Modelling Dependent Credit Risks Filip Lindskog, RiskLab, ETH Zürich 30 November 2000 Home page:http://www.math.ethz.ch/ lindskog E-mail:lindskog@math.ethz.ch RiskLab:http://www.risklab.ch
Modelling Dependent Credit Risks Copula Ideas Part 1 I II Dependence concepts, copula families Example - risky dependence Part 2 Common Poisson Shock Models III IV Model description Example - large loan portfolios c 2000 (F. Lindskog, RiskLab) 1
Drawbacks of linear correlation Linear correlation of random variables X, Y is not defined if the variance of X or Y is infinite. Linear correlation can easily be misinterpreted. Linear correlation is not invariant under non-linear strictly increasing transformations T : R R, i.e., ρ (T (X),T(Y )) ρ (X, Y ). Given distribution functions F and G for X and Y, all linear correlations between 1 and 1 can in general not be obtained by a suitable choice of the joint distribution. c 2000 (F. Lindskog, RiskLab) 2
Correlation values -0.5 0.0 0.5 1.0 0 1 2 3 4 5 Sigma Upper and lower bounds for ρ(x, Y ), where X Lognormal(0, 1) and Y Lognormal(0,σ 2 ). Note: For σ =4,ρ(X, Y )=0.01372 in fact means that Y = T (X), with T strictly increasing! c 2000 (F. Lindskog, RiskLab) 3
Copulas Definition A copula, C :[0, 1] n [0, 1], is a joint distribution function (d.f.) of n random variables uniformly distributed on [0, 1]. Let H be an n-dimensional d.f. with continuous margins F 1,...,F n. Then H(x 1,...,x n )=C(F 1 (x 1 ),...,F n (x n )). Conversely, if C is a copula and F 1,...,F n are d.fs, then H is an n-dimensional d.f. with margins F 1,...,F n. Hence the copula of (X 1,...,X n ) H is the d.f. of (F 1 (X 1 ),...,F n (X n )). If F 1,...,F n are strictly increasing d.f.s, then for every u =(u 1,...,u n )in[0, 1] n, C(u) =H(F 1 1 (u 1),...,F 1 n (u n )). c 2000 (F. Lindskog, RiskLab) 4
Special copulas M n (u) = min(u 1,u 2,...,u n ) W n (u) = max(u 1 + u 2 + + u n n +1, 0) Π n (u) = u 1 u 2...u n Note: M n and Π n are copulas for all n 2 but W n is a copula only for n =2. Definition 1. X 1,...,X n comonotonic (X 1,...,X n ) has copula M n (X 1,...,X n )= d (α 1 (Z),...,α n (Z)), α 1,...,α n increasing and Z is some real valued random variable. 2. X, Y countermonotonic (X, Y ) has copula W 2 (X, Y )= d (α(z),β(z)), α inc., β dec. and Z is some real valued r.v. 3. X 1,...,X n independent (X 1,...,X n ) has copula Π n. c 2000 (F. Lindskog, RiskLab) 5
Properties of copulas Bounds For every u [0, 1] n we have W n (u) C(u) M n (u). These bounds are the best possible. Concordance ordering If C 1 and C 2 are copulas, we say that C 2 is more concordant than C 1 (C 1 c C 2 )if C 1 (u) C 2 (u) andc 1 (u) C 2 (u) for all u [0, 1] n. C 1 and C 2 are joint survival functions. Strictly increasing transformations If α 1,α 2,...,α n are strictly increasing, then α 1 (X 1 ),α 2 (X 2 ),...,α n (X n )havethesame copula as X 1,X 2,...,X n. c 2000 (F. Lindskog, RiskLab) 6
Rank correlations Let (X, Y ) be a random vector with continuous margins F and G and copula C. Kendall s tau of (X, Y ) is given by τ(x, Y ) := P[(X X )(Y Y ) > 0] P[(X X )(Y Y ) < 0] = 4 C(u, v)dc(u, v) 1, [0,1] 2 where (X, Y )and(x,y ) are independent copies. Spearman s rho of (X, Y ) is given by ρ S (X, Y ) = ρ(f (X),G(Y )) = 12 uv dc(u, v) 3. [0,1] 2 Kendall s tau and Spearman s rho are called rank correlations. c 2000 (F. Lindskog, RiskLab) 7
