a theory of monitoring credit risk

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APRIL 2011

A Theory of Monitoring Credit RiskDouglas Dwyer1

Moodys Analytics

April 2011

Abstract

On any given day, credit analysts monitor multiple names. Some names will have been reviewed recently,but not all. Some names are easily traded out of, while some names are more difficult to hedge. Somenames represent large exposures while others represent small. Some are known high credit risks whileothers are low credit risks. The risk profile of some exposures may have changed recently while othersremained unchanged. How to triage? Which names should the analyst review first? Next? Ultimately, howdoes the institution value a credit review? This paper derives an optimal monitoring strategy, under theassumption that there is a fixed cost for reviewing a loan. The framework can be embedded within a

dynamic competitive equilibrium in which prices reflect public information. We calibrate the frameworkusing parameter values with empirical interpretations. In the dynamic setting, we show that in specificcircumstances, a mid-year review can produce incremental value equal to 1.3% of the exposure size.

1douglas.dwyer@moodys com. We would like to thank Ivo Antonov, Heather Russell, Roger Stein, and Jing Zhang

for valuable feedback on this paper. We also appreciate feedback from the Moodys Academic Advisory Board andparticipants at the Moodys Analytics Risk Practitioner Conference. Andrew Caplin, Boyan Jovanovic, and JohnLeahy provided valuable direction.


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1 IntroductionMonitoring credit risk is costly, and one cost is the collection and analysis of information. In the pastdecade, financial institutions have invested in systems that provide credit managers access to regularlyupdated credit risk information. With such information systems in place, credit managers can access

timely information at a marginal cost of close to zero. Depending on the asset class, available informationmay include credit spreads, financial statements, and the borrowers conduct on the account. Informationmay also include quantitative models designed to assess the borrowers credit risk. Models can include aprobability of default based on a structural model for larger firms with listed equity. For medium-sized,private firms, a financial statement-based probability of default (PD) derived from a quantitative modelmay be available. For small firms and consumer credit, measures of credit risk based on behavioralinformation may be available (e.g., a credit score).

Informations value lies in its potential to change action, and credit managers make decisions and actbased upon updated information. Actions may include originating new loans, extending additional credit toexisting borrowers, not refinancing a maturing term loan, cutting lines of credit, buying insurance againsta borrower defaulting, selling the exposure to another financial institution, working with the borrower toestablish an alternative source of financing, and finally, securitizing the exposure off the balance sheet via

a collateralized loan obligation. In consumer credit, managers make loan decisions using only quantitativerisk metrics (i.e., a credit score). In contrast, financial institutions use a combination of quantitative riskmetrics and a review of qualitative information to assess large exposures. Performing additional analyses

requires resources that have a direct cost.2

Risk management departments review credits periodically according to specific requirements set byregulators and financial institutions. The OCC prescribes reviewing high risk credits more often and allowsfor smaller, performing credits to be reviewed less often (page 9, Rating Credit Risk, OCC, 2001).Depending on resources, credit analysts may conduct an annual review of each credit, and they mayconduct an additional review whenever the quantitative measure of risk differs from the internal rating bya significant margin. A major change in the borrowers business or balance sheet may trigger a review.The reviews outcome may be a new internal rating or a new PD; the new rating may require the bank totake an action on the loan.

In this paper, we focus on measuring the economic value of a review and deriving an optimal reviewpolicy within a theoretical setting. In the core framework, prior to the review, the credit analyst has apartial information PDwith which to assess the risk-return trade-off of the exposure. If the credit analystchooses to conduct the review, the credit analyst has a full information PD with better discriminatorypower. The outcome of the review is both a new and more informative PD , as well as a decisionregarding whether or not to take action on the exposure; the lenders action is to sell the exposure. If the

lender sells the exposure, he loses the spread income while preserving the principal.3

Ex post, the review adds value when it leads a manager to a different action to either hold a name theywould otherwise sell or to sell a name they would otherwise hold. The ex antevalue of the review per unitof exposure is the difference in the expected return on the loan under full information with optimal sellingversus partial information with optimal selling, given partial information. The value of the review is largeston the names that the portfolio manager is indifferent between holding and selling under partialinformation.

2Dwyer and Russell (2010) present one approach to combining qualitative factors with a quantitative risk metric.

3When an institution decides to limit their exposure to a particular borrower, they have multiple actions they can take

(enforcing covenants, cutting lines of credits, buying default insurance, securitization, and so forth). The availableactions depend upon the specifics of the relationship as well as the asset class. The most straightforward of theseactions to model is the simple selling of the loan (or not extending credit in the first place), which is why we begin bymodeling this scenario.


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We extend the framework to include a time dimension; the information content of a review erodes overtime since the last review. We derive an optimal review policy in which some relationships are reviewedregularly and others periodically. We also show how the framework can be embedded in a competitivegeneral equilibrium. In this setting, the spread charged to the borrower is agreed upon prior to the reviewbeing conducted. The spread must be large enough to cover both a transaction cost associated withmanaging the relationship as well as the credit risk associated with the loan. In this equilibrium, thespread on the loan reflects public information, but not private information. Based on public information,the riskier loans require higher spreads. Conditional on public information, the safest loans are reviewedless often than the riskier loans. For a given level of public information, there is a U-Shaped relationshipbetween the frequency of review and the level of credit risk implied by the private information.

We contrast the value of the relationship under an optimal review policy with the value of the relationshipunder a fixed review cycle of every year. The incremental value of the review is greatest when the optimalreview policy is to review the loan more than once each year. This multi-annual review occurs for loanswith elevated risk according to public information and a more favorable level of risk under full information.We find that, in some circumstances, the incremental value of a mid-year review can exceed 1% of theexposure amount.

The building blocks of the framework include probabilities of default, average default rates for apopulation, and cumulative accuracy profiles (CAP). Financial institutions now collect such information as

part of the Basel II initiative. Therefore, the framework can potentially be calibrated to an asset classesspecific settings. The base case we use throughout the paper is calibrated to be consistent with threeempirical benchmarks: (i) a partial information CAP curve comparable to that which can be achievedthrough a commercially available, financial statement-based probability of default model; (ii) the amountof information available utilizing both public and private information is comparable to that which can beachieved using a structural model of default that incorporates equity prices and is commercially available;and (iii) the average default rate of the population is 2% per annum. These benchmarks are reasonablestarting points for analyzing the optimal review policy for debt issued by private firms large enough tohave listed equity, but do not have listed equity. Different asset classes would likely require differentcalibrations.

Relation to the Literature

This paper draws upon two different bodies of literature: power curves and the economics of inaction.Researchers use power curvesto quantify the predictive power of a medical test or a credit risk model.We draw from the economics of inaction literature to derive the optimal dynamic policy for reviewing acredit.

In medical testing, the outcome of a test can be a continuous variable, for which higher levels areassociated with higher incidences of a disease (e.g., higher blood pressure is associated with a higherlikelihood of heart disease). One may want to determine a threshold for the test outcome. If the testoutcome is above this threshold, the disease is treated (e.g., medication is given to prevent heartdisease) and not otherwise. In determining this threshold, one considers the rates of true positives(treating someone who does have the disease), false positives, true negatives, and false negativesimplied by the threshold, as well as their associated costs and benefits, and then optimizes accordingly.

Power curves are analytic tools used in the biostatistics literature for this purpose.4

Roger Stein (2005)drew upon this literature to determine optimal lending policies in the context of managing credit risk. Healso showed how to compute the benefit to a financial institution of using one model relative to another ina variety of settings. This paper uses the same tools, but takes a step back. It views the test as having acost to administer, and it considers whether or not the value of knowing the outcome of this test justifiesthe cost of administering the test.

