dependence modeling and credit risk -...

Dependence Modeling and Credit Risk

Paola Mosconi

Banca IMI

Bocconi University, 20/04/2015 and 27/04/2015

Paola Mosconi Lecture 6 1 / 112

Disclaimer

The opinion expressed here are solely those of the author and do not represent inany way those of her employers


Main References

Vasicek Model

Vasicek, O. (2002) The Distribution of Loan Portfolio Value, Risk, December

Granularity Adjustment

Pykhtin, M. and Dev, A. (2002) Credit risk in asset securitisations: an analyticalmodel, Risk, May

Multi-Factor Merton Model

Pykhtin, M. (2004), Multi-Factor Adjustment, Risk, March


Outline

1 IntroductionCredit RiskPortfolio Models

2 Vasicek Portfolio Loss ModelIntroductionLimiting Loss DistributionProperties of the Loss Distribution

3 Granularity Adjustment4 Multi-Factor Merton Model

IntroductionVaR ExpansionComparable One-Factor ModelMulti-Factor AdjustmentApplications

5 Capital Allocation6 Conclusions7 Appendix8 Selected References


Introduction

Outline







Introduction Credit Risk

Credit Risk

Credit risk is the risk due to uncertainty in a counterparty’s ability to meet itsfinancial obligations (default or downgrade of the obligor).

Measurement of credit risk is based on three fundamental parameters:

Probability of Default (PD)What is the likelihood that the counterparty will default on its obligation either overthe life of the obligation or over some specified horizon, such as a year?

Loss Given Default (LGD = 1− Rec):In the event of a default, what fraction of the exposure may be recovered throughbankruptcy proceedings or some other form of settlement?

Exposure at Default (EAD)In the event of a default, how large will the outstanding obligation be when thedefault occurs?



Sources of Risk

Default risk

Migration risk

Spread riskRisk of changes in the credit spreads of the borrower, for example due to marketconditions (should not result in a change in the credit rating)

Recovery riskRisk that the actual recovery rate is lower than previously estimated

Sovereign riskRisk that the counterparty will not pay due to events of political or legislativenature



Expected Loss (EL)

The Expected Loss is the average loss in value over a specified time horizon.

For a single exposure:

EL = PD · LGD · EAD

The Expected Loss of a portfolio, beingan additive measure, is given by the sumof individual losses.

Figure: Portfolio Expected Loss



Unexpected Loss (UL)

The Unexpected Loss represents the variability of the loss distribution aroundits mean value EL.

Portfolio diversification:

does not impact the EL:

EL portfolio = sum of expected losses of the individual positions

but typically reduces the UL:

UL portfolio < sum of UL of the individual positions.

The Unexpected Loss is used to define the Economic Capital.



Quantile Function

Given a random variable X with continuous and strictly monotonic probability densityfunction f (X ), a quantile function Qp assigns to each probability p attained by f the valuex for which P(X ≤ x) = p.

The quantile function

Qp = infx∈R

{x : P(X ≤ x) ≥ p}

returns the minimum value of x from amongst all those values whose cumulativedistribution function (cdf) value exceeds p.

If the probability distribution is discrete rather than continuous then there may begaps between values in the domain of its cdf

if the cdf is only weakly monotonic there may be flat spots in its range



Inverse Distribution Function

Given a random variable X with continuous and strictly monotonic probability densityfunction f (X ), if the cumulative distribution function F = P(X ≤ x) is strictly increasingand continuous then, F−1(y) with y ∈ [0, 1] is the unique real number x such thatF (x) = y . In such a case, this defines the inverse distribution function or quantile function.

However, the distribution does not, in general, have an inverse. One may define,for y ∈ [0, 1], the generalized inverse distribution function:

F−1(y) = inf{x ∈ R |F (x) ≥ y}This coincides with the quantile function.

Example 1 : The median is F−1(0.5).

Example 2 : Put τ = F−1(0.95). τ is the 95% percentile



VaR and Expected Shortfall (ES) I

Value at Risk

The Value at Risk of the portfolio loss L at confidence level q is given by the followingquantile function:

VaRq = infℓ∈R

{ℓ : P(L > ℓ) ≤ 1− q}

= infℓ∈R

{ℓ : P(L ≤ ℓ) ≥ q}

Expected Shortfall

The Expected Shortfall of the portfolio loss L at confidence level q is given by:

ESq(L) = E[L | L ≥ VaRq(L)]

Typically, for credit risk, the confidence level is q = 99.9% and the time horizon is T = 1y.



VaR and Expected Shortfall (ES) II

VaR: the best of worst (1− q)%losses

ES: the average of worst (1− q)%losses

Figure: VaR vs ES



Economic Capital (EC)

Banks are expected to hold reserves against expected credit losses which are con-sidered a cost of doing business.

The Economic Capital is given by theUnexpected Loss, defined as:

EC = VaRq − EL

The EC is not an additive measure: atportfolio level, the joint probability distri-bution of losses must be considered (cor-relation is crucial).

Figure: Economic Capital



Diversification of Credit Risk

Risk diversification in a credit portfolio is determined by two factors:

granularity of the portfolio: i.e. the number of exposures inside the portfolio andthe size of single exposures (idiosyncratic or specific risk)

systematic (sector) risk, which is described by the correlation structure of obligorsinside the portfolio

Figure: Risk diversification vs portfolio concentration


Introduction Portfolio Models

Portfolio Models

The risk in a portfolio depends not only on the risk in each element of the portfolio,but also on the dependence between these sources of risk.

Most of the portfolio models of credit risk used in the banking industry are based onthe conditional independence framework. In these models, defaults of individualborrowers depend on a set of common systematic risk factors describing thestate of the economy.

Merton-type models, such as PortfolioManager and CreditMetrics, have becomevery popular. However, implementation of these models requires time-consumingMonte Carlo simulations, which significantly limits their attractiveness.



Asymptotic Single Risk Factor (ASRF) Model

Among the one-factor Merton-type models, the so called Asymptotic Single RiskFactor (ASRF) model has played a central role, also for its regulatory applicationsin the Basel Capital Accord Framework.

