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 in any 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 analytical model, Risk, May Multi-Factor Merton Model Pykhtin, M. (2004), Multi-Factor Adjustment, Risk, March Paola Mosconi Lecture 6 3 / 112

Outline 1 Introduction Credit Risk Portfolio Models 2 Vasicek Portfolio Loss Model Introduction Limiting Loss Distribution Properties of the Loss Distribution 3 Granularity Adjustment 4 Multi-Factor Merton Model Introduction VaR Expansion Comparable One-Factor Model Multi-Factor Adjustment Applications 5 Capital Allocation 6 Conclusions 7 Appendix 8 Selected References Paola Mosconi Lecture 6 4 / 112

Introduction Outline 1 Introduction Credit Risk Portfolio Models 2 Vasicek Portfolio Loss Model Introduction Limiting Loss Distribution Properties of the Loss Distribution 3 Granularity Adjustment 4 Multi-Factor Merton Model Introduction VaR Expansion Comparable One-Factor Model Multi-Factor Adjustment Applications 5 Capital Allocation 6 Conclusions 7 Appendix 8 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 its financial 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 over the 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 through bankruptcy 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 the default occurs? Paola Mosconi Lecture 6 6 / 112

Introduction Credit Risk Sources of Risk Default risk Migration risk Spread risk Riskofchangesinthecreditspreadsoftheborrower,forexampleduetomarket conditions (should not result in a change in the credit rating) Recovery risk Risk that the actual recovery rate is lower than previously estimated Sovereign risk Risk that the counterparty will not pay due to events of political or legislative nature 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, being an additive measure, is given by the sum of 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 around its 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 density function f(x), a quantile function Q p assigns to each probability p attained by f the value x for which P(X x) = p. The quantile function Q p = inf {x : P(X x) p} x R returns the minimum value of x from amongst all those values whose cumulative distribution function (cdf) value exceeds p. If the probability distribution is discrete rather than continuous then there may be gaps 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 density function f(x), if the cumulative distribution function F = P(X x) is strictly increasing and continuous then, F 1 (y) with y [0,1] is the unique real number x such that F(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 following quantile function: VaR q = inf {l : P(L > l) 1 q} l R = inf {l : P(L l) q} l R Expected Shortfall The Expected Shortfall of the portfolio loss L at confidence level q is given by: ES q(l) = E[L L VaR q(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 considered a cost of doing business. The Economic Capital is given by the Unexpected Loss, defined as: EC = VaR q EL The EC is not an additive measure: at portfolio level, the joint probability distribution of losses must be considered (correlation 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 and the size of single exposures (idiosyncratic or specific risk) systematic (sector) risk, which is described by the correlation structure of obligors inside 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. Mostoftheportfoliomodelsofcreditriskusedinthebankingindustryarebasedon the conditional independence framework. In these models, defaults of individual borrowers depend on a set of common systematic risk factors describing the state of the economy. Merton-type models, such as PortfolioManager and CreditMetrics, have become very popular. However, implementation of these models requires time-consuming Monte 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 Risk Factor (ASRF) model has played a central role, also for its regulatory applications in the Basel Capital Accord Framework. ASRF (Vasicek, 1991) The model allows to derive analytical expressions for VaR and ES, by relying on 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 small size loans 4 dependence structure among different obligors described by the gaussian copula Paola Mosconi Lecture 6 17 / 112

Introduction Portfolio Models ASRF Extensions Violations of the hypothesis underlying the ASRF model give rise to corrections which are explicitly taken into account by the BCBS (2006) under the generic name 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 systematic components of risk 3 Contagion: exposures to independent obligors that exhibit default dependencies, which exceed what one should expect on the basis of their sector 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 account both name concentration and sector concentration Paola Mosconi Lecture 6 19 / 112