5

4Power curves include Receiver Operator Characteristic curves (ROC curves) and Cumulative Accuracy Profiles

(CAP curves).5

Biostatisticians study this question as well. Mandelbatt et al. (2009) provide a recent and controversial example.


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Economists have developed theories in which it is optimal for agents to do nothing for periods of time in adynamic setting (c.f., Stokey, 2009). When an action has a fixed cost, one can typically derive an optimalrange of inaction. Economists have used such theories to explain stickiness in prices and wages,investment spikes, and the slow adoption of new technologies. In this line of research, one paperexplicitly models the fixed cost as the cost of acquiring and analyzing updated information (Reis, 2006).Reis models the fixed cost of changing a price as the cost of updating one's information, and the newinformation is used to reset the price to an optimal level. He focuses on explaining the dynamics ofinflation. While the contexts are different, the structures of the dynamic programming problems aresimilar: we both model the optimal wait time to update ones information in an infinite horizon setting.

The remainder of the paper is organized in the following way.

Section 2 presents the basic framework of the model, in which there is both a partial information PDand a full information PD. We show how to derive the CAP plots as well as the value of information ina static setting.

Section 3 incorporates a time dimension and shows how to solve for the optimal review policy in adynamic setting.

Section 4 argues that the framework can be embedded into a dynamic competitive generalequilibrium, in which prices reflect public information.

Section 5 calibrates the framework to actual examples and characterizes the equilibrium spreads aswell as the optimal review policy.

Section 6 looks at the question: how much value does pursuing an optimal review strategy create?

Section 7 provides concluding remarks.

Appendix A is a table of the symbols used in the paper.

Appendix B derives the distribution of PDs and the Cumulative Accuracy Profiles implied by thisframework.

Appendix C describes the algorithm used to calibrate the model parameters for the benchmark case.


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2 The ModelWe model how to determine the value of information for specific names in a portfolio. We presume theresources required to execute a review are costly, and the outcome of the review may be a decision tohedge the exposure for a specific cost. The credit analyst should choose to hedge the exposure if thecredit risk is very elevated relative to the sum of the spread and the transaction cost of selling the loan.The credit analyst can make this decision using either partial information or full information. The marginalcost of the partial information is assumed to be zero, whereas, obtaining full information on a specificcredit has a cost. We compute the value of information per unit of exposure as the difference between theexpected return under full information, where the credit analyst acts optimally, and the return based onpartial information, where that the credit analyst acts optimally using the available information. In thissetting, information is valuable when it leads an analyst to act differently.

2.1 The Basic Framework

Suppose you have an exposure with a full information PD defined by:

0( )N +X

In this equation,Xis a row vector, the elements of the row vector are standard normal variablesindependent of each other, is a column vector of coefficients, and 0 is a scalar, i.e., the intercept. Thefunction,N, is the standard cumulative normal distribution. Without loss of generality, we assume that allthe elements ofare positive so that large values ofXcorrespond to elevated credit risk. We will assume

that the intercept,0 , is negative, which implies that the median credit has a probability of default of lessthan 50%. We can give a latent variable interpretation to the full information PD as follows. Let be alatent variable that has a standard normal distribution. Default occurs if:

0 < +X

The probability of being less than 0 +X is given by 0( )N +X , i.e., default occurs when there is

a sufficiently bad realization of the latent variable . Partition X into Y Zso that Y Z = +X , where

Yis known andZis unknown. Using this notation, we define the full information probability of default as:

0 0( , ; , , ) ( )

FIPD Y Z N Y Z = + + (1)

Now we derive the partial information PD. Default occurs when:

0 Y Z < + +

or

0

1 1

YZ

+


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By the same argument, if both YandZare unknown, then the PD is given by 0

1N

+

. For

convenience, we refer to this quantity as the CDT, the Central Default Tendency. The CDTis the averagedefault rate of the population.

It will be convenient to define ( ) ( )1/2 1/2

0 0 1 ; 1

= + = + so the partial information PD can

be written as0 ( )N Y + .

2.2 Distribution of PDs and Cumulative Accuracy Profiles

In Appendix B, we construct the distribution of the partial information PDsand the full information PDs.We also derive an expression for the Cumulative Accuracy Profile (CAP curves) and the correspondingAccuracy Ratio (AR). These constructs allow us to calibrate the parameters using empirical referencepoints. We show that the quantile functions of the full information and partial information PDs are givenby:

( )1 0( ) ( ) 'FIF q N N q = + and ( )1

0 ( ) ( ) '

PIF q N N q = +

These functions take a percentile as an input and produce the PD associated with that percentile. Forexample, if q=0.975, then 97.5% of the population has a full information PD of less than FFI(0.975). TheCAP curve for the full information and partial information PDs are given by:

( )1 00

( ) ( ) 'x

FICAP x N N q dq CDT = +

and

( )1

00

( ) ( ) '

= +

x

PICAP x N N q dq CDT

Figure 1 presents the quantile function for full information and partial information PDs based on a CDT of

2% and of 0.9 and of 0.5. The figure compares the distribution of the full information PD with that ofthe partial information PD. We find that the distribution of the full information PD is more dispersed than

the distribution of partial information PDs. Figure 1s right hand panel focuses on upper percentiles of thedistribution. Under this parameterization, the AR of the partial information PDis 77.8%, and for the fullinformation PD, the AR is 85.5%. An AR of 85.5% is roughly comparable with the accuracy ratio of theMoodys KMV public firm model on a large sample at a one-year horizon and exceeds the accuracy ratiostypically found in private firm models. An AR of 77.8% is roughly comparable to the AR that can beachieved using a financial statement-based model on a comparable sample. A population default rate of

2% is a reasonable value for the long-run average default rate for a Commercial and Industrial portfolio.6

Note that the partial information PD is larger than the full information PD for 91% of the sample, but forthe riskiest names, the full information PD is larger than the partial information PD. Also note that themedian partial information PD (about 0.40%) is three times larger than the median full information PD(about 0.13%).

6See Sections 2.4 and 4.8 of Dwyer and Zhao (2009).


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Figure 1 Quantile Function for Full and Partial Information PD

2.3 Determining the Value of Information

Consider an exposure that pays back the principal plus a coupon in one time period (e.g., one-year) fromnow. We will refer to the principal as the exposure or size of the loan. The coupon is equal to s + rper unitexposure, where ris the risk free rate and s is the spread over the risk free rate. The transaction costassociated with selling the loan is h. For convenience, we assume that the coupon is paid whether or notdefault occurs, and, if default occurs, the LGD is 100% of the principal. We assume that the risk manageris risk-neutral, and that the risk manager currently holds the exposure. The risk manager hedges if theprobability of default exceeds the spread, plus the transaction cost. Under full information, the riskmanager hedges if the risk of holding the exposure (the full information PD) exceeds the return on holdingthe exposure, plus the transaction cost (s+h):

( )0N Y Z s h + + > +

If the complete information PD is not available, the risk manager hedges if:

( )0 N Y s h + > + .

Before deriving the value of information, in general, going through a specific example is instructive.Suppose that the transaction cost is 0%, the spread earned on the loan equals 2%, and the partialinformation PD is 2.01%. If full information is not available, then the risk manager hedges, in which case

he loses 0% versus an expected loss of 0.01% associated when holding the position.

Let the vectors YandZeach have one element, and let =0.9 and =0.5. We can solve for0so that thedefault rate of the population is 2%:

0 25 50 75 1000.01

0.10

1.00

10.00

100.00

Percent of Population

PD(%)

Full

Partial

80 85 90 95 1001.00

10.00

100.00


PD(%)

Full

Partial


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1 2 2

0 1 1

00

2

1

11

2

1

(.02) 1 2.948

2.6361

0.8051

N

= + + =

= = +

= =+

.