Properties of rank correlation Let X and Y be continuous random variables with copula C, andletδ denote Kendall s tau or Spearman s rho. The following properties are not shared by linear correlation. If T is strictly monotone, then δ(t (X),Y) =δ(x, Y ), T increasing, δ(t (X),Y) = δ(x, Y ), T decreasing. δ(x, Y )=1 C = M 2 δ(x, Y )= 1 C = W 2 δ(x, Y ) depends only on the copula of (X, Y ). Given a rank correlation matrix there is always a multivariate distribution with this rank correlation matrix, regardless of the choice of margins. This is not true for linear correlation. c 2000 (F. Lindskog, RiskLab) 8
Tail dependence Let (X, Y ) be a random vector with continuous margins F and G and copula C. The coefficient of upper tail dependence of (X, Y )is λ U := lim P[Y > G 1 (u) X >F 1 (u)] u 1 = lim C(u, u)/(1 u) u 1 provided that the limit λ U [0, 1] exists. If λ U > 0, then (X, Y ) has upper tail dependence. If lim C(u, u)/u = λ L > 0 u 0 exists, then (X, Y ) has lower tail dependence. Tail dependence is a copula property. c 2000 (F. Lindskog, RiskLab) 9
Gaussian X1 Y1 0 10 20 30 40 0 10 20 30 40 Gumbel X2 Y2 0 10 20 30 40 0 10 20 30 40 Two bivariate distributions with standard lognormal margins and Kendall s tau 0.7, but different dependence structures. Gumbel copulas have upper tail dependence, but Gaussian copulas have not. c 2000 (F. Lindskog, RiskLab) 10
Elliptical copulas A spherical distribution is an extension of the multivariate normal distribution N n (0, I n ) and an elliptical distribution is an extension of N n (µ, Σ). Recall that N n (µ, Σ) can be defined as the distribution of X = µ + AY, where Y N n (0, I n )andσ=aa T. A random vector X has a spherical distribution if X = d RU for some positive random variable R independent of the random vector U uniformly distributed on the unit hypersphere. A random vector X has an elliptical distribution with parameters µ and Σ if X = d µ + AY, where Y is spherical and Σ = AA T. c 2000 (F. Lindskog, RiskLab) 11
Let Z =(Z 1,...,Z n ) N n (0, Σ) with nondegenerate margins. 1 (Φ(Z 1)),...,Fn 1 (Φ(Z n)) has the Gaussian copula (F 1 Cρ Ga (u) =Φ n ρ(φ 1 (u 1 ),...,Φ 1 (u n )), where ρ is the linear correlation matrix corresponding to the covariance matrix Σ. Let X = ν Z, where Z and S χ 2 S ν independent. (F 1 1 (t ν(x 1 )),...,F 1 n (t ν (X n )) has the t ν -copula are Cν,ρ(u) t =t n ν,ρ(t 1 ν (u 1 ),...,t 1 ν (u n )). Cν,ρ t has upper and lower tail dependence. For all elliptical copulas τ(i, j) = 2 π arcsin ρ(i, j). c 2000 (F. Lindskog, RiskLab) 12
Gaussian X1 Y1-4 -2 0 2 4-4 -2 0 2 4 t X2 Y2-4 -2 0 2 4-4 -2 0 2 4 Two bivariate distributions with standard normal margins and τ =0.6. Gaussian and t 2 -copulas. c 2000 (F. Lindskog, RiskLab) 13
Archimedean copulas Let ϕ be a continuous strictly decreasing convex function from [0, 1] to [0, ] such that ϕ(1) = 0. Then ϕ [ 1] (ϕ(u)+ϕ(v)), u,v [0, 1], is a copula with generator ϕ, where ϕ [ 1] (t) = { ϕ 1 (t), t ϕ(0), 0, t ϕ(0). If ϕ(0) =, thenϕ [ 1] = ϕ 1. Gumbel copula: Take ϕ(t) =( ln t) θ with θ [1, ), C θ (u, v) = exp( [( ln u) θ +( ln v) θ ] 1/θ ) Clayton copula: Take ϕ(t) =(t θ 1)/θ with θ [ 1, )\{0}, C θ (u, v) =max([u θ + v θ 1] 1/θ, 0) c 2000 (F. Lindskog, RiskLab) 14
Gumbel X1 Y1 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 Clayton X2 Y2 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 Two bivariate distributions with Gamma(3, 1) margins and τ =0.5. Gumbel copulas have upper tail dependence, Clayton copulas have lower tail dependence. c 2000 (F. Lindskog, RiskLab) 15