Vasicek Portfolio Loss Model Outline 1 Introduction Credit Risk Portfolio Models 2 Vasicek Portfolio Loss Model Introduction Limiting Loss Distribution Properties of the Loss Distribution 3 Granularity Adjustment 4 Multi-Factor Merton Model Introduction VaR Expansion Comparable One-Factor Model Multi-Factor Adjustment Applications 5 Capital Allocation 6 Conclusions 7 Appendix 8 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 single systematic factor. The probability distribution of portfolio losses has a number of important applications: 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 debt obligations (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 on the probability distribution of the portfolio loss. Consider a portfolio of loans, each of which is subject to default resulting in a loss to the lender. Suppose the portfolio is financed partly by equity capital and partly by borrowed funds. The credit quality of the lender s notes will depend on the probability that the loss on the portfolio exceeds the equity capital. To achieve a certain credit rating of its notes (say Aa on a rating agency scale), the lender needs to keep the probability of default on the notes at the level corresponding to that rating (about 0.001 for the Aa quality). It means that the equity capital allocated to the portfolio must be equal to the percentile of the distribution of the portfolio loss that corresponds to the desired 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 the value of the borrower s assets at the loan maturity T falls below the contractual value B of its obligations payable. Asset value process Let A i be the value of the i-th borrower s assets, described by the process: da i = µ i A i dt +σ i A i dx i The asset value at T can be obtained by integration: loga i (T) = loga i +µ i T 1 2 σ2 i T +σ i T Xi (1) where X i 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: p i = P[A i (T) < B i ] = P[X i < ζ i ] = N(ζ i ) where N(.) is the cumulative normal distribution function and represents the default threshold. ζ i = logb i loga i µ i T + 1 2 σ2 i T σ i T 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 L i be the gross loss (before recoveries) on the i-th loan, so that L i = 1 if the i-th borrower defaults and L i = 0 otherwise. Let L be the portfolio percentage gross loss: L = 1 n L i n i=1 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 each other, 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 limit theorem 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 {X i} i=1,...,n in eq. (1) are jointly standard normal with equal pair-wise correlations ρ, and can be expressed as: X i = ρy + 1 ρξ i where Y and ξ 1,ξ 2,...,ξ n are mutually independent standard normal variables. The variable Y can be interpreted as a portfolio common (systematic) factor, such as an economic index, over the interval (0,T). Then: the term ρy is the company s exposure to the common factor the term 1 ρξ i 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 factor Y, 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: [ N 1 (p) ] ρy p(y) = P(L i = 1 Y) = P(X i < ζ i Y) = N 1 ρ The quantity p(y) provides the loan default probability under the given scenario. The unconditional default probability p is the average of the conditional probabilities over the 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 L i are independent equally distributed variables with a finite variance. Conditional Portfolio Loss The portfolio loss conditional on Y converges, by the law of large numbers, to its expectation 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 s derivation (1991). Since p(y) is a strictly decreasing function of Y i.e. it follows that: p(y) x Y p 1 (x) 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 distribution function of loan losses on a very large portfolio is in the limit: [ ] 1 ρn 1 (x) N 1 (p) P(L x) = N 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 holds even for portfolios with unequal weights. Let the portfolio weights be w 1,w 2,...,w n with wi = 1. The portfolio loss: n L = w i L i i=1 conditional on Y converges to its expectation p(y) whenever (and this is a necessary and sufficient condition): n wi 2 0 i=1 In other words, if the portfolio contains a sufficiently large number of loans without it being dominated by a few loans much larger than the rest, the limiting distribution provides 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 parameters 0 < p, ρ < 1. The cumulative distribution function is continuous and concentrated on the interval 0 x 1: [ 1 ρn 1 (x) N 1 ] (p) F(x;p,ρ) := N 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: { 1 ρ f(x;p,ρ) = exp 1 [ ] 2 1 ρn 1 (x) N 1 1 [ (p) + N 1 (x) ] } 2 ρ 2ρ 2 which is: unimodal with the mode at L mode = N [ ] 1 ρ 1 2ρ N 1 (p) when ρ < 1 2 monotone when ρ = 1 2 U-shaped when ρ > 1 2 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 concentrated at L = p. Figure: Probability density function (left) and cumulative distribution function (right) for p = 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 with probabilities 1 p and p, respectively. Figure: Probability density function (left) and cumulative distribution function (right) for p = 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) for ρ = 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) for ρ = 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: s 2 = var(l) = E { [L E(L)] 2} = E(L 2 ) [E(L)] 2 = N 2 (N 1 (p),n 1 (p),ρ) p 2 where N 2 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: L α = F(α;1 p;1 ρ) Figure: Source: Vasicek Risk (2002) The table lists the values of the α-percentile L α expressed as the number of standard deviations from the mean, for several values of the parameters. The α-percentiles of the standard 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 asset correlation is ρ = 0.4 and whose probability of default is p = 0.01. The portfolio expected loss is E(L) = 0.01 and the standard deviation is s = 0.0277. If the lender wishes to hold the probability of default on his notes at 1 α = 0.001, he will need enough capital to cover 11.0 times the portfolio standard deviation. If the 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 a reasonably 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 w2 i = 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 = VaR q (L) EL = F(q;1 p;1 ρ) p [ ρn 1 (q) N 1 (1 p) = N ] p 1 ρ [ ρn 1 (q)+n 1 (p) = N ] p 1 ρ where N 1 (1 x) = N 1 (x). The formula has been obtained under the assumption 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 is given by: n K = 8% RW i EAD i where, the individual risk weight RW i is: [ [ N 1 (p ] ] i) ρ i N 1 (0.1%) RW i = 1.06 LGD i N p i MF(M i,p i) 1 ρi i=1 where: MF is a maturity factor adjustment, depending on the effective maturity M i of loan i p i is individual probabilities of default of loan i q = 99.9% ρ i is a regulatory factor loading which depends on p i and the type of the loan (corporate, SMEs, residential mortgage etc...) Paola Mosconi Lecture 6 48 / 112