The partial information PD is given by:( 2.636 0.805 )N Y +

One can verify that if Yequals 0.726, the partial information PD is 2.01%. As Yhas a standard normaldistribution, this implies that 76.6% (N(0.726)) of the population has a partial information PD of less thanor equal to 2.01%. Suppose that the risk manager conducts a loan review and obtains Z, and that therealized value ofZturns out to be 0. Then the full information PD is given by N(-2.948+0.9*0.726+0.5*0)or 1.090%. Therefore, a neutral realizationforZlowers the PD of the exposure and leads a risk managernot to hedge the exposure. In this case, the expected return on the exposure is now 0.91%. Theadditional information saves the risk manager 0.91% (the expected return on the loan). For this example,

whether or not the fully-informed risk managerchooses to hedge the loan depends on the realization of Z.The fully-informed risk managerhedges if the full information PD exceeds 2%:

( 2.948 0.9 0.726 0.5 ) 2%N Z + + >

Taking the normal inverse of both sides and solving for Zreveals that the fully informed risk managerhedges wheneverZis greater than 0.480, approximately 31.6% of the time. In this example, fullinformation leads the risk manager to do what he would have done under partial information about 31.6%of the time and to do something different 68.4% of the time, i.e., he chooses not to hedge the loan. If hechooses to conduct a credit review, the value of the credit review equals the probability that he actsdifferently as a result of the review (he does not hedge the loan), N(0.480), multiplied by the expectedgain from not hedging the loan, given that he does not hedge the loan. The value of information is givenby:

0.480

( 2.948 0.9 0.726 0.5 ) ( )(0.480) 0.02(0.480)

N Z n Z dZNN

+ +

where nis the density function of a standard normal distribution. Using numerical techniques to evaluatethe integral, we find that the value of information is 0.860% for this example. Therefore, the risk manager

would pay up to 0.860% of the amount of the exposure for knowledge of Z. Figure 2 depicts this exampleplotting the full information PD and the excess spread, h+s-PDFI, as a function ofZ. Note that it is askewed distribution, in that the downside associated with large realizations ofZis larger than the upsideassociated with positive realizations ofZ. Under partial information, the value of the loan is close to zero.Under full information, the value of the loan equals the probability of the excess spread being positivemultiplied by the expectation of the distribution of the excess spread truncated to the positive region.


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Figure 2 Full Information PD and Excess Spread, as a function ofZ

The general solution to this example for the case that the partial information PD leads the risk manager tohedge (partial information PD>h + s) is:

( )0( ) ( )

( )( )

cv

N Y Z n Z dZN cv h s

N cv

+ + +

where

( )1 0( )N h s Ycv

+ += .

Here cv is the realized value of Zthat leads the risk manger to be indifferent between hedging and nothedging.N(cv) is the probability that the risk manager does not hedge (i.e., the probability of doing

something different), h + s is the spread on the loan plus the transaction cost and

0( ) ( )

( )

cv

N Y Z n Z dZ

N cv

+ +is the expected loss on the loan, conditional on a good outcome for the

review (i.e.,Z cv), this loss is saved as the loan is hedged, and the spread and the transaction cost aresubtracted as this money is lost by hedging.

-20%

-10%

0%

10%

20%

30%

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Realized Z

FullInformationPD/ExcessSpread

Full Information PD Excess Spread


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In the general case, the value of information can be expressed as:

( )

( )

0

0

0

( ) ( )( , , ; , , ) ( )

( )

( ) ( )(1 ( ))

( )

cv

PI

cv

PI

N Y Z n Z dZVI Y h s PD h s N cv h s

N cv

N Y Z n Z dZPD h s N cv h s

N cv

+ += > + +

+ ++ < +

(3)

where cv and PDPIare defined as above, and (x) takes on a value of 1, if the statementx is true and 0otherwise. The value of information is expressed as a proportion of the exposure.

Table 1 Value of the Review (bps)

PartialPartialPartialPartial Information PDInformation PDInformation PDInformation PD

0.10%0.10%0.10%0.10% 0.20%0.20%0.20%0.20% 0.50%0.50%0.50%0.50% 1.00%1.00%1.00%1.00% 2.00%2.00%2.00%2.00% 5.00%5.00%5.00%5.00% 10.00%10.00%10.00%10.00% 20.00%20.00%20.00%20.00% 50.00%50.00%50.00%50.00%

h+s

h+s

h+s

h+s(bps)

(bps)

(bps)

(bps)

10101010 6 4 2 1 0 0 0 0 0

20202020 4 11 6 3 1 0 0 0 0

50505050 2 6 25 16 7 1 0 0 0

100100100100 1 3 16 47 27 7 1 0 0

200200200200 0 1 7 27 86 33 8 1 0

500500500500 0 0 1 7 33 186 75 14 0

1000100010001000 0 0 0 1 8 75 320 94 1

2000200020002000 0 0 0 0 1 14 94 515 22

5000500050005000 0 0 0 0 0 0 1 22 738

Table 1 presents the value of information as a function of (h+s) and the partial information PD for a CDT

of 2% and of 0.9 and of 0.5. Note that holding the partial information PD constant (moving down acolumn), the value of information is greatest where the firm is indifferent between holding/hedging theexposure under partial information. Further, note that the maximum value of information for each partialinformation PD increases with the PD. The value of information has the potential to be the largest for theloans with the largest expected losses. The value of information is fairly sizable. For example, suppose afinancial institution has a $10mm exposure to a firm with a partial information PD of 2% and a spread of100bps. Under this framework, the financial institution should review the loan provided the cost of thereview is less than $27,000.

Such a table could be used to determine the exposures to review first to triage. Nevertheless, up to thispoint, the framework abstracts away from the fact that credit analysts manage a portfolio over time. Wenow turn to incorporating a time dimension into the framework.


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3 Incorporating a Time DimensionBanks manage credit risk through time. Whether or not a bank reviews a loan today depends upon thelast time the bank reviewed the loan. The value of the information collected during the review processdecays over time. In this section, we show how to incorporate a time dimension into the problem byassuming that both the terms of the loan and the public information are fixed over time, but the privateinformation varies over time. We also provide a general setup of the problem using transition matrices. Insubsequent sections, we show how the framework can be embedded in a general equilibrium. Wecharacterize the solution to the framework and the implications for equilibrium prices using three differentexamples.

The Dynamic Programming Problem

Suppose a bank has an option to lend to a specific borrower at a fixed spread until the lender defaults.The public information for the borrower is time-invariant, while the private information, St , changes overtime.

The timing of events within the period is as follows. Going into the period, there exists a value of St. By theend of the period, a new value of St+1is realized. At the start of the period, the lender first chooses

whether or not to review the loan, and then chooses whether or not to sell or buy back the loan.7

Then,

the default event is or is not realized, and the probability of default in period tis determined by St. Theprincipal is then refinanced/repaid if default has not occurred, and the principal and the coupon is lost inits entirety if default does occur. Finally, the new value of St+1 is realized at the end of the period. If thelender chooses to review the loan in period t, they know the exact value of St, and carry this knowledge

into the next period. Payments madej periods from now are discounted at j .

Let there beNpossible states, and let the i-th element of the vector S imply the value of Sfor state i with Sincreasing in i. Let stateN be the absorbing state that implies default. And letDt be anNelement rowvector denoting the distribution of the borrower across these states based on the information available tothe lender at the start of the period (prior to doing the review). Let D(i) be a vector such that the ithelement equals one and 0 otherwise. This vector represents the distribution of the borrower acrosspossible states when the borrower is known to be in state i. Letbe the transition matrix governing theprobability of moving from state i to statej in the next period. As the last state is the absorbing state, the

last column of this matrix is the probability of default from each state. For convenience, let PDdenote thelast column of ....

Let CoR be the cost of a review, which is constant over time.