ASRF (Vasicek, 1991)

The model allows to derive analytical expressions for VaR and ES, by relyingon a limiting portfolio loss distribution, based on the following assumptions:

1 default-mode (Merton-type) model

2 a unique systematic risk factor (single factor model)

3 an infinitely granular portfolio i.e. characterized by a large number of smallsize loans

4 dependence structure among different obligors described by the gaussiancopula



ASRF Extensions

Violations of the hypothesis underlying the ASRF model give rise to correctionswhich are explicitly taken into account by the BCBS (2006) under the genericname of concentration risk. They can be classified in the following way:

1 Name concentration: “imperfect diversification of idiosyncratic risk”, i.e.imperfect granularity in the exposures

2 Sector concentration: “imperfect diversification across systematiccomponents of risk”

3 Contagion: “exposures to independent obligors that exhibit defaultdependencies, which exceed what one should expect on the basis of theirsector affiliations”



Summary

In the following, we will introduce:

1 the original work by Vasicek on the ASRF model

2 hints to the granularity adjustment, via single factor models

3 multi-factor extension of the ASRF, which naturally takes into accountboth name concentration and sector concentration


Vasicek Portfolio Loss Model

Outline







Vasicek Portfolio Loss Model Introduction

Loan Portfolio Value

Using a conditional independence framework, Vasicek (1987, 1991 and 2002)derives a useful limiting form for the portfolio loss distribution with a singlesystematic factor.

The probability distribution of portfolio losses has a number of important applica-tions:

determining the capital needed to support a loan portfolio

regulatory reporting

measuring portfolio risk

calculation of value-at-risk

portfolio optimization

structuring and pricing debt portfolio derivatives such as collateralized debtobligations (CDOs)


Vasicek Portfolio Loss Model Introduction

Capital Requirement

The amount of capital needed to support a portfolio of debt securities depends onthe probability distribution of the portfolio loss.

Consider a portfolio of loans, each of which is subject to default resulting in aloss to the lender. Suppose the portfolio is financed partly by equity capital andpartly by borrowed funds. The credit quality of the lender’s notes will depend onthe probability that the loss on the portfolio exceeds the equity capital. To achievea certain credit rating of its notes (say Aa on a rating agency scale), the lenderneeds to keep the probability of default on the notes at the level corresponding tothat rating (about 0.001 for the Aa quality).

It means that the equity capital allocated to the portfolio must be equal to thepercentile of the distribution of the portfolio loss that corresponds to thedesired probability.


Vasicek Portfolio Loss Model Limiting Loss Distribution

Limiting Loss Distribution

1 Default Specification

2 Homogeneous Portfolio Assumption

3 Single Factor Approach

4 Conditional Probability of Default

5 Vasicek Result (1991)

6 Inhomogeneous Portfolio



Default Specification I

Following Merton’s approach (1974), Vasicek assumes that a loan defaults if thevalue of the borrower’s assets at the loan maturity T falls below the contractualvalue B of its obligations payable.

Asset value process

Let Ai be the value of the i-th borrower’s assets, described by the process:

dAi = µi Ai dt + σi Ai dxi

The asset value at T can be obtained by integration:

logAi (T ) = logAi + µiT − 1

2σ2i T + σi

√T Xi (1)

where Xi ∼ N (0, 1) is a standard normal variable.



Default Specification II

Probability of default

The probability of default of the i-th loan is given by:

pi = P[Ai (T ) < Bi ] = P[Xi < ζi ] = N(ζi )

where N(.) is the cumulative normal distribution function and

ζi =logBi − logAi − µiT + 1

2σ2i T

σi

√T

represents the default threshold.



Homogeneous Portfolio Assumption I

Consider a portfolio consisting of n loans characterized by:

equal dollar amount

equal probability of default p

flat correlation coefficient ρ between the asset values of any two companies

the same term T

Portfolio Percentage Gross Loss

Let Li be the gross loss (before recoveries) on the i-th loan, so that Li = 1 if thei-th borrower defaults and Li = 0 otherwise. Let L be the portfolio percentagegross loss:

L =1

n

n∑

i=1

Li



Homogeneous Portfolio Assumption II

If the events of default on the loans in the portfolio were independent of eachother, the portfolio loss distribution would converge, by the central limit theorem,to a normal distribution as the portfolio size increases.

Because the defaults are not independent, the conditions of the central limittheorem are not satisfied and L is not asymptotically normal.

Goal

However, the distribution of the portfolio loss does converge to a limiting form.In the following, we will derive its expression.



Single Factor Approach

The variables {Xi}i=1,...,n in eq. (1) are jointly standard normal with equal pair-wise cor-relations ρ, and can be expressed as:

Xi =√ρY +

√

1− ρ ξi

where Y and ξ1, ξ2, . . . , ξn are mutually independent standard normal variables.

The variable Y can be interpreted as a portfolio common (systematic) factor, such asan economic index, over the interval (0,T ). Then:

the term√ρY is the company’s exposure to the common factor

the term√1− ρ ξi represents the company’s specific risk



Conditional Probability of Default

The probability of the portfolio loss is given by the expectation, over the common factorY , of the conditional probability given Y . This is equivalent to:

assuming various scenarios for the economy

determining the probability of a given portfolio loss under each scenario

weighting each scenario by its likelihood

Conditional Probability of Default

When the common factor is fixed, the conditional probability of loss on any one loan is:

p(Y ) = P(Li = 1|Y ) = P(Xi < ζi |Y ) = N

[

N−1(p)−√ρY√

1− ρ

]

The quantity p(Y ) provides the loan default probability under the given scenario. Theunconditional default probability p is the average of the conditional probabilities overthe scenarios.



Vasicek Result (1991) I

Conditional on the value of Y , the variables Li are independent equally distributedvariables with a finite variance.

Conditional Portfolio Loss

The portfolio loss conditional on Y converges, by the law of large numbers, to itsexpectation p(Y ) as n → ∞:

L(Y ) → p(y) for n → ∞



Vasicek Result (1991) II

We derive the expression of the limiting portfolio loss distribution following Vasicek’sderivation (1991).

