442 00426

Rating 2 B -00681 620E-03

Rating 3 CCC-CC -00735 103E-05

Rating 4 CCC -00502 208E-04

Leverage LTDebt MV Equity -00515 00714

Size Book Value (logarithm) 01154 263E-03

Intangibility Intangible Total Assets 00600 00214

Liquidity Current Cssets Current Liabilities -00366 00251

Profitabilty Net Income Net Sales -659E-04 00230

Colllateral Rank Higher -gt Lower Quality 00306 307E-03

Debt Cushion Debt Below the Loan -02801 518E-06

Aggregate Speculative Grade Default Rate -09336 00635

Percent Bank Debt in the Capital Structure 02854 561E-06

Percent Secured Debt in the Capital Structure 01115 265E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 274E+15

04670

02040

748E-12

Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -

Moodys Rated Defaulted Revolvers (1985-2009)

455

Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

The Credit Risk Parameters for Stress Testing PD

bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock

bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that

determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given

rating may change (eg economic downturn leads to more defaults)

bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions

to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible

bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

The Credit Risk Parameters for Stress Testing PD (continued)

bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default

bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST

bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement

bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions

bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

PD Estimation for Credit Models Rating Agency Data

bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness

bull Information about firmsrsquo creditworthiness has historically been difficult to obtain

bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years

bull However it is common to take average default rates by ratings as PD estimates

bull The figure shows that agency ratings reflect market segmentations

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

PD Estimation Rating Agency Data ndash Migration amp Default Rates

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87395 8626 0602 0010 0027 0002 0002 0000 3336 0000AA 0971 85616 7966 0359 0045 0018 0008 0001 4996 0020A 0062 2689 86763 5271 0488 0109 0032 0004 4528 0054BBB 0043 0184 4525 84517 4112 0775 0173 0019 5475 0176BB 0008 0056 0370 5644 75759 7239 0533 0080 9208 1104B 0010 0034 0126 0338 4762 73524 5767 0665 10544 4230CCC 0000 0021 0021 0142 0463 8263 60088 4104 12176 14721CC-C 0000 0000 0000 0000 0324 2374 8880 36270 16701 35451

FromTo AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54130 24062 5209 0357 0253 0038 0038 0000 15832 0081AA 3243 50038 21225 3220 0521 0150 0030 0012 21374 0186A 0202 8545 52504 14337 2617 0831 0143 0023 20247 0551BBB 0231 1132 13513 46508 8794 2827 0517 0083 24763 1631BB 0043 0181 2325 12105 26621 10741 1286 0129 38668 7900B 0038 0062 0295 1828 6931 22064 4665 0677 43918 19523CCC 0000 0000 0028 0759 2065 7138 8234 1034 44365 36378CC-C 0000 0000 0000 0000 0208 2033 1940 2633 44352 48833

Moodys Letter Rating Migration Rates (1970-2010)Panel 1 One-Year Average Rates

Panel 2 Five-Year Average Rates

Source Moodys Investor Service Default Report Corporate Default and Recovery Rates (1920-2010) 17 Mar 2011

bull Migration matrices summarize the average rates of transition between rating categories

bull The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

bull Default rates are increasing for worse ratings amp as the time horizons increase

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

PD Estimation Rating Agency Data ndash Default Rates

0000

0200

0400

0600

0800

1000

1200

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade

Aaa

Aa

A

Baa

All Inv Grade

0000

20000

40000

60000

80000

100000

120000

Def

ault

Rate

()

Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade

Ba

B

Caa-C

All Spec Grade

00 01 02 03 04 05

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

SpecGradeDefaultRates

000

005

010

015

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv Grade

Mean 00000 00405 00493 02065 00928Median 00000 00000 00000 00000 00000St Dev 00000 01516 01089 03198 01420Min 00000 00000 00000 00000 00000Max 00000 06180 04560 10960 04610

Ba B Caa-C

All Spec Grade

Mean 12532 52809 240224 47098Median 10020 45550 200000 35950St Dev 11982 38827 197715 29758Min 00000 00000 00000 09590Max 48920 154700 1000000 131370

bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution

Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

PD Estimation Rating Agency Data ndash Performance of Ratings

bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010

bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average

bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

PD Estimation for Credit Models Kamakura Public Firm Model

bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques

bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables

bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)

bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

PD Estimation for Credit Models Kamakura Public Firm Model (cont)

bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model

bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM

bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance

Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

PD Estimation for Credit Models Bayesian Model

bull Jacobs amp Kiefer (2010) Bayesian 1 (Binomial ndash rating agencies) 2 (Basel II ASRF) amp 3-parameter extension (Generalized Linear Mixed Models) models

bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations

bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated

E(θ|R) σθ

95 Credible Interval E(ρ|R) σρ

95 Credible Interval E(τ|R) στ

95 Credible Interval

Acceptance Rate

Stressed Regulatory Capital (θ)1

Minimum Regulatory Capital2

Stressed Regulatory Capital Markup

1 Parameter Model 000977 000174

(000662 00134) 0245 653 529 2349

2 Parameter Model 00105 000175

(000732 00140) 00770 00194

(00435 0119) 0228 672 555 2106

3 Parameter Model 00100 000176

(00069 00139) 00812 00185

(0043 0132) 0162 00732

(-0006 0293) 0239 669 538 2452

1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD

Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)

Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

PD Estimation for Credit Models Bayesian Model (cont)

bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)

bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)

bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)

0000 0005 0010 0015 0020 0025 0030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

The Credit Risk Parameters for Stress Testing Correlations

bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it

bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography

bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)

bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy

bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors

bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

The Credit Risk Parameters for Stress Testing Correlations (cont)

bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)

bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers

bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes

bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans

bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Correlation Estimation for Credit Risk Models ndash Empirical Example

bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations

bull The plot shows that correlations are time-varying and can differ according to time horizon

bull The table shows how correlations amongst different sectorsrsquo indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500

-082

-062

-042

-022

-002

018

038

058

078

Date (YYYYMMDD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window

30yr T-bond for 3mo rolling window

30yr T-bond for 6mo rolling window

30yr T-bond for 1yr rolling window

30yr T-bond for 2yr rolling window

30yr T-bond for 3yr rolling window

SampP 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

SampP 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

SampP 600 Small Cap Equity Index

PLX Precious Metals Index

SampP 500 Equity Index - -00211 -01504 00056 -00602 -72E-04 08395 07852 07723 08071 00801

Golman Sachs Commodity Index 00456 - 00256 02520 08600 00257 00096 -00413 00188 00299 01849

10 Year Treasury Yield 339E-37 00382 - 00241 00632 05791 -00727 00302 -00509 01053 00881

CRB Precious Metals Index 06237 238E-112 00419 - 01528 -00414 00374 -00324 00649 00152 05978

CRB Energy Index 643E-06 000E+00 273E-06 112E-30 - 00145 -00255 -00467 -00356 00129 01538

1 Year Treasury Yield 09407 04185 000E+00 839E-05 02800 - 00785 01340 00757 01871 00086

SampP 400 Equity Index 000E+00 04478 127E-14 304E-03 00558 612E-10 - 08675 09224 09263 01232

NASDAQ Equity Index 000E+00 00025 00283 176E-02 643E-04 123E-22 000E+00 - 08701 08315 00512

Russsel 2000 Equity Index 000E+00 01211 127E-14 886E-08 763E-03 598E-10 000E+00 000E+00 - 09748 01353

SampP 600 Small Cap Equity Index 000E+00 01154 345E-08 04232 04972 493E-23 000E+00 000E+00 000E+00 - 01086

PLX Precious Metals Index 211E-09 426E-44 645E-11 000E+00 117E-30 05233 267E-20 173E-04 339E-24 966E-09 -

Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ate

s

P-Values

Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis

000 002 004 006 008 010

00

02

04

06

08

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0006 CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

006 007 008 009 010 011

00

00

05

01

00

15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions

PD=001 LGD=04EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=00610 CVaR=00800 CVaR=00971

Rho=01

Rho=015

Rho=02

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

The Credit Risk Parameters for Stress Testing Conclusion

bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)

bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well

bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these

bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios

bull They can be particularly interesting for investigations on economic capital with the help of portfolio models

bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Interpretation of Stress Test Results

bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk

bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements

bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases

bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail

bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Interpretation of Stress Test Results (continued)

bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance

bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original

portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets

too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Interpretation of Stress Test Results (concluded)

bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels

bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST

bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency

bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels

bull One can see the probability of occurrence or the plausibility of a ST as a related problem