Given the initial distribution, the distribution across the states in period j is given by:

j

t j t+ = D D

LetRkbe a column vector representing a given review strategy. The i-th element of this vector is thenumber of periods one waits to review an exposure that starts out in state i. Let Vkbe the value of havinga relationship with a borrower that is known to be in state i given review strategyRk. Define the i-thelement of the vectorRk+1as:

( )1

1

0

( ) argmax max( (1 ) ( ) ,0) ( ) CoRr

j j r r

k r k

j

R i s s i i

+=

= + +

D PD D V -

(4)

7By assuming that the lender can freely sell/buy back the loan, the only dynamic decision is whether or not to review

the loan. This assumption makes the problem more straightforward to solve, as whether or not the borrower sells orbuys back the loan during the current period does not impact cash flows in the next period.


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Define the i-th element of the vector Vk+1 as:

( )1

1

0

( ) sup max( (1 ) ( ) ,0) ( ) COR

+=

= + +

r

j j r r

k r k

j

V i s s i iD PD D V - (5)

In both equations, the term

1

0

max( (1 ) ( ) ,0)r

j j

j

s s i

=

+ D PD represents the present discounted value

of expected cash flows earned until the next review is conducted. If one chooses to lend in a given period,

the expected cash flows are (1 ) ( )j

s s i + D PD. The max(X,0) operator captures that one would not

lend if this quantity is negative. In both equations, the term ( )( ) CORr r ki D V - reflects the value ofthe loan after conducting the review in rperiods.

If Rk+1= Rk then the strategy is optimal, and the vector Vk+1 is the value of having a relationship with this

borrower for each possible initial state.8

One numerical approach to solve this value function is the

following. First, compute the value function using anR0, in which every element is infinity (i.e., never doinga review). This value function involves computing an infinite sum that can easily approximate byevaluating the first 500 terms. Using this value for V0,one can compute R1 directly. For each initial state,compute the value of the relationship if it was reviewed in periods (1, 2, 3.) and find the maximum. Thisprocess determines bothR1and V1. One then iterates until convergence to determine the optimal strategyand the corresponding value function.

Note that the value of the relationship is bounded below by zero and above by the present discountedvalue of s being paid for certain, every period, in perpetuity. Also, note that the value function does notexplicitly include cash flows associated with investing the principal in another instrument (e.g., a risk freesecurity), should the lender choose to not extend credit. Consequently, s should be interpreted as thespread in excess of the risk free rate.

8This result is an application of Theorem 9.2 of Stokey and Lucas (1989).


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4 A Competitive Dynamic General EquilibriumUp to this point, the framework shows how the lender can make an optimal decision, taking both theprobability of default by a borrower and the spread on a loan as given. In a competitive generalequilibrium setting with full information, there is a close linkage between the probability of default and thespread on the loan. In this section, we show how to embed the framework within a competitive dynamicgeneral equilibrium, in which prices (i.e., spreads) reflect public information but not private information.This section shows that such equilibrium is possible using a very minimal model.

In this setting, a borrower (who we term the entrepreneur) and a lender (who we term the venturecapitalist) agree to terms before the lender reviews the loan so that the coupon paid on the loan does notdepend on the full information probability of default. We assume the entrepreneur and the venturecapitalist possess symmetric information, in that, both can access the reviews outcome after it isconducted.

At the start of each period, there is a new cohort of entrepreneurs. Each entrepreneur of type i can eitherproduce one unit of consumption good for certain or an uncertain amount of consumption goods with afixed and known distribution (provided they do not default). A venture capitalist provides an office to theentrepreneur. The entrepreneur requires an office, which he cannot produce himself. An office costs xunits of consumption to produce, and the office of one entrepreneur cannot be reassigned to anotherentrepreneur. Further, the office provides no direct utility to the entrepreneur or the venture capitalist. Theventure capitalist can choose to offer the entrepreneur a retainer of one unit of consumption prior toconducting the review. If successful, the entrepreneur refinances the principal with the venture capitalistand pays the coupon. The venture capitalist will choose not to refinance the loan if the odds of failureexceed the coupon on the loan. This decision is reversible; the venture capitalist can hire back theentrepreneur during the next period if he chooses to do so. Once the entrepreneur defaults, they eitherleave the economy or produce one unit of consumption good for certain.

We set up this equilibrium so that the interpretation of the coupon is, in a sense, comparable to a spreadin excess of the risk free rate. We assume that all consumption occurs at the end of the period. Bymaking this assumption, the lender is made whole on the use of the retainer when the borrower repaysthe principal at the end of the period. The coupon is only used to pay for the office, the cost of the review,and the credit risk. After being assigned, the opportunity cost of the entrepreneur using the office is zero it is a sunk cost.

The spread on the loan is contracted based on public information. The venture capitalist can choose to doa review and then hedgethe relationship. Alternatively, the venture capitalist can choose not to conductthe review and then choose whether or not to hedgethe relationship. The review costs CoR units ofconsumption goods. Further, the outcome of the review is known only by the entrepreneur and theventure capitalist other venture capitalists do not have access to the outcome. If the venture capitalistchooses to hedge the relationship in that period at the contracted spread, the entrepreneur can make acounter offer that the venture capitalist must accept or reject. If the relationship is hedgedduring thecurrent period, the entrepreneur utilizes the risk free technology to produce one unit of consumptiongoods.

There is a mass of possible entrepreneurs and an arbitrary number of venture capitalists. The quality ofan entrepreneur is measured by the full information probability of default and the distribution of the outputthat he could produce. In each period, there is a new cohort of possible entrepreneurs. The distribution of

the quality of each new cohort of entrepreneurs in each period is constant over time. The probability ofdefault by any entrepreneur is bounded below by a positive number.

Let the representative agents utility be given by:

1

( )

=

t tt

u c


14/29

where u is defined for positive c and exhibits diminishing marginal utility. Specifically, u is continuous, withcontinuous first and second derivatives. Further, the first derivative is positive for all positive c, and thesecond derivative is negative for all positive c.

Proposition

There exists a competitive equilibrium in which:

1. All agents act in an optimal fashion, given the actions of others.2. Output produced is constant over time.3. The coupon charged to the entrepreneur sets the value of the relationship for the venture

capitalist, equal to the cost of producing an office in foregone consumption goods.4. A constant proportion of output is consumed and a constant proportion of output is used to

produce offices for new entrepreneurs, and the remainder is used to pay for the credit reviews.5. The expected value of a relationship with a new entrepreneur is equal tox consumption units.6. The coupon charged to the entrepreneur is either the contractually agreed upon spread or

renegotiated to the odds of default.7. The relative prices across time are determined by the discount factor, i.e.,pt/p=

(t-).

Proof

If output is constant, and a constant proportion of output is consumed, then the marginal utility isconstant, and the relative prices are determined by the discount factor. Therefore, the value of extendingcredit to an entrepreneur who has a specific value for Uis given by the solution to Equations (4) and (5).This value is continuous and increasing in the coupon charged. If this value is less than x units ofconsumption goods, then credit will not be extended. If this value is greater than x units of consumption,then another venture capitalist would bid down the spread. Therefore, in equilibrium the coupon chargedby the entrepreneur ensures that the value of the relationship to the venture capitalist is equal to x units ofconsumption goods.

As all cohorts of entrepreneurs are the same, the demand for new offices in each period would be thesame. All entrepreneurs have a positive probability of default in each period. Therefore, there will be asteady state distribution of entrepreneurs, which implies that output will be a constant. Further, thenumber of entrepreneurs being reviewed in each period is a constant as well.