Since p(Y ) is a strictly decreasing function of Y i.e.

p(Y ) ≤ x ⇐⇒ Y ≥ p−1(x)

it follows that:

P(L ≤ x) = P(p(Y ) ≤ x) = P(Y ≥ p−1(x))

= 1−P(Y ≤ p−1(x)) = 1− N(p−1(x)) = N(−p

−1(x))

where N(−x) = 1− N(x) =∫ −x

−∞f (y) dy and on substitution, the the cumulative distri-

bution function of loan losses on a very large portfolio is in the limit:

P(L ≤ x) = N

[√1− ρN−1(x)− N−1(p)√

ρ

]



Vasicek Result (1991) III

The portfolio loss distribution is highly skewed and leptokurtic.

Figure: Source: Vasicek Risk (2002)



Inhomogeneous Portfolio

The convergence of the portfolio loss distribution to the limiting form above actually holdseven for portfolios with unequal weights. Let the portfolio weights be w1,w2, . . . ,wn with∑

wi = 1. The portfolio loss:

L =n∑

i=1

wi Li

conditional on Y converges to its expectation p(Y ) whenever (and this is a necessary andsufficient condition):

n∑

i=1

w2i → 0

In other words, if the portfolio contains a sufficiently large number of loans without itbeing dominated by a few loans much larger than the rest, the limiting distributionprovides a good approximation for the portfolio loss.


Vasicek Portfolio Loss Model Properties of the Loss Distribution

Properties of the Loss Distribution

1 Cumulative distribution function

2 Probability density function

3 Limits

4 Moments

5 Inverse distribution function (or quantile function)

6 Comparison with Monte Carlo Simulation

7 Economic Capital

8 Regulatory Capital



Cumulative Distribution Function

The portfolio loss is described by two-parameter distribution with the parameters0 < p, ρ < 1.

The cumulative distribution function is continuous and concentrated on theinterval 0 ≤ x ≤ 1:

F (x ; p, ρ) := N

[√1− ρN−1(x)− N−1(p)√

ρ

]

The distribution possesses the following symmetry property:

F (x ; p, ρ) = 1− F (1− x ; 1− p, ρ)



Probability Density Function I

The probability density function of the portfolio loss is given by:

f (x ; p, ρ) =

√

1− ρ

ρexp

{

− 1

2 ρ

[

√

1− ρN−1(x)− N−1(p)]2

+1

2

[

N−1(x)]2}

which is:

unimodal with the mode at

Lmode = N

[√1− ρ

1− 2ρN

−1(p)

]

when ρ < 12

monotone when ρ = 12

U-shaped when ρ > 12



Probability Density Function II

Figure: Probability density function for ρ = 0.2 (left), ρ = 0.5 (center) and ρ = 0.8(right) and p = 0.3.



Limit ρ → 0

When ρ → 0, the loss distribution function converges to a one-point distribution concen-trated at L = p.

Figure: Probability density function (left) and cumulative distribution function (right) forp = 0.3



Limit ρ → 1

When ρ → 1, the loss distribution function converges to a zero-one distribution withprobabilities 1− p and p, respectively.

Figure: Probability density function (left) and cumulative distribution function (right) forp = 0.3



Limit p → 0

When p → 0 the distribution becomes concentrated at L = 0.

Figure: Probability density function (left) and cumulative distribution function (right) forρ = 0.3



Limit p → 1

When p → 1, the distribution becomes concentrated at L = 1.

Figure: Probability density function (left) and cumulative distribution function (right) forρ = 0.3



Moments

The mean of the distribution is

E(L) = p

The variance is:

s2 = var(L) = E{

[L− E(L)]2}

= E(L2)− [E(L)]2

= N2(N−1(p),N−1(p), ρ)− p2

where N2 is the bivariate cumulative normal distribution function.



Inverse Distribution Function/Percentile Function I

The inverse of the distribution, i.e. the α-percentile value of L is given by:

Lα = F (α; 1 − p; 1− ρ)


The table lists the values of the α-percentile Lα expressed as the number of standarddeviations from the mean, for several values of the parameters. The α-percentiles of thestandard normal distribution are shown for comparison.



Inverse Distribution Function/Percentile Function II

These values manifest the extreme non-normality of the loss distribution.

Example

Suppose a lender holds a large portfolio of loans to firms whose pairwise assetcorrelation is ρ = 0.4 and whose probability of default is p = 0.01. The portfolioexpected loss is E(L) = 0.01 and the standard deviation is s = 0.0277. If thelender wishes to hold the probability of default on his notes at 1− α = 0.001, hewill need enough capital to cover 11.0 times the portfolio standard deviation. Ifthe loss distribution were normal, 3.1 times the standard deviation would suffice.



Simulation I

Computer simulations show that the Vasicek distribution appears to provide areasonably good fit to the tail of the loss distribution for more general portfolios.

We compare the results of Monte Carlo simulations of an actual bank portfolio.The portfolio consisted of:

479 loans in amounts ranging from 0.0002% to 8.7%, with δ =∑n

i=1 w2i = 0.039

the maturities ranged from six months to six years

the default probabilities from 0.0002 to 0.064

the loss-given default averaged 0.54

the asset returns were generated with 14 common factors.



Simulation II

The plot shows the simulated cumulative distribution function of the loss in one year (dots)and the fitted limiting distribution function (solid line).




Economic Capital

The asymptotic capital formula is given by:

EC = VaRq(L)− EL

= F (q; 1 − p; 1− ρ)− p

= N

[√ρN−1(q) − N−1(1− p)√

1− ρ

]

− p

= N

[√ρN−1(q) + N−1(p)√

1− ρ

]

− p

where N−1(1− x) = −N−1(x). The formula has been obtained under theassumption that all the idiosyncratic risk is completely diversified away.



Regulatory Capital

Under the Basel 2 IRB Approach, at portfolio level, the credit capital charge K isgiven by:

K = 8%

n∑

i=1

RWi EADi

where, the individual risk weight RWi is:

RWi = 1.06 · LGDi ·[

N

[

N−1(pi)−√ρi N

−1(0.1%)√1− ρi

]

− pi

]

· MF(Mi , pi)

where:

MF is a maturity factor adjustment, depending on the effective maturity Mi ofloan i

pi is individual probabilities of default of loan i

q = 99.9%

ρi is a regulatory factor loading which depends on pi and the type of the loan(corporate, SMEs, residential mortgage etc...)