Risky dependence Risks X 1,...,X n. Thresholds k 1,...,k n. Default occurs for company i if X i >k i. What is the probability of at least l defaults? Assume historical data are available allowing estimation of marginal distributions, pairwise rank correlations. } N = { i {1,...,n} Xi >k i is the number of defaults. c 2000 (F. Lindskog, RiskLab) 16
The probability of all companies defaulting is given by P(N = n) = H(k 1,...,k n ) = Ĉ(F 1 (k 1 ),...,F n (k n )), where Ĉ is the survival copula of (X 1,...,X n ). We can evaluate P(N l) for various copulas. Illustration: l = n =3; X i Lognormal(0, 1); k i = k for all i; τ(x i,x j )=0.5 foralli j. We compare trivariate Gaussian and Gumbel copulas and use the relations ρ =sin(πτ/2) and θ = 1 1 τ to parametrisize the respective copulas so that they have a common Kendall s tau rank correlation matrix. c 2000 (F. Lindskog, RiskLab) 17
Probability of joint default 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 5 Threshold 0.0 0.010 0.020 0.030 6 8 10 12 14 Threshold Note that for k =5 VaR 0.95 (X i ) P Gumbel {N =3} P Gaussian {N =3} 2 and that for k =10 VaR 0.99 (X i ) P Gumbel {N =3} P Gaussian {N =3} 4. c 2000 (F. Lindskog, RiskLab) 18
Modelling Dependent Credit Risks Copula Ideas Part 1 I II Dependence concepts, copula families Example - risky dependence Part 2 Common Poisson Shock Models III IV Model description Example - large loan portfolios c 2000 (F. Lindskog, RiskLab) 19
Independent Poisson shock processes {N (e) (t),t 0}, e =1,...,m, with intensities λ (e) > 0. N (e) (t) is the number of shocks of type e during the time period [0,t]. At the rth occurrence of shock e the indicators I (e) j,r Be(p(e) j ) indicates whether a loss of type j {1,...,n} occurs. I (e) r =(I (e) 1,r,...,I(e) n,r ) for r =1,...,N (e) (t) are considered to be independent and identically distributed with a multivariate Bernoulli distribution. For higher order marginal probabilities of we use the notation I (e) r P (I (e) j = i j,i (e) k = i k )=p (e) j,k (i j,i k ). c 2000 (F. Lindskog, RiskLab) 20
The processes The loss counters {N j (t),t 0}, j =1,...,n, counts losses of type j occurring in [0,t]. The counting processes for shock events and losses are thus linked by N j (t) = m e=1 N (e) (t) r=1 I (e) j,r. The process {(N 1 (t),...,n n (t)),t 0} is multivariate Poisson with E(N j (t)) = t m e=1 λ (e) p (e) j. The covariance structure is given by cov ( N j (t),n k (t) ) = t m e=1 λ (e) p (e) j,k (1, 1). c 2000 (F. Lindskog, RiskLab) 21
The total number of losses The total number of losses N(t) = n j=1 N j (t) is not Poisson but rather compound Poisson. Moments The moments of N(t) are given recursively by E (N(t) p ) = t where p 1 k=0 ( p 1) ( ϕ(k +1)E N(t) p k 1 ) k and ϕ(k) = E(N(t)) = t n j 1 =1 n n j k =1 e=1 m j=1 e=1 m λ (e) p (e) j λ (e) p (e) j 1,...,j k (1,...,1). Probability distribution The probabilities P (N(t) k) can be calculated using Panjer recursion. c 2000 (F. Lindskog, RiskLab) 22
Times to first losses Let T j =inf{t : N j (t) > 0}. (T 1,...,T n ) has a multivariate exponential distribution whose survival copula is a Marshall-Olkin copula. P (T i >t i,t j >t j )=C αi,α j (F i (t i ), F j (t j )) where and F i (t i ) = exp t m e=1 λ (e) p (e) i C αi,α j (u, v) =min(u 1 α iv, uv 1 α j) with me=1 λ (e) p (e) i,j (1, 1) me=1 λ (e) p (e) i,j (1, 1) α i = me=1 λ (e) p (e),α j = me=1 i λ (e) p (e). j Linear and Kendall s tau correlation structure: α i α j ρ(t i,t j )=τ(t i,t j )=. α i + α j α i α j c 2000 (F. Lindskog, RiskLab) 23