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp

EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different

bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the

estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have

to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased

on the basis of an increased LGD

bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio

after stressing the PDs

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of

default provision - typically given by exposure (EAD) times LGD

bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in

economic variables sensitivity analysis is statistically founded

bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as

mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset

value correlations or dependencies amongst systematic risk drivers

bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp

we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any

event or context amp for all loans without respect to individual properties

bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial

institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital

requirements but it does not help for portfolio and risk management

bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains

the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk

parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the

benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp

are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential

effects resulting from possible correlations of risk factors

bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors

bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting

points hence scenarios are chosen which involve risk factors having the largest impact

bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past

historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be

realistic amp generally not possible to specify the probability of the scenario

bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage

that if tells us the probability of a scenario occuring

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of

unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution

as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management

bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have

a macro-economic model of the dependence of the risk parameters

bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk

factors and stress events need intricate methods of estimation and validation

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the

potential need to specify probability distributions for events not in our reference data-sets

bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events

bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management

bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)

bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Procedures for Conducting Stress Tests Uniform ST

bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile

from a posterior distribution of PD

bull Develop stressed rating migration migrations (eg increase decrease downgrade upgrade rates) and derive stressed rating grades

bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings

bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role

bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Procedures for Conducting Stress Tests Risk Factor Sensitization

bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity

indices credit spreads exchange rates GDP oil prices credit losses

bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later

bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis

bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital

bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests

bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Procedures for Conducting Stress Tests Historical Scenarios

bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation

bull Though risk management implications is a backward looking approach there are good reasons to use it

bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of

LTCM and the Russian default or the 1994 global bond price crash

bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability

bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios

bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Procedures for Conducting Stress Tests Statistical Scenarios

bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA

bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate

bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST

bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Procedures for Conducting Stress Tests Hypothetical Scenarios

bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis

bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used

bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target

bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects

bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST

which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch

in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC

bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates

bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

A Simple Stress Testing Example Default amp Transition Rate Data

0

02

04

06

08

1

12

1979

1231

1980

0930

1981

0630

1982

0331

1982

1231

1983

0930

1984

0630

1985

0331

1985

1231

1986

0930

1987

0630

1988

0331

1988

1231

1989

0930

1990

0630

1991

0331

1991

1231

1992

0930

1993

0630

1994

0331

1994

1231

1995

0930

1996

0630

1997

0331

1997

1231

1998

0930

1999

0630

2000

0331

2000

1231

2001

0930

2002

0630

2003

0331

2003

1231

2004

0930

2005

0630

2006

0331

2006

1231

2007

0930

2008

0630

2009

0331

2009

1231

2010

0930

Def

ault

Rate

US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)

DR_Baa-A

DR_Ba

DR_B

DR_C-Caa

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

A Simple Stress Testing Example Default amp Transition Rate Data

(contrsquod)bull Collapse the best ratings due to

paucity of defaultsbull DR increase exponentially amp

diagonals smaller as ratings worsenbull Correlations higher between adjacent

than more separated ratings

Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumBaa-A 00930 00000 02140 230 00000 12926 10000 1588 1073 1390Ba 11341 07246 13265 117 00000 67460 10000 7088 5535B 61652 52326 57116 093 00000 330645 10000 3981Caa-C 311884 200000 291442 093 00000 1000000 10000

Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

Correlations

Baa-A Ba B Caa-C Default

Baa-A 9794 162 036 004 004

Ba 129 8723 952 062 134

B 013 511 8382 525 569

Caa-C 023 144 810 6834 2189

Through-the-Cycle Annual Transition Matrix US Domiciled Industrial Obligors

(Moodys DRS 1980-2011)

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

A Simple Stress Testing Example Bond Index Return Data

-012

-007

-002

003

008

Loga

rithm

ic R

etur

ns

Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)

BondUSCorpBaa-A BondUSCorpBa

BondUSCorpB BondUSCorpC-Caa

Sector Rating Mean MedianStandard Deviation

Coefficient of

Variation Minimum MaximumAa-Aaa 00377 00274 07167 1899 -123977 116545 10000 3607 884 826Baa-A 00433 00331 05247 1211 -115403 74375 10000 868 1646B-Ba 00372 00418 05308 1427 -60864 108899 10000 7883C-Caa 00194 00425 04478 2312 -47283 83753 10000

Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )

Correlations

Portfolio 1 -Industrials

bull Note the high variability relative to the mean of these

bull Higher ratings actually return amp vary more but CV is U-shaped

bull Highest correlations between adjacent ratings at the high amp low end

bull Some of the correlations are lower and some higher than Basel II prescribed

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

A Simple Stress Testing Example Risk Factor Data

bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate

bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad

index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro

factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Level

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

VIX Volatility Index 239 218 112 4673 100 619 10000 -112 423 -1452 2393 -1078 2208 -2608 -275 3405 -1197Fama-French Size 000 000 008 3812 -020 017 - 10000 1547 -1629 -967 1313 1184 622 -1040 725 642Fama-French Value 002 001 010 630 -029 035 - - 10000 -3786 1034 -1699 310 -509 1223 641 -1096Fama-French Market 003 004 013 532 -037 028 - - - 10000 -518 -1871 -340 -786 -1348 054 -892Fama-French Risk-Free Rate 002 002 001 064 000 006 - - - - 10000 1554 -2552 -7915 1488 7791 083Fama-French Momentum 003 003 012 393 -058 034 - - - - - 10000 -998 -824 1139 317 867CampI Charegoff Rates 001 091 058 6284 010 254 - - - - - - 10000 -998 -2766 528 -1174GDP - Annual Change 003 300 232 8846 -503 848 - - - - - - - - 10000 -2386 179CPI - Annual Change 004 300 265 6604 115 1296 - - - - - - - - - 10000 -712Oil Price - Annual Change 010 333 3471 36494 -5614 13093 - - - - - - - - - - 10000