Outline









The asymptotic capital formula implied by the Vasicek distribution (1991):

EC = N

[√ρN−1(q) + N−1(p)√

1− ρ

]

− p

is strictly valid only for a portfolio such that the weight of its largest exposure is in-finitesimally small.

All real-world portfolios violate this assumption and, therefore, one might question the rel-evance of the asymptotic formula. Indeed, since any finite-size portfolio carries some undi-versified idiosyncratic risk, the asymptotic formula must underestimate the “true”capital.

The difference between the “true”capital and the asymptotic capital is known asgranularity adjustment.



Granularity Adjustment in Literature

Various extensions for non-homogeneous portfolios have been proposed in literature.

The granularity adjustment technique was introduced by Gordy (2003)

Wilde (2001) and Martin and Wilde (2002) have derived a general closed-formexpression for the granularity adjustment for portfolio VaR

More specific expressions for a one-factor default-mode Merton-type model havebeen derived by Pykhtin and Dev (2002)

Emmer and Tasche (2003) have developed an analytical formulation for calculatingVaR contributions from individual exposures

Gordy (2004) has derived a granularity adjustment for ES



Outline








Outline

1 Introduction

2 VaR Expansion

3 Comparable One-Factor Model

4 Multi-Factor Adjustment

5 Applications


Multi-Factor Merton Model Introduction

Introduction

The Multi-Factor Merton model has been introduced by Pykhtin (2004) in order toaddress the issue of both

name concentration and

sector concentration

in a portfolio of credit loans.

The model allows to derive analytical expressions for VaR and ES of theportfolio loss and turns out to be very convenient for capital allocation purposes.



Multi-Factor Set-Up: Portfolio

We consider a multi-factor default-mode Merton model.

The portfolio consists of:

loans associated to M distinct borrowers. Each borrower has exactly one loancharacterized by exposure EADi , whose weight in the portfolio is given by

wi =EADi

∑M

i=1 EADi

each obligor is assigned a probability of default pi and a loss given default LGDi .The loss given default is described by means of a stochastic variable Q (with meanµi and standard deviation σi ), whose independence of other sources of randomnessis assumed



Multi-Factor Set-Up: Time Horizon and Threshold

Time horizon

Borrower i will default within a chosen time horizon (typically, one year) withprobability pi . Default happens when a continuous variable Xi describing the finan-cial well-being of borrower i at the horizon falls below a threshold.

Default threshold

We assume that variables {Xi} (which may be interpreted as the standardized assetreturns) have standard normal distribution. The default threshold for borrower i isgiven by N−1(pi), where N

−1(.) is the inverse of the cumulative normal distributionfunction.



Multi-Factor Set-Up: Systematic Risk Factors

We assume that asset returns depend linearly on N normally distributed system-atic risk factors with a full-rank correlation matrix. Systematic factors representindustry, geography, global economy or any other relevant indexes that mayaffect borrowers’ defaults in a systematic way.

Borrower i ’s standardized asset return is driven by a certain borrower-specificcombination of these systematic factors Yi (known as a composite factor):

Xi = riYi +√

1− r2i ξi (2)

where ξi ∼ N (0, 1) is the idiosyncratic shock. Factor loading ri measures borroweri ’s sensitivity to the systematic risk.



Multi-Factor Set-Up: Independent Systematic Risk Factors

Since it is more convenient to work with independent factors, we assume that N originalcorrelated systematic factors are decomposed into N independent standard normal system-atic factors Zk ∼ N (0, 1) (k = 1, . . . ,N). The relation between {Zk} and the compositefactor is given by

Yi =N∑

k=1

αik Zk

where αik must satisfy the relation

N∑

k=1

α2ik = 1

to ensure that Yi has unit variance. Asset correlation between distinct borrowers i and j

is given by

ρij = ri rj

N∑

k=1

αikαjk



Multi-Factor Set-Up: Portfolio Loss

If borrower i defaults, the amount of loss is determined by its loss-given default stochasticvariable Qi . No specific assumptions about the probability distribution of Qi is made, exceptfor its independence of all the other stochastic variables.

The portfolio loss rate L is given by the weighted average of individual loss rates Li

L =

M∑

i=1

wi Li =

M∑

i=1

wi Qi 1{Xi≤N−1(pi )} (3)

where 1{.} is the indicator function. This equation describes the distribution ofthe portfolio losses at the time horizon.



Limiting Loss Distribution I

A traditional approach to estimating quantiles of the portfolio loss distribution in themulti-factor framework is Monte Carlo simulation.

Limiting Loss Distribution

In the case of a large enough, fine-grained, portfolio, most of the idiosyncratic risk isdiversified away and portfolio losses are driven primarily by the systematic factors.

In this case, the portfolio loss can be replaced by the limiting loss distribution of aninfinitely fine-grained portfolio, given by the expected loss conditional on thesystematic risk factors (see Gordy (2003) for details):

L∞ = E [L|{Zk}] =

M∑

i=1

wi µi N

[

N−1(pi )− ri∑N

k=1 αikZk√

1− r2i

]

(4)



Limiting Loss Distribution II

Although equation (4) is much simpler than equation (3), it still requires Monte Carlosimulation of the systematic factors {Zk} when the number of factors is greater than one.

Moreover, it is not clear how large the portfolio needs to be for equation (4) to becomeaccurate.

Goal

To design an analytical method for calculating tail quantiles and tailexpectations of the portfolio loss L given by equation (3).

The method has been devised by Pykhtin (2004) and is based on a Taylorexpansion of VaR, introduced by Gourieroux, Laurent and Scaillet (2000) andperfected by Martin and Wilde (2002).


Multi-Factor Merton Model VaR Expansion

VaR Expansion: Assumptions

The Value at Risk of the portfolio loss L at a confidence level q is given by thecorresponding quantile, which is denoted by tq(L).