Granularity Adjustment Outline 1 Introduction Credit Risk Portfolio Models 2 Vasicek Portfolio Loss Model Introduction Limiting Loss Distribution Properties of the Loss Distribution 3 Granularity Adjustment 4 Multi-Factor Merton Model Introduction VaR Expansion Comparable One-Factor Model Multi-Factor Adjustment Applications 5 Capital Allocation 6 Conclusions 7 Appendix 8 Selected References Paola Mosconi Lecture 6 49 / 112

Granularity Adjustment Granularity Adjustment The asymptotic capital formula implied by the Vasicek distribution (1991): [ ρn 1 (q)+n 1 (p) EC = N ] p 1 ρ is strictly valid only for a portfolio such that the weight of its largest exposure is infinitesimally small. All real-world portfolios violate this assumption and, therefore, one might question the relevance of the asymptotic formula. Indeed, since any finite-size portfolio carries some undiversified idiosyncratic risk, the asymptotic formula must underestimate the true capital. The difference between the true capital and the asymptotic capital is known as granularity 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-form expression for the granularity adjustment for portfolio VaR More specific expressions for a one-factor default-mode Merton-type model have been derived by Pykhtin and Dev (2002) Emmer and Tasche (2003) have developed an analytical formulation for calculating VaR 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 Introduction Credit Risk Portfolio Models 2 Vasicek Portfolio Loss Model Introduction Limiting Loss Distribution Properties of the Loss Distribution 3 Granularity Adjustment 4 Multi-Factor Merton Model Introduction VaR Expansion Comparable One-Factor Model Multi-Factor Adjustment Applications 5 Capital Allocation 6 Conclusions 7 Appendix 8 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 to address 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 the portfolio 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 loan characterized by exposure EAD i, whose weight in the portfolio is given by w i = EAD i M i=1 EADi each obligor is assigned a probability of default p i and a loss given default LGD i. The loss given default is described by means of a stochastic variable Q (with mean µ i and standard deviation σ i), whose independence of other sources of randomness is 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) with probability p i. Default happens when a continuous variable X i describing the financial well-being of borrower i at the horizon falls below a threshold. Default threshold We assume that variables {X i } (which may be interpreted as the standardized asset returns) have standard normal distribution. The default threshold for borrower i is given by N 1 (p i ), where N 1 (.) is the inverse of the cumulative normal distribution function. 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 systematic risk factors with a full-rank correlation matrix. Systematic factors represent industry, geography, global economy or any other relevant indexes that may affect borrowers defaults in a systematic way. Borrower i s standardized asset return is driven by a certain borrower-specific combination of these systematic factors Y i (known as a composite factor): X i = r i Y i + 1 ri 2 ξ i (2) where ξ i N(0,1) is the idiosyncratic shock. Factor loading r i measures borrower i 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 original correlated systematic factors are decomposed into N independent standard normal systematic factors Z k N(0,1) (k = 1,...,N). The relation between {Z k } and the composite factor is given by N Y i = α ik Z k where α ik must satisfy the relation k=1 N α 2 ik = 1 k=1 to ensure that Y i has unit variance. Asset correlation between distinct borrowers i and j is given by N ρ ij = r ir j α ik α jk k=1 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 stochastic variableq i.nospecific assumptionsabouttheprobabilitydistributionofq i ismade,except for its independence of all the other stochastic variables. The portfolio loss rate L is given by the weighted average of individual loss rates L i L = M w i L i = i=1 M w i Q i 1 {Xi N 1 (p i)} (3) i=1 where 1 {.} is the indicator function. This equation describes the distribution of the 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 the multi-factor framework is Monte Carlo simulation. Limiting Loss Distribution In the case of a large enough, fine-grained, portfolio, most of the idiosyncratic risk is diversified 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 an infinitely fine-grained portfolio, given by the expected loss conditional on the systematic risk factors (see Gordy (2003) for details): [ M L N 1 ] (p N i) r i k=1 = E[L {Z k }] = w i µ i N α ikz k (4) 1 r 2 i i=1 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 Carlo simulation of the systematic factors {Z k } 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 become accurate. Goal To design an analytical method for calculating tail quantiles and tail expectations of the portfolio loss L given by equation (3). The method has been devised by Pykhtin (2004) and is based on a Taylor expansion of VaR, introduced by Gourieroux, Laurent and Scaillet (2000) and perfected 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 the corresponding quantile, which is denoted by t q(l). The calculation of t q(l) goes through the following steps: 1 assume that we have constructed a random variable L such that its quantile at level q, t q(l), can be calculated analytically and is close enough to t q(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 L ε 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 t q(l ε) is obtained through a series expansion in powers of ε around t q(l). Up to the second order, t q(l) t q(l ε=1) reads: t q(l) t dtq(lε) q(l)+ + 1 d 2 t q(l ε) (5) dε 2 dε 2 ε=0 ε=0 where: dt q(l ε) dε d 2 t q(l ε) dε 2 = E[U L = t q(l)] ε=0 = 1 f L (l) ε=0 d ( fl (l)var[u L = l] ) dl l=t q(l) f L (.) 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 Merton framework Merton (1974) i.e. L := l(y) = M w i µ i ˆp i (Y) i=1 where, it is implicitly assumed that: X i = a iy + 1 a 2 i ζ i ζ i N(0,1) and ˆp i(y) is the probability of default of borrower i, conditional on Y = y: [ ] N 1 (p i) a iy ˆp i(y) = N 1 a 2 i 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 of L at level q can be calculated analytically (see Castagna, Mercurio and Mosconi, 2009): t q (L) = l(n 1 (1 q)) Remark Let us note that the derivatives of t q(l) in the VaR expansion are given by expressions conditional on L = t q(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 t q(l) = l: dt q(l ε) = E[U Y = N 1 (1 q)] dε ε=0 Paola Mosconi Lecture 6 66 / 112