In this equilibrium, a venture capitalist may choose to hedge a loan revealed as a high credit risk throughthe review process. This outcome could occur even if the expected value of the output of theentrepreneur is large. Therefore, there could potentially be a mutually advantageous renegotiation of theloan terms after conducting the review. The entrepreneur could choose to counter offer so that s=PD/ (1-PD), where the PD is determined using all information available to the entrepreneur and the venturecapitalist. The venture capitalist would be indifferent between accepting and rejecting the counter offer,and we assume that he accepts it. The entrepreneur would only make such a counter offer if the expectedoutput of his production exceeds 1+s. As the venture capitalist does not derive any incremental benefitfrom the counter offer, the value of the loan to the venture capitalist is still given by Equations (4) and (5).

This completes our outline of a proof.

Discussion

A reasonable question regarding this equilibrium: why doesnt another venture capitalist offer funding at alower spread to borrowers suspected of having a good review? One explanation: the fear of a biddingwar. If outside venture capitalist A offers a 3% spread to entrepreneur B, given credit at 4%, then venturecapitalist B could counter offer with better terms, or not, and venture capitalist A would either be a victim


15/29

of winners curse, or not. As long as venture capitalist A knows more about the entrepreneur than venturecapitalist B, then venture capitalist B can only lose a bidding war via Bertrand competition.

In this setting, the entrepreneur maintains the equity portion of the capital structure, and the venturecapitalist maintains the senior capital structure position (after paying the entrepreneur the retainer). Thisspecification helps keep the problem tractable. If the venture capitalist were to share in a portion of theupside potential of the entrepreneurs output, then the decision to review the credit depends upon the

venture capitalists beliefs regarding the profit function of the entrepreneur, and the venture capitalistsdecision to review the loan becomes more involved. Often, the venture capitalist receives equity in theentrepreneurs firm, but not always. One should note that in this equilibrium, the amount of resourcesdevoted to a review is not likely to be optimal from the social planners perspective, as the value of thereview depends upon the upside potential of the entrepreneur, which the venture capitalist does not factorinto his decision to review the loan. This finding is reminiscent of Grossman and Stiglitz (1980), whichargues that competitive markets are intrinsically inefficient when information is costly.

Note, we assume the venture capitalist acts as if he is risk neutral, but not necessarily the entrepreneur,and we do not explicitly specify how the entrepreneurs and venture capitalists income is distributed tothe representative agent. The venture capitalist acts to guarantee that the entrepreneur does at least aswell as his next best alternative, which is producing one unit of consumption goods for certain. Onejustification for the venture capital acting in a risk neutral fashion could be diversification: the venture

capitalists risk can be diversified either by one venture capitalist funding many entrepreneurs or therepresentative agents owning the income of many different venture capitalists.

5 An ExampleIn this section, we characterize the solution to the dynamic programming problem for three specificscenarios. These examples allow us to demonstrate how the equilibrium spreads vary with credit risk indifferent scenarios. They also allow us to show how both credit risk and model calibration influenceoptimal review policy.

In the base case scenario, we calibrate the model to match the CAP curves and CDT used in Figure 2.We construct the empirical reference points for this figure with respect to a one-year horizon. In thissection, we assume that the decision to review a loan is made once a quarter, and we choose parametervalues consistent with Figure 2 over a four-quarter horizon (i.e., a one-year horizon). We assume that thepublic information variable (U) is fixed. We use a discrete approximation for a stochastic process to derivea transition matrix for the time-varying private information (St). We solve for the equilibrium spread thatsets the value of the relationship equal to a fixed percentage of the value of the principal (which would be

x in the context of Section 4).

Suppose the full information PD is given by:

( ), 0 0, ; , , ( )FI t t t PD U S N U S = + + (6)

where Uhas a standard normal distribution independent of St. We allow St to vary with time, and St is whatthe credit analyst learns when conducting the review at time t. Uis the public information. The privateinformation St, follows a Ornstein-Uhlenbeck process that has a zero mean and an unconditional variance

of (2)1:

1 = +t t tdS S dW

where dWis a standard Wiener process.


16/29

This process is the continuous time analogue to an AR1 process, sometimes termed an elastic randomwalk, in that Stis always being pulled back to the long term mean of 0 (c.f., Vasicek, 1977). Given St-x, theexpectation and variance of Stare given by:

( ) exp( ) = t t x t xE S S S x and ( )1 exp( 2 )

2

=t t x

xVar S S

Following the equilibrium defined in Section 4, the spread the lender pays the borrower on the loan, s, is

fixed, there is a fixed cost of review, CoR, and a fixed discount factor. In order to value a loan, one

needs the parameters {0, , ,}. Further, in order to use the setup of Equations (4) and (5) to solve forthe value of loan, one must be able to represent the evolution of the probability of default as a transitionmatrix.

One can represent a discreet approximation to an Ornstein-Uhlenbeck process as follows. Subdivide atime interval Tinto n equal partitions. Let the time increment, n, be defined as T/n. Define a step size as

= n nh , and a grid { }..., 2 , ,0, , 2 ,...n n nh h h h= nX . If the probability of moving fromxi toxi+1 is

equal to 0.52

ix

h

+ and the probability of moving from xi toxi-1is given by 0.5

2i

xh

then this process

converges to an Ornstein-Uhlenbeck process with a mean of 0 and an unconditional variance of 1/(2) asn goes to infinity (c.f., Stokey, 2009, pages 27-28) . In order to specify this process as a transition matrix,one first needs to define a state space with a finite number of elements. One approach is to choose thesmallest S1 and the largest SMsuch that the state space contains plus or minus three standard deviationsof the unconditional variance of the process. Then the transition matrix,T , can be defined as

1 0.52

iiiS

h

= +T for { }2,...,i M , 1 0.5

2iii

Sh

+

= T for { }1,..., 1i M , 11 0.5

2i

Sh

= +T ,

0.52

iMMS

h

= T , and zero elsewhere. We use this transition matrix to build the transition matrix G.

The timing of events is important. We assume that T =1, and we interpret this time interval to represent aquarter of a year. We assume that the probability of default within the quarter is determined by the stateat the beginning of the quarter. Further, the probability of transition from one state to another state

conditional upon not defaulting is represented byn

T . We now construct the matrix G. Define the columnvector PD=[PDFI(S1), PDFI(S2), , PDFI(SL)]. Let 1 and 0 be row vectors of ones and zeros withMelements, respectively. The transition vector G can then be constructed as:

( )

1

ndiag =

T 1- PD PDG

0

The last row represents the probability of transitioning from default to another state, and the probability oftransitioning from a default state to a default state is 1. In order to ensure that the other rows sum to 1, wemust multiple each element of the matrix Tn by one minus the probability of default, so that Gij now has

the interpretation of the likelihood of transition from state i to statej during the quarter and not defaulting.


17/29

Calibration

Table 2 Calibration Scenarios

Parameters Base Case

Strong

Private

Information

Weak

Public

Information

0 -3.27 -3.83 -2.98 0.85 1.34 0.52

0.27 0.60 0.62

0.16 0.16 0.16

Public Information AR 77.8% 77.8% 35.0%

Full Information AR 85.5% 93.0% 85.5%

CDT 2% 2% 2%

This frameworks calibration requires values for the parameters: {0, , ,}. To a first order, theparameter0 determines the average default rate for the quarter, the parameter is the sensitivity of theprobability of default to public information, and the parameter is the sensitivity of the probability of

default to private information. The parameter determines the information decay rate. In Appendix B, we

argue that the parameter values for the base case in Table 2 are consistent with a CDTof 2% and apublic information accuracy ratio of 77.8%, a full information accuracy ratio of 88.5%, and the assumption

that private information decays over time with a half-life of four quarters.9

Table 2 presents two other scenarios as well. In the second scenario, the private information is moreinformative, in that the full information AR is 93%, but the public information is the same. In the thirdscenario, the public information is weak; the public information AR is 35%, while the full information AR iscomparable to the base scenario.