Variable

CorrelationsUS Historical Macroeconomic Risk Factor Variables Quarterly Data 1980-2010 (Source Various)

MaximumMinimum

Coefficient of

VariationStandard DeviationMedianMean

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-060

-040

-020

000

020

040

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

R

etur

n

Fama-French Equity Market Pricing Factor Returns

Fama-French Size Fama-French Value Fama-French Market Fama-French Risk-Free Rate Fama-French Momentum

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

A Simple Stress Testing Example Risk Factor Data (continued)

bull

-6000

-4000

-2000

000

2000

4000

6000

8000

10000

12000

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

Annu

al

Cha

nge

Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes

GDP - Annual Change CPI - Annual Change Oil Price - Annual Change

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

A Simple Stress Testing Example Risk Factor Data (continued)

000

100

200

300

400

500

600

700

Dat

e19

8006

3019

8103

3119

8112

3119

8209

3019

8306

3019

8403

3119

8412

3119

8509

3019

8606

3019

8703

3119

8712

3119

8809

3019

8906

3019

9003

3119

9012

3119

9109

3019

9206

3019

9303

3119

9312

3119

9409

3019

9506

3019

9603

3119

9612

3119

9709

3019

9806

3019

9903

3119

9912

3120

0009

3020

0106

3020

0203

3120

0212

3120

0309

3020

0406

3020

0503

3120

0512

3120

0609

3020

0706

3020

0803

3120

0812

3120

0909

3020

1006

30

VIX Volatility Index and CampI Charge-off Rates

VIX Volatility Index CampI Charegoff Rates

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

A Simple Stress Testing Example Default Rate Regression Model

Default Rate

VIX Volatility Index

Fama-French Size

Fama-French Value

Fama-French Market

Fama-French Risk-Free Rate

Fama-French Momentum

CampI Charegoff Rates

GDP - Annual Change

CPI - Annual Change

Oil Price - Annual Change

R-Squared Statistic

F Statistic P-Value

Baa-A Coeffi cient Estimate 00665 -0118 -03047 -02055 09276 -02872 002354 -001956 -001936 01654P-Value 298E-04 642E-03 153E-02 190E-02 774E-03 717E-03 526E-03 569E-03 229E-01 762E-03

Ba Coeffi cient Estimate 01973 -1047 -1055 -164 08095 -06578 07042 -02123 -04336 01351P-Value 500E-03 461E-03 375E-03 625E-03 178E-05 458E-02 925E-04 295E-04 414E-06 659E-03

B Coeffi cient Estimate 02129 -2249 -5488 -4443 1706 -5184 15415 -07663 -1396 01267P-Value 648E-03 734E-03 323E-03 277E-02 453E-03 211E-02 183E-03 463E-03 114E-03 375E-02

Caa-C Coeffi cient Estimate 1041 -5332 3242 -8875 3208 -5797 858 -3246 -4908 01743P-Value 727E-07 610E-03 172E-02 309E-03 375E-03 742E-03 618E-05 495E-03 695E-03 492E-03

denotes statistical significance at the 01 1 and 5 confidence levels respectively

447E-043719

Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)

119E-124478

579E-054228

259E-083880

bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad

market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more

precisely estimated for lower ratings

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

A Simple Stress Testing Example Results of Alternative Scenarios

bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect

bull Generally economic has a bigger stressed capital than reg-EC

bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux

Expected Loss - Credit Metrics

Economic Credit Capital - Credit Metrics

Regulatory Credit Capital - Basel 2 IRB

Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527

Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

A Simple Stress Testing Example Results of Alternative Scenarios

CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-010 -008 -006 -004 -002 000

05

00

15

00

25

00

B2-cVar999=929

CM-cVar999=717

EL=263

CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses

Pro

ba

bili

ty

-015 -010 -005 000

05

00

15

00

25

00

B2-cVar999=1527

CM-cVar999=1738

EL=503

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits

and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial

loans An 18-year internal study The Journal of the Risk Management Association May 28-35

bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228

bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework

bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)

bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189

bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of

Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on

bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper

bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper

bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper

bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)

bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470

bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro

Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro