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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
Paola Mosconi Lecture 6 2 / 112
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
Paola Mosconi Lecture 6 3 / 112
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
Paola Mosconi Lecture 6 4 / 112
Introduction
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
Paola Mosconi Lecture 6 5 / 112
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?
Paola Mosconi Lecture 6 6 / 112
Introduction Credit Risk
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
Paola Mosconi Lecture 6 7 / 112
Introduction Credit Risk
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
Paola Mosconi Lecture 6 8 / 112
Introduction Credit Risk
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.
Paola Mosconi Lecture 6 9 / 112
Introduction Credit Risk
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
Paola Mosconi Lecture 6 10 / 112
Introduction Credit Risk
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
Paola Mosconi Lecture 6 11 / 112
Introduction Credit Risk
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.
Paola Mosconi Lecture 6 12 / 112
Introduction Credit Risk
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
Paola Mosconi Lecture 6 13 / 112
Introduction Credit Risk
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
Paola Mosconi Lecture 6 14 / 112
Introduction Credit Risk
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
Paola Mosconi Lecture 6 15 / 112
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.
Paola Mosconi Lecture 6 16 / 112
Introduction Portfolio Models
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
Paola Mosconi Lecture 6 17 / 112
Introduction Portfolio Models
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”
Paola Mosconi Lecture 6 18 / 112
Introduction Portfolio Models
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
Paola Mosconi Lecture 6 19 / 112
Vasicek Portfolio Loss Model
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
Paola Mosconi Lecture 6 20 / 112
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)
Paola Mosconi Lecture 6 21 / 112
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.
Paola Mosconi Lecture 6 22 / 112
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
Paola Mosconi Lecture 6 23 / 112
Vasicek Portfolio Loss Model Limiting Loss Distribution
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.
Paola Mosconi Lecture 6 24 / 112
Vasicek Portfolio Loss Model Limiting Loss Distribution
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.
Paola Mosconi Lecture 6 25 / 112
Vasicek Portfolio Loss Model Limiting Loss Distribution
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
Paola Mosconi Lecture 6 26 / 112
Vasicek Portfolio Loss Model Limiting Loss Distribution
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.
Paola Mosconi Lecture 6 27 / 112
Vasicek Portfolio Loss Model Limiting Loss Distribution
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
Paola Mosconi Lecture 6 28 / 112
Vasicek Portfolio Loss Model Limiting Loss Distribution
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.
Paola Mosconi Lecture 6 29 / 112
Vasicek Portfolio Loss Model Limiting Loss Distribution
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 → ∞
Paola Mosconi Lecture 6 30 / 112
Vasicek Portfolio Loss Model Limiting Loss Distribution
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
]
Paola Mosconi Lecture 6 31 / 112
Vasicek Portfolio Loss Model Limiting Loss Distribution
Vasicek Result (1991) III
The portfolio loss distribution is highly skewed and leptokurtic.
Figure: Source: Vasicek Risk (2002)
Paola Mosconi Lecture 6 32 / 112
Vasicek Portfolio Loss Model Limiting Loss Distribution
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.
Paola Mosconi Lecture 6 33 / 112
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
Paola Mosconi Lecture 6 34 / 112
Vasicek Portfolio Loss Model Properties of the Loss Distribution
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)
Paola Mosconi Lecture 6 35 / 112
Vasicek Portfolio Loss Model Properties of the Loss Distribution
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
Paola Mosconi Lecture 6 36 / 112
Vasicek Portfolio Loss Model Properties of the Loss Distribution
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.
Paola Mosconi Lecture 6 37 / 112
Vasicek Portfolio Loss Model Properties of the Loss Distribution
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
Paola Mosconi Lecture 6 38 / 112
Vasicek Portfolio Loss Model Properties of the Loss Distribution
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
Paola Mosconi Lecture 6 39 / 112
Vasicek Portfolio Loss Model Properties of the Loss Distribution
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
Paola Mosconi Lecture 6 40 / 112
Vasicek Portfolio Loss Model Properties of the Loss Distribution
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
Paola Mosconi Lecture 6 41 / 112
Vasicek Portfolio Loss Model Properties of the Loss Distribution
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.
Paola Mosconi Lecture 6 42 / 112
Vasicek Portfolio Loss Model Properties of the Loss Distribution
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)
Figure: Source: Vasicek Risk (2002)
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.
Paola Mosconi Lecture 6 43 / 112
Vasicek Portfolio Loss Model Properties of the Loss Distribution
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.
Paola Mosconi Lecture 6 44 / 112
Vasicek Portfolio Loss Model Properties of the Loss Distribution
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.
Paola Mosconi Lecture 6 45 / 112
Vasicek Portfolio Loss Model Properties of the Loss Distribution
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).
Figure: Source: Vasicek Risk (2002)
Paola Mosconi Lecture 6 46 / 112
Vasicek Portfolio Loss Model Properties of the Loss Distribution
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.
Paola Mosconi Lecture 6 47 / 112
Vasicek Portfolio Loss Model Properties of the Loss Distribution
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...)
Paola Mosconi Lecture 6 48 / 112
Granularity Adjustment
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
Paola Mosconi Lecture 6 49 / 112
Granularity Adjustment
Granularity Adjustment
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.
Paola Mosconi Lecture 6 50 / 112
Granularity 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
Paola Mosconi Lecture 6 51 / 112
Multi-Factor Merton Model
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
Paola Mosconi Lecture 6 52 / 112
Multi-Factor Merton Model
Outline
1 Introduction
2 VaR Expansion
3 Comparable One-Factor Model
4 Multi-Factor Adjustment
5 Applications
Paola Mosconi Lecture 6 53 / 112
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.
Paola Mosconi Lecture 6 54 / 112
Multi-Factor Merton Model Introduction
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
Paola Mosconi Lecture 6 55 / 112
Multi-Factor Merton Model Introduction
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.
Paola Mosconi Lecture 6 56 / 112
Multi-Factor Merton Model Introduction
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.
Paola Mosconi Lecture 6 57 / 112
Multi-Factor Merton Model Introduction
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
Paola Mosconi Lecture 6 58 / 112
Multi-Factor Merton Model Introduction
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.
Paola Mosconi Lecture 6 59 / 112
Multi-Factor Merton Model Introduction
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)
Paola Mosconi Lecture 6 60 / 112
Multi-Factor Merton Model Introduction
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).
Paola Mosconi Lecture 6 61 / 112
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
Paola Mosconi Lecture 6 62 / 112
Multi-Factor Merton Model VaR Expansion
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 .
Paola Mosconi Lecture 6 63 / 112
Multi-Factor Merton Model VaR Expansion
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
]
Paola Mosconi Lecture 6 64 / 112
Multi-Factor Merton Model VaR Expansion
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).
Paola Mosconi Lecture 6 65 / 112
Multi-Factor Merton Model VaR Expansion
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)]
Paola Mosconi Lecture 6 66 / 112
Multi-Factor Merton Model VaR Expansion
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)
Paola Mosconi Lecture 6 67 / 112
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}.
Paola Mosconi Lecture 6 68 / 112
Multi-Factor Merton Model Comparable One-Factor Model
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
Paola Mosconi Lecture 6 69 / 112
Multi-Factor Merton Model Comparable One-Factor Model
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 .
Paola Mosconi Lecture 6 70 / 112
Multi-Factor Merton Model Comparable One-Factor Model
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)
Paola Mosconi Lecture 6 71 / 112
Multi-Factor Merton Model Comparable One-Factor Model
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 .
Paola Mosconi Lecture 6 72 / 112
Multi-Factor Merton Model Comparable One-Factor Model
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
Paola Mosconi Lecture 6 73 / 112
Multi-Factor Merton Model Comparable One-Factor Model
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}.