We solve for the equilibrium spread for different scenarios under the assumption that the initial value ofthe relationship is 2% of the principal, the cost of review is 0.5% of the principal, and the quarterlydiscount factor is 0.98. We also look at the case where the cost of review is relatively small (0.10%) andrelatively large (2.0%). We present five different values of U: {-2,-1,0,1,2}.

Table 3 and Figure 3 below show the equilibrium spread for the three different scenarios. The first columnis the annualized probability of default. In all cases, the equilibrium spread increases in risk. Further, thespread is positive for the safest names. In order for the value of the relationship to be 2%, a positive

spread is required.10

The riskiest names have a spread less than the public information probability ofdefault. For the riskiest names, the strategy is to conduct the review, but only to extend credit if theoutcome of the review is favorable. Consequently, it can be profitable to review credits that look badunder public information, given the market spread. The relationship between PD and the spread is thesteepest for the base case scenario. Under the base case scenario, the incremental predictive power ofthe private information is the least; therefore, the equilibrium coupon better reflects the public information.

9By a half-life of approximately four quarters, we mean that

4( ) 0.5

t t tE S S S

+ .

10We compute equilibrium spreads as low as 20bps per annum or 5bps per quarter. The value of 5bps per quarter in

perpetuity discounted at 2% is about 2.5%, close to the minimum required value of the relationship.


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Table 3 Equilibrium Spread (bps per annum) for a 0.50% COR

Base CaseBase CaseBase CaseBase Case Strong Private InformationStrong Private InformationStrong Private InformationStrong Private Information Weak Public InformationWeak Public InformationWeak Public InformationWeak Public Information

U

PublicInformation

PD Spread (bps)

Public

Information

PD Spread (bps)

PublicInformation

PD Spread (bps)

-2 0.00% 23 0.00% 20 0.58% 51

-1 0.03% 41 0.01% 29 2.09% 71

0 0.51% 101 0.84% 57 6.13% 103

1 5.08% 288 13.21% 158 14.85% 158

2 26.58% 798 59.77% 489 29.83% 251

Figure 3 Spread and Public Information PD for different scenarios with a 0.5% review cost

Table 4 and Figure 4 present the equilibrium spread for the base case scenario under different reviewcosts. When the cost of review is low, the spreads are relatively flat, and the largest spread fallssignificantly below the full information PD. In this case, the lender is likely to exercise their option toreview and choose not to extend credit, unless the review comes back favorably. As a result, spreads arelow for even borrowers whom appear very risky, but only the safe borrowers are extended credit. Whenthe cost of review is large, the spread profile becomes stepper, as reviews will be conducted less often,making it necessary for the spread to better reflect the public information PD. The spread on the riskiestloan remains less than the full information PD, which reflects that, for the high risk loans, the lender willperiodically exercise their option to review and only extend credit with a favorable review outcome.

0.01 0.10 1.00 10.00 100.000

100

200

300

400

500

600

700

800

900

1000

Public Information Annualized PD (%)

Spreadinbpsperannum

Spread and Public Information PD

Base CaseStrong Private

Weak Public


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Table 4 Equilibrium Coupon and Public Information PD for Different Review Costs

U

Public

Information

PD 0.10% 0.50% 2%

-2 0.001% 19 23 36

-1 0.028% 29 41 69

0 0.512% 65 101 176

1 5.082% 166 288 509

2 26.582% 591 798 1475

Figure 4 Spread and Public Information PD for Different Review Costs

0.01 0.10 1.00 10.00 100.000

200

400

600

800

1000

1200

1400

1600

Public Information Annualized PD (%)

Spreadinbpsperannum

Spread and Public Information PD

Base Case & COR is 10bps




20/29

Figure 5 Value of the Relationship and Time to Next Review for the Base Case with a 0.50% Cost of Review

The top panel of Figure 5 presents the equilibrium spreads for different values of U(public information) as

vertical lines. The middle panel presents time to review against the full information PD for different valuesof U. The bottom panel represents the loan value expressed as a percentage of the principal. For allvalues of U, the value of the relationship obeys a smooth pasting condition where, as the full informationPDbecomes large, the value becomes close to zero, and the derivative of the value of the loan withrespect to the full information PDapproaches zero. The time to the next review has a U-shapedrelationship with the outcome of the last review. The riskiest loans are potentially the most valuable, butare also the most sensitive to the review outcome. The safest loans are relatively insensitive to the fullinformation PD, because the credit riskcomponent of the loan is small relative to the coupons information

0.001 0.01 0.10 1.00 10.00

0

10

20

30

40Full Information PD and Value of Relationship

ValueofRelatio

nship(%)

Full Information Annualized PD (%)

0.001 0.01 0.10 1.00 10.000

5

10

15

20Full Information PD and Time to Next Review

TimetoNextRe

view(QTRS)

0.001 0.01 0.10 1.00 10.000

5

10

15

20Equilibrium Annualized Spread

U of -2

U of -1

U of 0

U of 1

U of 2


21/29

cost. Consequently, the riskiest loans are reviewed the most often. A relationship with a high riskborrower (according to public information) can be rather valuable, if the review comes back favorable.

In contrast to the single period case, the loans one is indifferent to holding are not the loans reviewedmost often. The intuition follows. Conditional upon extending credit, the expected cash flows within the

current period are a concave function of St (Figure 2 shows that the downside losses are bigger than theupside returns). As time passes and a review is not conducted, Stbecomes more uncertain, and theexpected value of cash flows declines, one may choose not to extend credit, but the value of resolving theuncertainty in Stincreases, making the review more valuable. If the expected cash flows are very close tozero but positive during the current period, the optimal strategy is to extend credit today, not extend creditduring the next several periods, and wait until enough time passes so that Stmay improve enough to payfor the cost of the review. If the review outcome is favorable, one extends credit again.

6 Value of a Review in a Dynamic SettingAn important question remains: how much economic value does pursuing an optimal review strategycreate? In the dynamic setting, one can express the value of pursuing an optimal review strategy as the

difference between the value of the relationship under the optimal review strategy and the value of therelationship if the lender pursues some other policy. In this section, we use reviewing each borrowerevery four quarters as the alternative strategy. One reason for choosing this strategy as a reference pointis that often regulatory guidelines require an annual review. Also, the work flow of the review process isoften tied to the receipt of the new, full-year financial statements.

We solve for the value of the relationship under a fixed annual review policy by setting every element of Rto 4 in Equations (4) and (5), and we place a lower bound of 0 on the value of the relationship. We theniterate to solve for the value of the relationship for every value of S. The lower bound of 0 has theinterpretation that, if the cost of reviewing the loan once a year exceeds the potential upside associatedwith extending such a borrower credit, then the lender will walk away from the relationship, and they willno longer be required to review the borrower every year. This places a smooth pasting condition on thevalue of the relationship under a fixed review cycle: As Sbecomes larger, the value of the relationship

approaches zero, and the change in the value of the relationship with respect to Sapproaches zero.We examine the value of the review under the base case with a 50bps cost of review. We focus on a firmwith a Uof 1, which corresponds to a PD based on public information of 5.1% and a spread of 288bps

(Table 3). We focus on a Uof 1, because it is often optimal to review more often than every four quartersfor this level of risk (Figure 5). In this case, the incremental value of the review is likely elevated.The top panel of Figure 6 compares the optimal review policy for different full information PDs with theevery four quarters rule. The middle panel shows the value of the relationship under the optimal policyand the value under the every four quarters rule. The bottom panel shows the difference between the two,the value of pursuing an optimal review policy. For a full information PD of 0.73%, the value of a mid-yearreview equals 1.32% of the exposure size. The incremental value of an optimal review policy is humpedshaped. For the safer firms, time to the next review is four quarters, so the optimal policy correspondswith the fixed review cycle, and the difference in value is smaller. For the riskiest firms, both value

functions asymptote to zero, which results in small differences as well.