The calculation of tq(L) goes through the following steps:

1 assume that we have constructed a random variable L such that its quantile at levelq, tq(L), can be calculated analytically and is close enough to tq(L)

2 express the portfolio loss L in terms of the new variable L

L ≡ L+ U ,

where U = L− L plays the role of a perturbation

3 make explicit the dependence of L on the scale of the perturbation and write

Lε ≡ L+ εU

with the understanding that the original definition of L is recovered for ε = 1



VaR Expansion: Result

Main result (Martin and Wilde, 2002)

For high enough confidence level q, the quantile tq(Lε) is obtained through a seriesexpansion in powers of ε around tq(L). Up to the second order, tq(L) ≡ tq(Lε=1) reads:

tq(L) ≈ tq(L) +dtq(Lε)

dε

∣

∣

∣

∣

∣

ε=0

+1

2

d2tq(Lε)

dε2

∣

∣

∣

∣

∣

ε=0

(5)

where:dtq(Lε)

dε

∣

∣

∣

∣

∣

ε=0

= E[U|L = tq(L)]

d2tq(Lε)

dε2

∣

∣

∣

∣

∣

ε=0

= − 1

fL(l)

d

dl

(

fL(l) var[U|L = l ])

∣

∣

∣

∣

∣

l=tq(L)

fL(.) being the probability density function of L and var[U|L = l ] the variance of U

conditional on L = l .



VaR Expansion: L

The key point consists in choosing the appropriate L.

L is defined as the as the limiting loss distribution in the one-factor Mertonframework Merton (1974) i.e.

L := l(Y ) =

M∑

i=1

wi µi pi (Y )

where, it is implicitly assumed that:

Xi = aiY +√

1− a2i ζi ζi ∼ N (0, 1)

and pi(y) is the probability of default of borrower i , conditional on Y = y :

pi(y) = N

[

N−1(pi)− aiy√

1− a2i

]



VaR Expansion: Quantile of L

Quantile of L

Since L is a deterministic monotonically decreasing function of Y , the quantile ofL at level q can be calculated analytically (see Castagna, Mercurio and Mosconi,2009):

tq(L) = l(N−1(1− q))

RemarkLet us note that the derivatives of tq(L) in the VaR expansion are given by expressions

conditional on L = tq(L). Since L is a deterministic monotonically decreasing function of

Y this conditioning is equivalent to conditioning on Y = N−1(1− q).



VaR Expansion: First Order Term

The first order derivative of VaR is expressed as the expectation of U = L− L,conditional on tq(L) = l :

dtq(Lε)

dε

∣

∣

∣

∣

∣

ε=0

= E[U|Y = N−1(1− q)]



VaR Expansion: Second Order Term

The second order derivative can be rewritten

d2tq(Lε)

dε2

∣

∣

∣

∣

∣

ε=0

= − 1

n(y)

d

dy

(

n(y)ν(y)

l ′(y)

)

∣

∣

∣

∣

∣

y=N−1(1−q)

where ν(y) ≡ var(U|Y = y) is the conditional variance of U, l ′(.) is the first derivative ofl(.) and n(.) is the standard normal density.

By carrying out the derivative with respect to y explicitly and using the fact thatn′(y) = −y n(y), the second order term becomes:

d2tq(Lε)

dε2

∣

∣

∣

∣

∣

ε=0

= − 1

l ′(y)

[

ν′(y)− ν(y)

(

l ′′(y)

l ′(y)+ y

)]

∣

∣

∣

∣

∣

y=N−1(1−q)


Multi-Factor Merton Model Comparable One-Factor Model

Comparable One-Factor Model: Y vs {Zk}

Goal

To relate random variable L to the portfolio loss L, we need to relate the effectivesystematic factor Y to the original systematic factors {Zk}.

We assume a linear relation given by:

Y =

N∑

k=1

bkZk bk ≥ 0

where the coefficients must satisfy∑N

k=1 b2k = 1 to preserve unit variance of Y .

In order to complete the specification of L, we need to specify the set of M effective factorloadings {ai} and N coefficients {bk}.



Key Requirement

In order to determine the coefficients {ai} and {bk}, we enforce the requirement that Lequals the expected loss conditional on Y :

L = E[L|Y ]

for any portfolio composition.

Besides being intuitively appealing, this requirement guarantees that the first-order termin the Taylor series vanishes for any confidence level q, i.e.

dtq(Lε)

dε

∣

∣

∣

∣

∣

ε=0

= E[U|Y = N−1(1− q)] = E[L − L|Y ]

= E[L|Y ]− E[L|Y ] = L− L = 0



Composite Factors Yi

To calculate E[L|Y ], we represent the composite risk factor for borrower i , Yi , as:

Yi = ρiY +√

1− ρ2i ηi

where ηi ∼ N (0, 1) is a standard normal random variable independent of Y (but in contrastto the true one-factor case, variables {ηi} are inter-dependent), and ρi is the correlationbetween Yi and Y given by:

ρi := corr(Yi ,Y ) =N∑

k=1

αikbk

Using this notations, asset return can be written as

Xi = ri ρi Y +√

1− (ri ρi )2 ζi (6)

where ζi ∼ N (0, 1) is a standard normal random variable independent of Y .



Effective Factor Loadings {ai}The conditional expectation of L results in:

E[L|Y ] =

M∑

i=1

wi µi N

[

N−1(pi)− ri ρi Y√

1− (ri ρi )2

]

This equation must be compared with the limiting loss distribution:

L =M∑

i=1

wi µi N

[

N−1(pi)− aiY√

1− a2i

]

Effective factor loadings ai

L equals E[L|Y ] for any portfolio composition if and only if the effective factor loadingsare defined as:

ai = ri ρi = ri

N∑

k=1

αikbk (7)



Conditional Asset Correlation

Even though the second term in the asset return equation (6) is independent of Y , it givesrise to a non-zero conditional asset correlation between two distinct borrowers i and j .

This becomes clear if we rewrite the asset return equation as:

Xi = aiY +N∑

k=1

(ri αik − ai bk)Zk +√

1− r2i ξi

where the second term is independent of Y .

Conditional Asset Correlation

This term is responsible for the conditional asset correlation, which turns out to be:

ρYij =

ri rj∑N

k=1 αikαjk − aiaj√

(1− a2i )(1− a2j )

Although ρYij has the meaning of the conditional asset correlation only for distinctborrowers i and j , we extend it in order to include the case j = i .



Choice of the Coefficients {bk} I

Given equation (7) which defines the factor loadings ai , the choice of the coefficients {bk}is not unique.

The set {bk} specifies the zeroth-order term tq(L) in the Taylor expansion and manyalternative specifications of {bk} are plausible, provided that the associated tq(L) is closeenough to the unknown target function value tq(L).

Goal

Ideally, we aim at finding a set {bk} that minimizes the difference between the twoquantiles. Intuitively, one would expect the optimal single effective risk factor Y to haveas much correlation as possible with the composite risk factors {Yi}, i.e.

max{bk}

(

M∑

i=1

ci corr(Y ,Yi )

)

such that

N∑

k=1

b2k = 1



Choice of the Coefficients {bk} II

Considering that

corr(Y ,Yi ) =

N∑

k=1

αik bk

the solution to this maximization problem is given by:

bk =

M∑

i=1

ciλαik (8)

where positive constant λ is the Lagrange multiplier chosen so that {bk} satisfythe constraint.

Unfortunately, it is not clear how to choose the coefficients {ci}.



Choice of the Coefficients {ci}

Some intuition about the possible form of the coefficients {ci} can be developed by mini-mization of the conditional variance ν(y). Under an additional assumption that all ri aresmall, this minimization problem has a closed-form solution given by eq. (8) with

ci = wi µi n[N−1(pi)]

Even though the assumption of small ri is often unrealistic and the performance of thissolution is sub-optimal, it may serve as a starting point in a search of optimal {ci}.

Coefficients {ci}One of the best-performing choices is represented by:

ci = wi µi N

[

N−1(pi) + ri N−1(q)

√

1− r2i

]


Multi-Factor Merton Model Multi-Factor Adjustment

Multi-Factor Adjustment: Total VaR

Recalling the VaR expansion formula (5) and considering that first-order contributionscancel out, the total VaR, approximated up to second order, is given by:

tq(L) ≈ tq(L) + ∆tq (9)

where

∆tq = − 1

2 l ′(y)

[

ν′(y)− ν(y)

(

l ′′(y)

l ′(y)+ y

)]

∣

∣

∣

∣

∣

y=N−1(1−q)

where l(y) =∑M

i=1 wi µi pi(y) and ν(y) = var[L|Y = y ] is the conditional variance of Lon Y = y .

If, conditional on Y individual loss contributions were independent, the term ∆tq wouldbe equivalent to Wilde’s granularity adjustment (Wilde, 2001). However, due to non-zeroconditional asset correlation between distinct borrowers, the correction term contains alsosystematic effects.



Multi-Factor Adjustment: Conditional Variance

Conditional variance decomposition

Conditional on {Zk}, asset returns are independent, and we can decompose theconditional variance ν(y) in terms of its systematic and idiosyncratic components:

ν(y) = ν∞(y) + νGA(y)

whereν∞(y) = var[E(L|{Zk})|Y = y ]

νGA(y) = E[var(L|{Zk})|Y = y ]

The same decomposition1 holds for the quantile correction (multi-factor adjustment):

∆tq = ∆t∞q +∆t

GAq

1See the Appendix for the decomposition based on the Law of Total Variance.Paola Mosconi Lecture 6 77 / 112


Sector Concentration Adjustment

The conditional variance of the limiting portfolio loss L∞ = E[L|{Zk}] on Y = y quantifiesthe difference between the multi-factor and one-factor limiting loss distributions (wewill denote this term as ν∞(y)) and is given by:

ν∞(y) = var[E(L|{Zk})|Y = y ]

=M∑

i=1

M∑

j=1

wiwj µiµj

[

N2(N−1[pi (y)],N

−1[pj (y)], ρYij )− pi (y)pj(y)

] (10)

where N2(·, ·, ·) is the bivariate normal cumulative distribution function.




The granularity adjustment νGA(y) describes the effect of the finite number of loans inthe portfolio. It vanishes in the limit M → ∞, provided that

M∑

i=1

w2i → 0 while

M∑

i=1

wi = 1

νGA(y) = E[ var(L|{Zk})|Y = y ]

=M∑

i=1

w2i

(

µ2i

[

pi (y)− N2(N−1[pi (y)],N

−1[pi(y)], ρYii )]

+ σ2i pi(y)

) (11)

where ρYii is obtained by replacing the index j with i in the conditional asset correlation.

In the special case of homogeneous LGDs and default probabilities pi , it becomes propor-tional to the Herfindahl-Hirschman index HHI =

∑M

i=1 w2i (see Gordy, 2003).



Multi-Factor Adjustment: Summary

The effects of concentration risk are encoded into eq.s (9), (10) and (11):

Sector concentration

It affects both the zeroth order term tq(L), in an implicit way and by con-struction, and the second order correction depending on ν∞(y). The latter,obtained in the limit of an infinitely fine-grained portfolio, represents the sys-tematic component of risk which cannot be diversified away

Single name concentration

It is described by the granularity adjustment νGA(y). For a large enoughnumber of obligors M (ideally, M → ∞) and under the condition of a suffi-ciently homogeneous distribution of loans’ exposures (in mathematical terms∑M

i=1 w 2i → 0, while

∑M

i=1 wi = 1) the granularity contribution vanishes


Multi-Factor Merton Model Applications

Applications

Goal

We want to test the performance of the multi-factor adjustment approximation.

In the following, we focus on two test cases:

1 two-factor set-up:

homogeneous case, M = ∞ (systematic part of the multi-factor adjustment)non-homogeneous case, M = ∞ (systematic part of the multi-factoradjustment)non-homogeneous case, finite M

2 multi-factor set-up:

homogeneous case, M = ∞ (systematic part of the multi-factor adjustment)non-homogeneous case



Two Factor Examples: Assumptions

We assume that:

loans are grouped into two buckets A and B, indexed by u.

Bucket u contains Mu identical loans characterized by a single probability of de-fault pu , expected LGD µu , standard deviation of LGD σu, composite factor Yu andcomposite factor loading ru.

The asset correlation inside bucket u is r2u .

Buckets are characterized by weights ωu defined as the ratio of the net principal ofall loans in bucket u to the net principal of all loans in the portfolio. Individual loanweights are related to bucket weights as ωu = wu Mu .

the composite factors are correlated with correlation ρ and the asset correlationbetween the buckets is ρ rA rB .