Multi-Factor Merton Model VaR Expansion VaR Expansion: Second Order Term The second order derivative can be rewritten d 2 t q(l ε) = 1 d dε 2 n(y) dy ε=0 ( 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 of l(.) and n(.) is the standard normal density. By carrying out the derivative with respect to y explicitly and using the fact that n (y) = y n(y), the second order term becomes: d 2 t q(l ε) = 1 [ ( )] l ν (y) (y) ν(y) dε 2 l (y) l (y) +y ε=0 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 {Z k } Goal To relate random variable L to the portfolio loss L, we need to relate the effective systematic factor Y to the original systematic factors {Z k }. We assume a linear relation given by: Y = N b k Z k b k 0 k=1 where the coefficients must satisfy N k=1 b2 k = 1 to preserve unit variance of Y. In order to complete the specification of L, we need to specify the set of M effective factor loadings {a i} and N coefficients {b k }. Paola Mosconi Lecture 6 68 / 112

Multi-Factor Merton Model Comparable One-Factor Model Key Requirement In order to determine the coefficients {a i} and {b k }, we enforce the requirement that L equals the expected loss conditional on Y: for any portfolio composition. L = E[L Y] Besides being intuitively appealing, this requirement guarantees that the first-order term in the Taylor series vanishes for any confidence level q, i.e. dt q(l ε) = E[U Y = N 1 (1 q)] = E[L L Y] dε ε=0 = 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 Y i To calculate E[L Y], we represent the composite risk factor for borrower i, Y i, as: Y i = ρ i Y + 1 ρ 2 i η i whereη i N(0,1)isastandardnormal randomvariable independentof Y (butincontrast to the true one-factor case, variables {η i} are inter-dependent), and ρ i is the correlation between Y i and Y given by: ρ i := corr(y i,y) = Using this notations, asset return can be written as N α ik b k k=1 X i = r i ρ i Y + 1 (r i ρ i ) 2 ζ i (6) where ζ i 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 {a i } The conditional expectation of L results in: E[L Y] = M w i µ i N i=1 [ ] N 1 (p i) r i ρ i Y 1 (ri ρ i ) 2 This equation must be compared with the limiting loss distribution: [ ] M N 1 (p i) a iy L = w i µ i N 1 a 2 i i=1 Effective factor loadings a i L equals E[L Y] for any portfolio composition if and only if the effective factor loadings are defined as: N a i = r i ρ i = r i α ik b k (7) k=1 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 gives rise 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: X i = a iy + N k=1 where the second term is independent of Y. (r i α ik a i b k )Z k + 1 ri 2 ξ i Conditional Asset Correlation This term is responsible for the conditional asset correlation, which turns out to be: N ρ Y ij = rirj k=1 α ikα jk a ia j (1 ai 2)(1 a2 j ) Although ρ Y ij has the meaning of the conditional asset correlation only for distinct borrowers 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 {b k } I Given equation (7) which defines the factor loadings a i, the choice of the coefficients {b k } is not unique. The set {b k } specifies the zeroth-order term t q(l) in the Taylor expansion and many alternative specifications of {b k } are plausible, provided that the associated t q(l) is close enough to the unknown target function value t q(l). Goal Ideally, we aim at finding a set {b k } that minimizes the difference between the two quantiles. Intuitively, one would expect the optimal single effective risk factor Y to have as much correlation as possible with the composite risk factors {Y i}, i.e. ( M ) N max c i corr(y,y i) such that bk 2 = 1 {b k } i=1 k=1 Paola Mosconi Lecture 6 73 / 112

Multi-Factor Merton Model Comparable One-Factor Model Choice of the Coefficients {b k } II Considering that N corr(y,y i ) = α ik b k k=1 the solution to this maximization problem is given by: b k = M i=1 c i λ α ik (8) where positive constant λ is the Lagrange multiplier chosen so that {b k } satisfy the constraint. Unfortunately, it is not clear how to choose the coefficients {c i }. Paola Mosconi Lecture 6 74 / 112