22/29

Figure 6 Value of a Review in a Dynamic Setting

Table 5 provides the numbers behind Figure 6. It also provides the Normalized S, which is Sdivided byits unconditional standard deviation. The excess annualized spread on the loan is provided (theannualized spread less the full information annualized PD). Note that the excess spread is negative for apositive S, which implies that one-half of the borrowers in this risk group will not receive credit aftercompletion of their review. Also, note that the value of the review peaks for a normalizedSof -1.38. This

implies is that the incremental value of an optimal review is greatest for the borrowers whom are relativelyrisky according to public information (a 5.3% PD according to public information), but much less riskyaccording to full information (a 0.73% full information PD). In this case, the optimal strategy is a mid-yearreview, which generates 1.32% of value relative to the size of the exposure.

0.1 10

5

10

15

Different Review Strategies

TimetoNext

Review(QTRS)

Optimal Review Strategy

Review Every 4 Quarters

0.1 10

5

10

15

20Value of Different Review Strategies (%)

ValueoftheRelationship

0.1 10

0.5

1

1.5Difference in Value of Review Strategies (%)

Full Information Annualized PD (%)

DifferenceinValue


23/29

Table 5 Value of a Review in a Dynamic Setting

NormalizedS

Annualized

Full

Information

PD (%) Time To Next Review Value of Relationship (%)

Annualized

Excess

Spread (Bps)

Optimal Constrained Optimal Constrained Difference

-2.07 0.32 3 4 12.51 11.52 0.99 255.2

-1.90 0.40 3 4 11.21 10.11 1.10 247.7

-1.72 0.49 2 4 9.80 8.60 1.20 238.5

-1.55 0.60 2 4 8.33 7.03 1.29 227.6

----1.381.381.381.38 0.730.730.730.73 2222 4444 6.826.826.826.82 5.515.515.515.51 1.321.321.321.32 214.5214.5214.5214.5

-1.21 0.88 2 4 5.34 4.12 1.22 198.8

-1.03 1.07 2 4 3.95 2.92 1.03 180.1

-0.86 1.28 4 4 2.79 1.89 0.90 158.0

-0.69 1.54 9 4 1.96 1.07 0.88 131.9

-0.52 1.84 14 4 1.39 0.47 0.92 101.2

-0.34 2.19 20 4 0.96 0.06 0.90 65.2

-0.17 2.60 26 4 0.61 0.00 0.61 23.1

0.00 3.08 33 4 0.38 0.00 0.38 -25.8

0.17 3.63 40 4 0.25 0.00 0.25 -82.5

In this section, we demonstrated the value of the review by contrasting the value of the relationship underan optimal review cycle versus a fixed review cycle. The value of the review is largest when the optimal

strategy is to review the loan more often than each year. It is also largest when the public informationimplies a highly elevated risk level relative to the full information PD.

7 ConclusionThis paper presents a basic framework for computing the value of a credit review. Our framework isconsistent with some basic characteristics of PDs. For example, the distribution of PD sin a portfoliotypically skews to the right as default probabilities are bounded between 0 and 1, and the largest PDs in apopulation are typically many times larger than the CDT.

Our core assumption is that the outcome of the credit review is a full information PDwith betterdiscriminatory power than a partial information PD, and the full information PDleads to a decision onwhether or not to sell the loan. In this framework, the relationship between the value of a review and thecharacteristics of the exposure is multidimensional: it depends on the partial information PD, the spread,the hedging costs on the loan, the central default tendency, and the accuracy ratios of the full and partialinformation PDs. Simple policy rules, such as review each loan each year or review each loan whose PDmoves outside a certain threshold range, applied without regarding to the spread on the loan, are notoptimal within this framework.


24/29

For reasonable baseline parameters, the value of a review can be as high as 200bps. In general, thevalue of a review is lower when the decision to hedge (or not) is a foregone conclusion. The highestvalues of a review occur near the point where the lender is indifferent between hedging and not hedging,given partial information. Holding the excess spread constant, the value of a review increases as thepartial information probability of default increases.

Our framework provides a theoretical interpretation for practices found in the review process. For certain

credit classes, a financial institution may take action without a formal review, based on the outcome of aquantitative scoring model (e.g., a pre-approved credit card), whereas, for other asset classes, an actionis never taken solely on the basis of a quantitative scoring model (e.g., the renewal of a line of credit to alarge corporation). As the cost of review per unit of exposure is likely higher in consumer credit thancorporate credit, different practices could be justified within this framework. Another common practice is towatch listthe riskiest loans and review them more often. In this framework, one should review loan Amore often than loan B if it is riskier and both are in the portfolio (spread exceeds expected loss underpartial information) and pay the same spread. Therefore watch listingthe riskiest credits can be justifiedunder this framework.

The realized value of a review for a good credit (one for which the excess spread is positive) isasymmetric. In most cases, the ex postvalue of a review equals zero. Most reviews determine that the fullinformation PD is less than the partial information PD, and no action is taken on the loan. In some cases,

however, the full information PD turns out to be worse than the partial information PD, enough worse thatthe institution acts on the loan. In these relatively rare cases, the savings to the lender are substantial. Asa result, one can underestimate a reviews value if looking at the ex postrealized values on a relativelysmall sample. This scenario speaks to the difficulty of properly assessing the value that analysts bring tomanaging a credit portfolio.

We show how the framework can be extended to incorporate a time dimension and show how it can beembedded in a competitive equilibrium, in which prices reflect public information. In this setting, wecompute the partial information PD utilizing public information and the full information PD utilizingboth public information and private information. The relevance of the information acquired during areview process decays with time. The optimal time until the next review depends on the spread on theloan, the cost of the review, the public information on the borrower, and the outcome of the most recentreview. It also depends on the characteristics of the asset class: how informative is the partial informationPD? The full information PD? How fast does information acquired during the review process change? For

reasonable parameter values, it may be optimal to review the loan during the next quarter and, in othercases, it may be optimal to rely solely upon public information. In the dynamic setting, the spread on aloan is more closely tied to the public information PD when the cost of review is higher or the incrementalinformation content of private information is smaller.

We assess the value of an optimal review policy by comparing it to a fixed review cycle of once every fourquarters. We show that the incremental value of the review is largest for the loans that should bereviewed more often than every four quarters. In fact, the value of a mid-year review can be as much as1.3% of the exposure amount.


25/29

Appendix ATable of Symbols

Symbol Meaning

PD Probability of Default

X Row Vector of both partial and full information default drivers

0 and The intercept and a column vector of coefficients for the full information PD

Yand Z The partial information and full information default drivers

and The coefficients for the partial and full information default drivers

CDT The average default rate of the population

and 0 The coefficient for the partial information driver and the intercept under partialinformation

CAP and AR Cumulative Accuracy Profile and Accuracy Ratio

s and h The spread on the loan and the transaction cost of selling

Dynamic Version of the Framework

St The time-varying private information

U The public information that is fixed for each borrower

The discount factor

N The number of possible values for S(includingthe absorbing state)

Dt A row vector representing the distribution of a borrower across different statesfrom the perspective of the lender

D(i) A row vector of zeros except for a one at the ithelement. Indicates that aborrowers current value of Sis Si

and G For a given borrower, represents the transition matrix governing the evolutionof S (the private information). G is the calibrated version of this matrix employed

in Section 5.