Two Factor Examples: Homogeneous Case, M = ∞ I

Input

pA = pB = 0.5%, µA = µB = 40%, σA = σB = 20% and rA = rB = 0.5.

Goal

To compare (see Figure 1(a) – homogeneous case):

the approximated t99.9%(L) + ∆t∞99.9% (dashed blue curves) and

the exact 99.9% quantile of L∞, calculated numerically (solid red curves)

Method

The quantile is plotted:

as a function of the correlation ρ between the composite risk factors

at three different bucket weights ωA



Two Factor Examples: Homogeneous Case, M = ∞ II

Results

The method performs very well except for the case of equal bucket weights(ωA = ωB = 0.5) at low ρ.

For all choices of bucket weights, performance of the method improves with ρ.

At any given ρ, performance of the method improves as one moves away from theωA = ωB case.

Conclusions

This behavior is natural because any of the limits ρ = 1, ωA = 0 and ωA = 1 correspondsto the one-factor case where the approximation becomes exact.

As one moves away from one of the exact limits, the error of the approximation isexpected to increase.

The performance of the approximation is the worst when one is as far from the limits aspossible – the case of equal bucket weights and low ρ.



Two Factor Examples: Non-Homogeneous Case, M = ∞ I

Input

Bucket A is characterized by the PD pA = 0.1% and the composite factor loading rA = 0.5,while bucket B has pB = 2% and rB = 0.2. The LGD parameters are left at the samevalues as before.This choice of parameters (assuming one-year horizon) is reasonable if we interpret:

bucket A as the corporate sub-portfolio (lower PD and higher asset correlation)

bucket B as the consumer sub-portfolio (higher PD and lower asset correlation)

Goal

To compare (see Figure 1(b) – non-homogeneous case):

the approximated t99.9%(L) + ∆t∞99.9% (dashed blue curves) and

the exact 99.9% quantile of L∞, calculated numerically (solid red curves)



Two Factor Examples: Non-Homogeneous Case, M = ∞ II

Results

From figure 1(b), one can see that:

the performance of the systematic part of the multi-factor adjustment is excel-lent for all choices of the bucket weights and the risk factor correlation

the method in general performs much better in non-homogeneous casesthan it does in homogeneous ones.



Two Factor Examples: M = ∞

Figure: Exact (solid red curves) vs approximated (dashed blue curves) systematiccontributions to VaR99.9%. (a) Homogeneous case, (b) non-homogeneous case. Source:Pykhtin (2004)



Two Factor Examples: Non-Homogeneous Case, Finite M

Input

Two cases: wA = 0.3 and wA = 0.7, assuming the risk factor correlation ρ = 0.5.

Goal

To test the effects of name concentration on two portfolios (respectively, with wA = 0.3and wA = 0.7) by varying the bucket population.

The 99.9% quantile calculated with the approximated method is compared with the samequantile calculated via a Monte Carlo simulation.

Results

As with Wilde’s one-factor granularity adjustment, performance of the granularity adjust-ment generally improves as the number of loans in the portfolio increases. However,this improvement is not uniform across all bucket weights and population choices.



Two Factor examples: Finite M

Figure: Effects of name concentration in two portfolios. Approximated solution vs MonteCarlo. Source: Pykhtin (2004)



Multi Factor Examples: Assumptions

All Mu loans in bucket u are characterized by the same PD pu, expected LGD µu ,standard deviation of LGD σu , composite systematic risk factor Yu and compositefactor loading ru. Bucket weights ωu are defined as the ratio of the net principal ofall loans in bucket u to the net principal of all loans in the portfolio.

Systematic factors:

N − 1 industry-specific (independent) systematic factors {Zk}k=1,...,N−1

one global systematic factor ZN

composite systematic factors:

Yi = αi ZN +√

1− α2i Zk(i)

where k(i) denotes the industry that borrower i belongs to. The weight of theglobal factor is assumed to be the same for all composite factors: αi = α.

Correlations:

between any pair of composite systematic factors is ρ = α2

asset correlation inside bucket u is r2uasset correlation between buckets u and v is ρ ru rv



Multi Factor Examples: Homogeneous Case, M = ∞ I

Input

We assume that the buckets are identical and populated by a very large number of identicalloans, with pu = 0.5%, µu = 40%, σu = 20% and ru = 0.5.

Goal

To show the accuracy of the approximation as a function of ρ for several values of N.

The accuracy is defined as the ratio of t99.9%(L) + ∆t∞99.9% to the 99.9% quantile of L∞

obtained via Monte Carlo simulation.



Multi Factor Examples: Homogeneous Case, M = ∞ II

Results

The accuracy quickly improves as ρ increases.

This behavior is universal because in the limit of ρ = 1 the model is reduced to theone-factor framework.

At any given ρ, the approximation based on a one-factor model works better asthe number of factors increases.

Conclusions

In the homogeneous case with composite risk factor correlation ρ, the limit N − 1 = M isequivalent to the one-factor set-up with the factor loading ru

√ρ.

When we increase the number of the systematic risk factors, we move towards thisone-factor limit and the quality of the approximation is bound to improve.



Multi Factor Examples: Homogeneous Case, M = ∞ III

Figure: Accuracy of the approximation. Source: Pykhtin (2004)



Multi Factor Examples: Non-Homogeneous Case I

Goal

To compare 99.9% quantiles of the portfolio loss calculated using the multi-factor ad-justment approximation with the ones obtained from a Monte Carlo simulation for thecase of 10 industries at several values of the composite risk factor correlation ρ.

Input

The parameters of the buckets are shown in Table B.

All buckets have equal weights ωu = 0.1, so the net exposure is the same for each bucket.We compare three portfolios (denoted as I, II and III), which only differ by the numberof loans in the buckets.



Multi Factor Examples: Non-Homogeneous Case II

Figure: Input. Source: Pykhtin (2004)



Multi Factor Examples: Non-Homogeneous Case II

Results

The performance of the method for the calculation of the quantiles for theasymptotic loss (L∞ is the same for all three portfolios) is excellent even forvery low levels of ρ

the performance of the approximation for L∞ in non-homogeneous cases istypically much better than it is in homogeneous cases



Multi Factor Examples: Non-Homogeneous Case III

Results

Portfolio I:

the approximated method performs as impressively as it does for the asymptotic lossat all levels of ρ. This is because the largest exposure in the portfolio is rathersmall – only 0.2% of the portfolio exposure.

