Portfolio Credit Risk Models Paul Embrechts London School of Economics Department of Accounting and Finance AC 402 FINANCIAL RISK ANALYSIS Lent Term, 2003 c Paul Embrechts and Philipp Schönbucher, 2003
Default Correlation Basics two obligors A and B time horizon T (fixed) p A, p B : (individual) default probabilities Not yet sufficient to determine either the probability, that A and B default before T : the joint default probability p AB i.e. the probability that A defaults before T, given that B has defaulted before T : the conditional default probabilities p A B and p B A the linear correlation coefficient ϱ AB between the default events 1 {A} and 1 {B}. c Paul Embrechts and Philipp Schönbucher, 2003 1
Connection is given by Bayes rule: p A B = p AB p B, p B A = p AB p A (1) and by the definition of the lineare correlation coefficient ϱ AB = p AB p A p B pa (1 p A )p B (1 p B ). (2) c Paul Embrechts and Philipp Schönbucher, 2003 2
Where is Default Correlation Relevant? Portfolios of defaultable bonds, CDOs basket structures (especially beyond first-to-default) When trying to diversify credit risk: management Portfolio management and credit risk Letter of Credit backed debt, Credit Guarantees Counterparty risk, especially in credit derivatives c Paul Embrechts and Philipp Schönbucher, 2003 3
Causes of Default Correlation direct relationship between both parties e.g. one firm creditor of another using same inputs general state of a certain industry / region general state of the economy Historical examples USA: Oil Industry: 22 companies defaulted 1982-1986 c Paul Embrechts and Philipp Schönbucher, 2003 4
Railroad Conglomerates: One default each year 1970-1977 Airlines: 3 defaults 1970-1971 5 defaults 1989-1990 Thrifts: (Savings and Loan Crisis) 19 defaults 1989-1990 Casinos / Hotel Chains: 10 defaults 1990 Retailers: >20 defaults 1990-1992 Construction / Real Estate: 4 defaults 1992 c Paul Embrechts and Philipp Schönbucher, 2003 5
Are Default Correlations Important? Default correlations are very important because default probabilities are very small. ϱ AB can have a much larger effect than usual (e.g. for equities etc). Orders of magnitude: ϱ AB = ϱ = O(1) is not very small, p A = p B = p 1 small. Joint default probability: p AB = p A p B + ϱ AB pa (1 p A )p B (1 p B ) p AB p 2 + ϱp ϱp c Paul Embrechts and Philipp Schönbucher, 2003 6
Conditional default probability: p A B = p A + ϱ AB pa p B (1 p A )(1 p B ) p A B ϱ. Correlation dominates joint default probabilities and conditional default probabilities. Numerical example: ϱ AB = 10% and p = 1%. p AB = 0.01 0.01 + 0.1 0.01 0.99 = 0.00109 p A B = 0.01 + 0.1 0.99 = 0.109 c Paul Embrechts and Philipp Schönbucher, 2003 7
Missing Data need to find structural models: to reduce dimensionality (dimension easily >1000) to compensate missing data and most importantly: The specification of full joint default probabilities is too complex: For N names have 2 N joint default events N marginal distributions (individual default probabilities) Note: c Paul Embrechts and Philipp Schönbucher, 2003 8
This is different from normally distributed random variables: There the N(N 1)/2 elements of the correlation matrix are sufficient to describe the full dependency structure. Individual default modelling and correlation modelling can be separated. (Buzzword: Copula Functions) c Paul Embrechts and Philipp Schönbucher, 2003 9
Market Variables and Data Sources Actual rating and default correlations Advantages: Objective, direct Disadvantages: Sparse data sets, long time ranges, need to aggregate Spread correlations Advantages: Reflect information in markets Disadvantages: Data quality problems, liquidity, availability Equity correlations: Advantages: liquid, easily available, good quality data Disadvantages: link to credit quality less obvious, needs a lot of pre-processing. c Paul Embrechts and Philipp Schönbucher, 2003 10