Multi-Factor Merton Model Comparable One-Factor Model Choice of the Coefficients {c i } Some intuition about the possible form of the coefficients {c i} can be developed by minimization of the conditional variance ν(y). Under an additional assumption that all r i are small, this minimization problem has a closed-form solution given by eq. (8) with c i = w i µ i n[n 1 (p i)] Even though the assumption of small r i is often unrealistic and the performance of this solution is sub-optimal, it may serve as a starting point in a search of optimal {c i}. Coefficients {c i} One of the best-performing choices is represented by: [ ] N 1 (p i)+r i N 1 (q) c i = w i µ i N 1 r 2 i 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 contributions cancel out, the total VaR, approximated up to second order, is given by: t q(l) t q(l)+ t q (9) where t q = 1 [ ( )] ν l (y) (y) ν(y) 2l (y) l (y) +y y=n 1 (1 q) where l(y) = M i=1 wi µi ˆpi(y) and ν(y) = var[l Y = y] is the conditional variance of L on Y = y. If, conditional on Y individual loss contributions were independent, the term t q would be equivalent to Wilde s granularity adjustment (Wilde, 2001). However, due to non-zero conditional asset correlation between distinct borrowers, the correction term contains also systematic effects. Paola Mosconi Lecture 6 76 / 112

Multi-Factor Merton Model Multi-Factor Adjustment Multi-Factor Adjustment: Conditional Variance Conditional variance decomposition Conditional on {Z k }, asset returns are independent, and we can decompose the conditional variance ν(y) in terms of its systematic and idiosyncratic components: ν(y) = ν (y)+ν GA (y) where ν (y) = var[e(l {Z k }) Y = y] ν GA (y) = E[var(L {Z k }) Y = y] The same decomposition 1 holds for the quantile correction (multi-factor adjustment): t q = t q + t GA q 1 See 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 {Z k }] on Y = y quantifies the difference between the multi-factor and one-factor limiting loss distributions (we will denote this term as ν (y)) and is given by: ν (y) = var[e(l {Z k }) Y = y ] M M ] (10) = w iw j µ iµ j [N 2(N 1 [ˆp i(y)],n 1 [ˆp j(y)],ρ Y ij ) ˆp i(y)ˆp j(y) i=1 j=1 where N 2(,, ) 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 ν GA (y) describes the effect of the finite number of loans in the portfolio. It vanishes in the limit M, provided that M i=1 w 2 i 0 while M w i = 1 i=1 ν GA (y) = E[var(L {Z k }) Y = y ] M ( [ ] ) (11) = wi 2 µ 2 i ˆp i(y) N 2(N 1 [ˆp i(y)],n 1 [ˆp i(y)],ρ Y ii ) +σi 2 ˆp i(y) i=1 where ρ Y ii is obtained by replacing the index j with i in the conditional asset correlation. In the special case of homogeneous LGDs and default probabilities p i, it becomes proportional to the Herfindahl-Hirschman index HHI = M i=1 w2 i (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 t q (L), in an implicit way and by construction, and the second order correction depending on ν (y). The latter, obtained in the limit of an infinitely fine-grained portfolio, represents the systematic component of risk which cannot be diversified away Single name concentration It is described by the granularity adjustment ν GA (y). For a large enough number of obligors M (ideally, M ) and under the condition of a sufficiently homogeneous distribution of loans exposures (in mathematical terms M i=1 w2 i 0, while M i=1 w i = 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-factor adjustment) 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 M u identical loans characterized by a single probability of default p u, expected LGD µ u, standard deviation of LGD σ u, composite factor Y u and composite factor loading r u. The asset correlation inside bucket u is r 2 u. Buckets are characterized by weights ω u defined as the ratio of the net principal of all loans in bucket u to the net principal of all loans in the portfolio. Individual loan weights are related to bucket weights as ω u = w u M u. the composite factors are correlated with correlation ρ and the asset correlation between the buckets is ρr A r B. Paola Mosconi Lecture 6 82 / 112