Portfolio II:

the number of loans in each of the buckets has been decreased uniformly by a factorof five, which has brought the largest exposure to 1% of the portfolio exposure.The method’s performance is still very good at high to medium values of risk factorcorrelation, but is rather disappointing at low ρ.

Portfolio III:

it has the same largest exposure as portfolio II, but much higher dispersion ofthe exposure sizes than either portfolio I or portfolio II. Although the resulting lossquantile is very close to the one for portfolio I, the approximation does not performas well as it does for portfolio I because of the higher largest exposure.



Multi Factor Examples: Non-Homogeneous Case IV

Figure: 11-Factor non-homogeneous set-up. Source: Pykhtin (2004)


Capital Allocation

Outline







Capital Allocation

Introduction

Aside from the total portfolio’s VaR, there is a growing need for information about:

the marginal contribution of the individual portfolio components to the diversifiedportfolio VaR

the proportion of the diversified portfolio VaR that can be attributed to each of theindividual components constituting the portfolio

the incremental effect on VaR of adding a new instrument to the existing portfolio

The total portfolio VaR can be decomposed in partial VaRs that can be attributed tothe individual instruments comprised in the portfolio. These component VaRs have theappealing property that they aggregate linearly into the diversified portfolio VaR.


Capital Allocation

Incremental VaR

Marginal VaR

The marginal VaR is the change in VaR resulting from a marginal change in the relativeposition in instrument i . Hence, the marginal VaR of component i , MVaRi , equals:

MVaRi =∂VaR

∂wi

Incremental VaR

Incremental VaR provides information regarding the sensitivity of portfolio risk tochanges in the position holding sizes in the portfolio.

IVaRi =∂VaR

∂wi

wi


Capital Allocation

Capital Allocation

Capital Allocation

An important property of incremental VaR is subadditivity. That is, the sum of theincremental risks of the positions in a portfolio equals the total risk of the portfolio:

VaR =M∑

i=1

∂VaR

∂wi

wi =M∑

i=1

IVaRi

This property has important applications in the allocation of risk to different units, wherethe goal is to keep the sum of the risks equal to the total risk.

Subadditivity descends from Euler’s homogeneous function theorem2 and the fact thatVaR is a homogeneous function of degree k = 1 in the loans’ weights.

2See the Appendix.Paola Mosconi Lecture 6 102 / 112

Conclusions

Outline







Conclusions

conclusions

We have introduced Portfolio Models based on the conditional independenceframework:

the Vasicek model, based on the Asymptotic Single Risk Factor (ASRF) as-sumptions, provides an analytical description of the limiting loss distribution,but neglects risk concentration by construction

the Multi-Factor Merton model has been introduced by Pykhtin (2004)to take into account both granularity and sector concentration risk. Themodel, being based on an analytical, though approximated, formula for thequantile, turns out to be very useful for capital allocation purposes.


Appendix

Outline







Appendix

Law of Total Variance

The law of total variance says:

var(Y ) = E(var(Y | X )) + var(E(Y | X ))

Then the two components are:

the average of the variance of Y about the prediction based on X, as X varies

the variance of the prediction based on X, as X varies


Appendix

Derivatives of l(y)

In order to calculate ∆tq we need explicit expressions for the derivatives of l(y) and ν(y).They are given by:

l′(y) =

M∑

i=1

wi µi p′i (y)

l′′(y) =

M∑

i=1

wi µi p′′i (y)

p′i (y) = − ai

√

1− a2in

[

N−1(pi)− aiy√

1− a2i

]

p′′i (y) = − a2i

1− a2i

N−1(pi )− aiy√

1− a2in

[

N−1(pi)− aiy√

1− a2i

]


Appendix

Derivatives of ν(y)

ν′∞(y) = 2

M∑

i=1

M∑

j=1

wiwj µiµj p′i (y)

N

N−1[pj(y)]− ρYij N−1[pi (y)]

√

1− (ρYij )2

− pj(y)

ν′GA(y) =

M∑

i=1

w2i p

′i (y)

(

µ2i

[

1− 2N

(

N−1[pi (y)]− ρYii N−1[pi(y)]

√

1− (ρYii )2

)]

+ σ2i

)


Appendix

Euler’s Homogeneous Function Theorem

If f (x1, . . . , xn) is a function with the property

f (λ x1, . . . , λ xn) = λk f (x1, . . . , xn)

it is said to be homogeneous of order k .

Then, according to Euler’s homogeneous function theorem:

n∑

i=1

(

∂f

∂xi

)

xi = k f (x1, . . . , xn)


Selected References

Outline







Selected References

Selected References I

BSCS (2006). Studies on credit risk concentration, Working Paper No. 15

Castagna, A., Mercurio, F. and Mosconi, P. (2009). Analytical credit VaR withstochastic probabilities of default and recoveries. Bloomberg Portfolio ResearchPaper No 2009-05-Frontiers

Emmer, S. and Tasche, D. (2003). Calculating credit risk capital charges with theone-factor model, Working paper

Gordy, M. (2003). A risk-factor model foundation for ratings-based bank capitalrules, Journal of Financial Intermediation, 12, July, pages 199-232

Gordy, M. (2004). Granularity In New Risk Measures for Investment andRegulation, edited by G. Szego, Wiley

Gourieroux, C., Laurent, J.-P. and Scaillet, O. (2000). Sensitivity analysis of valuesat risk, Journal of Empirical Finance, 7, pages 225-245


Selected References

Selected References II

Martin, R. and Wilde, T. (2002). Unsystematic credit risk, Risk, November

Merton, R. (1974). On the pricing of corporate debt: The risk structure of interestrates. J. of Finance 29, 449-470

Pykhtin, M. (2004). Multi-factor adjustment. Risk Magazine, (3):85-90

Vasicek, O. (1987). Probability of loss on a loan portfolio. Working Paper, KMVCorporation

Vasicek, O. (1991). Limiting loan loss probability distribution, KMV Corporation

Wilde, T. (2001) Probing granularity, Risk, August


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