Paola Mosconi Lecture 6 74 / 112
Multi-Factor Merton Model Comparable One-Factor Model
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
]
Paola Mosconi Lecture 6 75 / 112
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.
Paola Mosconi Lecture 6 76 / 112
Multi-Factor Merton Model Multi-Factor Adjustment
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
Multi-Factor Merton Model Multi-Factor Adjustment
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.
Paola Mosconi Lecture 6 78 / 112
Multi-Factor Merton Model Multi-Factor Adjustment
Granularity Adjustment
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).
Paola Mosconi Lecture 6 79 / 112
Multi-Factor Merton Model Multi-Factor Adjustment
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
Paola Mosconi Lecture 6 80 / 112
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
Paola Mosconi Lecture 6 81 / 112
Multi-Factor Merton Model Applications
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 .
Paola Mosconi Lecture 6 82 / 112
Multi-Factor Merton Model Applications
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
Paola Mosconi Lecture 6 83 / 112
Multi-Factor Merton Model Applications
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.
Paola Mosconi Lecture 6 84 / 112
Multi-Factor Merton Model Applications
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)
Paola Mosconi Lecture 6 85 / 112
Multi-Factor Merton Model Applications
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.
Paola Mosconi Lecture 6 86 / 112
Multi-Factor Merton Model Applications
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)
Paola Mosconi Lecture 6 87 / 112
Multi-Factor Merton Model Applications
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.
Paola Mosconi Lecture 6 88 / 112
Multi-Factor Merton Model Applications
Two Factor examples: Finite M
Figure: Effects of name concentration in two portfolios. Approximated solution vs MonteCarlo. Source: Pykhtin (2004)
Paola Mosconi Lecture 6 89 / 112
Multi-Factor Merton Model Applications
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
Paola Mosconi Lecture 6 90 / 112
Multi-Factor Merton Model Applications
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.
Paola Mosconi Lecture 6 91 / 112
Multi-Factor Merton Model Applications
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.
Paola Mosconi Lecture 6 92 / 112
Multi-Factor Merton Model Applications
Multi Factor Examples: Homogeneous Case, M = ∞ III
Figure: Accuracy of the approximation. Source: Pykhtin (2004)
Paola Mosconi Lecture 6 93 / 112
Multi-Factor Merton Model Applications
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.
Paola Mosconi Lecture 6 94 / 112
Multi-Factor Merton Model Applications
Multi Factor Examples: Non-Homogeneous Case II
Figure: Input. Source: Pykhtin (2004)
Paola Mosconi Lecture 6 95 / 112
Multi-Factor Merton Model Applications
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
Paola Mosconi Lecture 6 96 / 112
Multi-Factor Merton Model Applications
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.
Paola Mosconi Lecture 6 97 / 112
Multi-Factor Merton Model Applications
Multi Factor Examples: Non-Homogeneous Case IV
Figure: 11-Factor non-homogeneous set-up. Source: Pykhtin (2004)
Paola Mosconi Lecture 6 98 / 112
Capital Allocation
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
Paola Mosconi Lecture 6 99 / 112
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.
Paola Mosconi Lecture 6 100 / 112
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
Paola Mosconi Lecture 6 101 / 112
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
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
Paola Mosconi Lecture 6 103 / 112
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.
Paola Mosconi Lecture 6 104 / 112
Appendix
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
Paola Mosconi Lecture 6 105 / 112
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
Paola Mosconi Lecture 6 106 / 112
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
]
Paola Mosconi Lecture 6 107 / 112
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
)
Paola Mosconi Lecture 6 108 / 112
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)
Paola Mosconi Lecture 6 109 / 112
Selected References
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
Paola Mosconi Lecture 6 110 / 112
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
Paola Mosconi Lecture 6 111 / 112
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
Paola Mosconi Lecture 6 112 / 112
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