Homogeneous Portfolio, Independent Defaults (i) We consider default and survival of a portfolio until a fixed time-horizon of T. Interest-rates are set to zero. (ii) We have a portfolio of N exposures to N different obligors. (iii) The exposures are of identical size L, and have identical recovery rates of c. (iv) The defaults of the obligors happen independently of each other. Each obligor defaults with a probability of p before the time-horizon T. (v) Call X the number of defaults. c Paul Embrechts and Philipp Schönbucher, 2003 11
The Binomial Distribution The loss in default is X(1 c)l. Under independence, the probability of exactly X = n defaults until time T is P [ X = n ] = ( ) N p n (1 p) N n = n N! n!(n n)! pn (1 p) N n =: b(n; N, p), Distribution function: P [ X n ] = n m=0 ( ) N p m (1 p) N m =: B(n; N, p). m c Paul Embrechts and Philipp Schönbucher, 2003 12
prob 20% 18% 16% 14% 12% 10% 8% 6% 4% 2% 0% 0 10 20 30 n Number of obligors N = 100, individual default probability p = 5% c Paul Embrechts and Philipp Schönbucher, 2003 13
VaR Levels under Independence Default Prob. (%) 1 2 3 4 5 6 7 8 9 10 99.9% VaR Level 5 7 9 11 13 14 16 17 19 20 The Binomial distribution with independent defaults has an extremely thin tail and underestimates default risk. We need more model structure to generate realistic default loss distributions. c Paul Embrechts and Philipp Schönbucher, 2003 14
A Simplified Firm s Value Model (i) The default of each obligor n is triggered by the change of the value V n (t) of the assets of its firm. (ii) V n (T ) is normally distributed. W.l.o.g.: initial values V n (0) = 0 and standardised V n (T ) Φ(0, 1). (iii) Obligor n defaults if its firm s value falls below a barrier V n (T ) K n. (iv) The asset values of different obligors are correlated with each other. The variance-covariance matrix of the V 1,..., V N is denoted by Σ. This model is very similar to the JPMorgan Credit Metrics model. c Paul Embrechts and Philipp Schönbucher, 2003 15
One-Factor Model The values of the assets of the obligors are driven by: one common factor Y, and an idiosyncratic standard normal noise component ɛ n V n (T ) = ϱ Y + 1 ϱ ɛ n n N, where Y and ɛ n, n N are i.i.d. Φ(0, 1)-distributed. Conditional on the realisation of the systematic factor Y, the firm s values and the defaults are independent. The asset correlation between two obligors is ϱ c Paul Embrechts and Philipp Schönbucher, 2003 16
The distribution of the defaults All obligors have the same default barrier K n = K and the same exposure L n = 1. Probability of having n defaults: P [ X = n ] = P [ X = n Y = y ] φ(y)dy. Conditional on Y = y, the probability of having n defaults is P [ X = n Y = y ] = ( ) N (p(y)) n (1 p(y)) N n, n where we used the conditional independence of the defaults in the portfolio. c Paul Embrechts and Philipp Schönbucher, 2003 17
The individual conditional default probability p(y) is the probability that the firm s value V n (T ) is below the barrier K, given that the systematic factor Y takes the value y: p(y) = P [ V n (T ) < K Y = y ] [ ϱ ] = P Y + 1 ϱ ɛn < K Y = y [ = P ɛ n < K ] ϱ Y Y = y = Φ( K ϱ y ). 1 ϱ 1 ϱ Substituting yields P [ X = n ] = ( ) ( N Φ( K ) ϱ y n ( ) 1 Φ( K ϱ y N n )) φ(y)dy. n 1 ϱ 1 ϱ c Paul Embrechts and Philipp Schönbucher, 2003 18
20,00% 18,00% 16,00% 14,00% 12,00% 10,00% 8,00% 6,00% 4,00% 2,00% 0,00% 0 5 10 15 20 25 30 rho=0 rho=1 rho=10 rho=30 rho=50 Parameters: Number of obligors N = 100, individual default probability p = 5%, c Paul Embrechts and Philipp Schönbucher, 2003 19
100,00% 0 5 10 15 20 25 30 10,00% 1,00% 0,10% rho=0 rho=1 rho=10 rho=30 rho=50 0,01% Parameters: Number of obligors N = 100, individual default probability p = 5%, asset correlation rho in percentage points: 0, 1, 10, 30, 50 c Paul Embrechts and Philipp Schönbucher, 2003 20
