Dependence Modeling and Credit Risk

Dependence Modeling and Credit Risk Paola Mosconi Banca IMI Bocconi University, 20/04/2015 Paola Mosconi Lecture 6 1 / 53

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 / 53

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 / 53

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 Selected References Paola Mosconi Lecture 6 4 / 53

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 Selected References Paola Mosconi Lecture 6 5 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 Selected References Paola Mosconi Lecture 6 20 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 / 53

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 Selected References Paola Mosconi Lecture 6 49 / 53

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 / 53

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 / 53

Selected References 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 Selected References Paola Mosconi Lecture 6 52 / 53

Selected References Selected References I BSCS (2006). Studies on credit risk concentration, Working Paper No. 15 Emmer, S. and Tasche, D. (2003). Calculating credit risk capital charges with the one-factor model, Working paper Gordy, M. (2003). A risk-factor model foundation for ratings-based bank capital rules, Journal of Financial Intermediation, 12, July, pages 199-232 Gordy, M. (2004). Granularity In New Risk Measures for Investment and Regulation, edited by G. Szegö, Wiley Martin, R. and Wilde, T. (2002). Unsystematic credit risk, Risk, November Merton, R. (1974). On the pricing of corporate debt: The risk structure of interest rates. J. of Finance 29, 449-470 Vasicek, O. (1987). Probability of loss on a loan portfolio. Working Paper, KMV Corporation Vasicek, O. (1991). Limiting loan loss probability distribution, KMV Corporation Wilde, T. (2001) Probing granularity, Risk, August Paola Mosconi Lecture 6 53 / 53