Multi-Factor Merton Model Applications Two Factor Examples: Homogeneous Case, M = I Input p A = p B = 0.5%, µ A = µ B = 40%, σ A = σ B = 20% and r A = r B = 0.5. Goal To compare (see Figure 1(a) homogeneous case): the approximated t 99.9% (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 ω A 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 (ω A = ω B = 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 the ω A = ω B case. Conclusions This behavior is natural because any of the limits ρ = 1, ω A = 0 and ω A = 1 corresponds to the one-factor case where the approximation becomes exact. As one moves away from one of the exact limits, the error of the approximation is expected to increase. The performance of the approximation is the worst when one is as far from the limits as possible 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 p A = 0.1% and thecomposite factor loading r A = 0.5, while bucket B has p B = 2% and r B = 0.2. The LGD parameters are left at the same values 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 t 99.9% (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 excellent for all choices of the bucket weights and the risk factor correlation the method in general performs much better in non-homogeneous cases than 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) systematic contributions to VaR 99.9%. (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: w A = 0.3 and w A = 0.7, assuming the risk factor correlation ρ = 0.5. Goal To test the effects of name concentration on two portfolios (respectively, with w A = 0.3 and w A = 0.7) by varying the bucket population. The 99.9% quantile calculated with the approximated method is compared with the same quantile calculated via a Monte Carlo simulation. Results As with Wilde s one-factor granularity adjustment, performance of the granularity adjustment 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 Monte Carlo. Source: Pykhtin (2004) Paola Mosconi Lecture 6 89 / 112

Multi-Factor Merton Model Applications Multi Factor Examples: Assumptions All M u loans in bucket u are characterized by the same PD p u, expected LGD µ u, standard deviation of LGD σ u, composite systematic risk factor Y u and composite factor loading r u. Bucket weights ω u are defined as the ratio of the net principal of all loans in bucket u to the net principal of all loans in the portfolio. Systematic factors: N 1 industry-specific (independent) systematic factors {Z k } k=1,...,n 1 one global systematic factor Z N composite systematic factors: Y i = α i Z N + 1 α 2 i Z k(i) where k(i) denotes the industry that borrower i belongs to. The weight of the global factor is assumed to be the same for all composite factors: α i = α. Correlations: between any pair of composite systematic factors is ρ = α 2 asset correlation inside bucket u is r 2 u asset correlation between buckets u and v is ρr u r v Paola Mosconi Lecture 6 90 / 112

Multi-Factor Merton Model Applications Multi Factor Examples: Homogeneous Case, M = I Input We assume thatthe bucketsare identical and populatedby a very large number of identical loans, with p u = 0.5%, µ u = 40%, σ u = 20% and r u = 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 t 99.9% (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 the one-factor framework. At any given ρ, the approximation based on a one-factor model works better as the number of factors increases. Conclusions In the homogeneous case with composite risk factor correlation ρ, the limit N 1 = M is equivalent to the one-factor set-up with the factor loading r u ρ. When we increase the number of the systematic risk factors, we move towards this one-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 adjustment approximation with the ones obtained from a Monte Carlo simulation for the case 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 ω u = 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 number of 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 the asymptotic loss (L is the same for all three portfolios) is excellent even for very low levels of ρ the performance of the approximation for L in non-homogeneous cases is typically 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 loss at all levels of ρ. This is because the largest exposure in the portfolio is rather small only 0.2% of the portfolio exposure. Portfolio II: the number of loans in each of the buckets has been decreased uniformly by a factor of 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 factor correlation, but is rather disappointing at low ρ. Portfolio III: it has the same largest exposure as portfolio II, but much higher dispersion of the exposure sizes than either portfolio I or portfolio II. Although the resulting loss quantile is very close to the one for portfolio I, the approximation does not perform as 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