Effect of Asset Correlation on VaR Levels Asset Correlation 99.9% VaR Level 99% VaR Level 0% 13 11 1% 14 12 10% 27 19 20% 41 27 30% 55 35 40% 68 44 50% 80 53 c Paul Embrechts and Philipp Schönbucher, 2003 21
Results Increasing asset correlation (and thus default correlation) leads to a shift of the probability weight to the left ( good events) and to the tail on the right. Very good events (no or very few defaults) and very bad events (many defaults) become more likey. Already for ϱ = 10% we have strong effects on the tails. VaR levels are strongly increased. c Paul Embrechts and Philipp Schönbucher, 2003 22
The large portfolio approximation Assumption 1 (Large Uniform Portfolio) The portfolio consists of a very large N number of credits of uniform size. Let X now denote the fraction of the defaulted securities in the portfolio. The individual default probability is p(y) = Φ( K ϱ y 1 ϱ ). c Paul Embrechts and Philipp Schönbucher, 2003 23
The Law of Large Numbers the fraction of obligors that actually defaults is (almost surely) exactly equal to the individual default probability: P [ X = p(y) Y = y ] = 1. We do not know the realisation of Y yet, but we can invoke iterated expectations to reach P [ X x ] = E [ P [ X x Y ] ] = = P [ X = p(y) x Y = y ] φ(y)dy 1 {p(y) x} φ(y)dy = y φ(y)dy = Φ(y ) c Paul Embrechts and Philipp Schönbucher, 2003 24
The Critical Level y y is the min. level that the systematic factor must reach to avoid more than x defaults. y is chosen such that p( y ) = x, and p(y) x for y > y. Thus y is y = 1 ( ) 1 ϱ Φ 1 (x) K. ϱ c Paul Embrechts and Philipp Schönbucher, 2003 25
The distribution function: The Result F (x) := P [ X x ] = Φ ( 1 ϱ ( 1 ϱ Φ 1 (x) Φ 1 (p)) ). The probability density function f(x): f(x) = 1 ϱ ϱ exp { 1 2 (Φ 1 (x)) 1 2ϱ ( Φ 1 (p) ) } 2 1 ϱ Φ 1 (x). Quality of the approximation: remarkably good in most applications need minimum number of obligors need a certain level of asset correlation (> 1%) c Paul Embrechts and Philipp Schönbucher, 2003 26
20,00% 18,00% 16,00% 14,00% Probability 12,00% 10,00% 8,00% 6,00% 4,00% 2,00% 0,00% 0% 5% 10% 15% 20% 25% 30% Loss Fraction 1% 10% 30% 50% c Paul Embrechts and Philipp Schönbucher, 2003 27
100,00% 0% 5% 10% 15% 20% 25% 30% 10,00% Probability 1,00% 0,10% 0,01% Loss Fraction 1% 10% 30% 50% c Paul Embrechts and Philipp Schönbucher, 2003 28
References [1] Barry Belkin, Stephan Suchover, and Lawrence Forest. A one-parameter representation of credit risk and transition matrices. Credit Metrics Monitor, 1(3):46 56, 1998. [2] Credit Suisse First Boston. Credit Risk+. Technical document, Credit Suisse First Boston, 1997. URL: www.csfb.com/creditrisk. [3] Darrell Duffie. First-to-default valuation. Working paper, Graduate School of Business, Stanford University, 1998. [4] Christopher C. Finger. Conditional approaches for credit metrics portfolio distributions. Credit Metrics Monitor, 2(1):14 33, April 1999. [5] Greg Gupton, Christopher Finger, and Bhatia Mike. Credit Metrics technical document. Technical document, JPMorgan & Co. Inc., April 1997. URL: www.creditmetrics.com. [6] David X. Li. The valuation of basket credit derivatives. Credit Metrics Monitor, 2:34 50, April 1999. [7] Andre Lucas, Pieter Klaassen, Peter Spreij, and Stefan Staetmans. An analytic approach to credit risk of large corporate bond and loan portfolios. Research Memorandum 1999-18, Vrije Universiteit Amsterdam, February 1999. [8] Krishan M. Nagpal and Reza Bahar. An analytical approach for credit risk analysis under correlated defaults. Credit Metrics Monitor, 2:51 74, April 1999. [9] Oldrich Vasicek. The loan loss distribution. Working paper, KMV Corporation, 1997. c Paul Embrechts and Philipp Schönbucher, 2003 29