CoR and s The cost of review and spread on the loan

R and V Column vectors representing the number of periods until the next review and thevalue of the option to lend for each possible value of S

and The parameters governing the sensitivity of the PD to the public information, Uand private information, St

The parameter governing the persistence of the private information


26/29

Appendix BThe Distribution of PDs and CAP CurvesIn this Appendix, we show how to derive the distribution of the partial information PDsand the fullinformation PDs. We also derive an expression for the Cumulative Accuracy Profile (CAP curves) and thecorresponding Accuracy Ratio (AR). These constructs allow us to calibrate the parameters using

empirical reference points. We show how the framework enables comparative statics that change thepartial information AR, while holding the full information ARconstant and vice versa.

In validating a PD model, key workhorses are CAP curves and their corresponding ARs.11

Theseconstructs assess a credit risk measures ability to rank order firms in terms of separating out the badborrowers from the good borrowers. CAP curves are typically defined with respect to a specific sample. Inthis section, we work with the populationanalogues to these measures. The cumulative accuracy profile(CAP plot) of a credit risk measure can be defined as:

1

0 0 0( ) ( ) ( ) ( )

q q

CAP q D x dx D x dx D x dx CDT = =

Where D(q) is defined as follows. Consider a small interval, (q, q+), which is a subset of [0,1]. Theportion of the exposures in the population riskier (according to some credit risk measure) than theexposures within the interval is q. The portion of the exposures in the population safer than theexposures within the interval is q+. D(q) is defined to provide the rate at which exposures in this interval

default: ( )( )0( ) lim ( )q

qD q D x dx

+

= . The Accuracy Ratio is defined as

1

0( ) 0.5

0.5(1 )

CAP q dqAR

CDT

=

.

The AR is the ratio of the area between the CAP Curve and the 45 degree line, divided by the areabetween the CAP Curve of a Perfect Model and the 45 degree line. A perfect model has an AR of 1, anda random model has an AR of 0.

In our context, for the full information PD,1( )N PD has a normal distribution with a mean of 0 and a

variance of ' . Therefore, the quantile function of the full information PD is given by:

( )1 0( ) ( ) 'FIF q N N q = +

This function takes a percentile as an input and produces the PD associated with that percentile. Forexample, if q=0.975, then 97.5% of the population has a PD less then FFI(0.975). It is convenient to definePDFI(q) = 1-FFI(q), so that PDFI(0.025) has the interpretation that 2.5% of the population has PD of greaterthan PDFI(0.025). By symmetry of the normal distribution, we see that:

( ) ( )1 10 0( ) 1 ( ) ' ( ) 'FIPD q N N q N N q = + = +

Note that in this context, PDFI(q)is theD(q)used above in defining a CAP curve. Therefore, the CAPcurve for the full information PD is given by:

( )1 00

( ) ( ) 'x

FICAP x N N q dq CDT = +

11Some analysts use Receiver Operating Characteristic curves and Gini Coefficients. Engelmann et. al., (2003)

provide a description of the concept of a Cumulative Accuracy Profile, an Accuracy Ratio, and their relationship toReceiver Operator Characteristic curves. A note by Andrew Curtiss (2009) proves the equivalence of an AccuracyRatio and a Gini Coefficient.


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We evaluate this function using numerical integration. Computing the corresponding AR involvesevaluating a double numerical integral.

This result is useful for conducting comparative statics on the framework. For example, suppose we fix

2 2 + at 2B and 0 at1 2( ) 1N CDT B

+ . For any given , we can solve for 2 2B = such thatthe full information CAP curve, the full information AR, and the CDTare the same (provided is less than

B), but the partial information AR changes.The left-hand panel of Figure 7 shows how the CAP curve ofthe partial information PD can change, while the full information CAP plot remains constant. As the partialinformation PD becomes more informative, the partial information CAP curve approaches the fullinformation CAP curve.

Figure 7 Changing CAP Plots with Model Parameters

By identical arguments, the quantile function of the partial information PD is given by:

( )1

0 ( ) ( ) 'PIF q N N q

= +

and the partial information CAP plot is given by:

( )1 00

( ) ( ) 'q

PICAP q N N x dx CDT = +

Therefore, if we change and in such a way that holds ' , 0 and CDT constant, then we change

the discriminatory power of the full information PD without changing the discriminatory power of the partialinformation PD.

For example, fix0

at 02 2

0 01

B

+ +and

20

1

G

=

+. For any positive , if

2

20

1

1

G

+=

+and

1 2 2

0( ) 1N CDT = + + then the partial information CAP plot, the partial information AR, and the

CDT remain constant, but the full information AR increases in . The right-hand panel of Figure 7 showshow the full information PD CAP curve changes while the partial information PD CAP plot remainsconstant. The full information PD CAP curve approaches the CAP curve of a perfect model as the fullinformation PD becomes more informative.

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100


Percen

tofDefaults

Changing Paritial Information

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100


Percen

tofDefaults

Changing Full Information


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Appendix C Calibration

In Section 3, we state that specific values of 0{ , , , } imply specific values for the CDT, partialinformation AR, and full information AR at one-year horizon. In this section, we describe the calibration

procedure used to make this assertion. We make three assumptions. First, we assume a specificdistribution of the private information (S) of entrants, given their public information (U). Second, in orderfor a steady state to exist, it is necessary that all firms have a positive probability of default. Third, wecompute the average CDT, partial information AR, and full information AR, with respect to the wholepopulation, without regard to whether or not they are actually extended credit.

We describe a procedure that takes as inputs 0{ , , , } and produces {CDT, ARPI,ARFI} at the one-year horizon. In order to do so, we need to compute a steady state distribution of firms in terms of UandS. We then must compute the partial and full information probability of default at a one-year horizon.Finally, we must compute the average default rate, cumulative accuracy profiles, and the corresponding

accuracy ratios. Once we define the function, we can compute the parameters 0{ , , , } that areconsistent with the desired {CDT, ARPI,ARFI} by minimizing a loss function.

First, we approximate a standard normal distribution using a binomial distribution. Specifically, Ucan takeon values of { 0 , 1 ,..., 24 } U where the probability that Utakes on the value Uiis

251 1

25 2 2

i ii

. We set { }, to ensure that Uwill have a mean of zero and a variance of 1.

For each value of U, we solve for the steady state distribution of Sas follows. First, we assume that thereare entrants in each period. The new entrants are distributed according to the unconditional distribution ofan Ornstein-Uhlenbeck process using a discrete approximation. Using an n of 20, this is easily computed

as the first row of T500

, where Tis the transition matrix defined in Section 5. LetEbe this row vector with a0 added to the end to represent the absorbing state (i.e., firms in the default state do not enter).

For each value of U, we utilize1

iU

i

=EG to compute the steady state distribution. Note that each term inthe sum is a row vector that represents distribution in the current period of the cohort that entered iperiods ago. Finally, we convert it into a distribution by setting the population in the absorbing set to 0 andthen normalizing so that the row vector sums to one. Let SSUbe the steady state distribution associatedwith a specific value of U.

We now have the state steady distributions for both Sand U. Given {U,S}, the one-year full informationprobability of default is given by the last element of DGU

4, whereDis a row vector of zeros, with a 1indicating the current values of Sand GUis the transition matrix associated with the specific value of U.For a given U, the public information probability of default is given by the last element SSUGU

4. The

average default rate of the population can be computed as either the weighted sum of public informationprobabilities for each value of Uor the weighted sum of the full information one-year default probabilities

for each value pair of {U,S}. For the public information case, the weights are given by the binomialdistribution. For the full information case, the weights are the product of the probability of Ui and theprobability of Si given U, where the later is given by the steady state distribution of Sgiven U. Thecomputation of the partial information AR and the full information AR are then computed using thediscrete analogues to the definitions provided in Appendix C. By treatingas fixed, we can solve for the

values of 0{ , , } that produce specific values of {CDT, ARPI, ARFI} by minimizing a loss function.


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a theory of monitoring credit risk

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