Large loan portfolios Consider a loan portfolio consisting of n = 100000 obligors, divided into sectors and rating categories. We will consider a global shock process which could be thought of as recessions in the world economy; sector shock processes which represent adverse economic conditions for specific industries or geographical sectors; idiosyncratic bad management shock processes for each obligor. c 2000 (F. Lindskog, RiskLab) 24
Simplifying notation Denote the conditional default probability for an l-rated k-sector obligor by s k l given a k-sector shock and g k l given a global shock. For the jth obligor belonging to sector k = k(j) and rating class l = l(j) λ j = λ l(j) idio + sk(j) l(j) λk(j) sector + gk(j) l(j) λ global. Companies with the same rating l have the same overall default intensity λ l total where λ j = λ l(j) total. Let n l,k denote the number of companies in rating class l and sector k. c 2000 (F. Lindskog, RiskLab) 25
Parameter assumptions t =1year; two rating classes, four sectors; λ 1 total =0.005; λ 2 total =0.02; The sizes n l,k of the 8 homogeneous groups are n 1,1 = 10000, n 1,2 = 20000, n 1,3 = 15000, n 1,4 = 5000, n 2,1 = 10000, n 2,2 = 25000, n 2,3 = 10000, n 2,4 = 5000. The default indicators for a group of companies are conditionally independent given the occurrence of a global or sector shock. c 2000 (F. Lindskog, RiskLab) 26
Case 1 For both rating classes we attribute 40% of defaults to idiosyncratic shocks: λ l idio /λl total =0.4. To sector specific causes: 20% of defaults of sector 1 obligors, 50% of defaults of sector 2 obligors, 10% of defaults of sector 3 obligors, 40% of defaults of sector 4 obligors: s k l λk sector /λl total =0.2, 0.5, 0.1, 0.4. Moreover we believe that the frequencies of sector and global shocks are in the ratio λ 1 sector : λ2 sector : λ3 sector : λ4 sector : λ global =1:5:2:4:1. c 2000 (F. Lindskog, RiskLab) 27
The model is specified up to a factor f>0: Shock intensities: (λ 1 sector,λ2 sector,λ3 sector,λ4 sector,λ global) = f(0.2, 1.0, 0.4, 0.8, 0.2) Univariate conditional default probabilities given sector shocks: (s 1 1,s2 1,s3 1,s4 1,s1 2,s2 2,s3 2,s4 2 ) = 1 (0.5, 0.25, 0.125, 0.25, 2, 1, 0.5, 1)10 2 f Univariate conditional default probabilities given global shocks: (g 1 1,g2 1,g3 1,g4 1,g1 2,g2 2,g3 2,g4 2 ) = 1 f (1, 0.25, 1.25, 0.5, 4, 1, 5, 2)10 2. How important is the choice of f? c 2000 (F. Lindskog, RiskLab) 28
0 200 400 600 800 0 100 300 500 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 0 100 300 500 0 200 400 600 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 Density estimates (histograms) plotted by row for f =1, 2, 4, 8. c 2000 (F. Lindskog, RiskLab) 29
Case 2 Suppose we are able to quantify the probabilities with which sector or global shocks cause the default of individual obligors. Set (s 1 1,s2 1,s3 1,s4 1,s1 2,s2 2,s3 2,s4 2 ) =(0.25, 0.08, 0.05, 0.1, 1, 0.3, 0.25, 0.25)10 2 and (g1 1,g2 1,g3 1,g4 1,g1 2,g2 2,g3 2,g4 2 ) =(0.25, 0.1, 0.4, 0.1, 1, 0.5, 1.5, 1)10 2. We vary the shock intensities (λ 1 idio,λ2 idio,λ1 sector,λ2 sector,λ3 sector,λ4 sector,λ global) progressively in the following way (0.005, 0.02, 0.0, 0.0, 0.0, 0.0, 0.0) (0.004, 0.016, 0.2, 1.0, 0.4, 0.8, 0.2) (0.002, 0.008, 0.6, 3.0, 1.2, 2.4, 0.6) (0.0, 0.0, 1.0, 5.0, 2.0, 4.0, 1.0). c 2000 (F. Lindskog, RiskLab) 30
Hence we start with the special case of no common shocks and a situation where the individual default counters N j (1) are independent Poisson and the total number of defaults N(1) is Poisson. In the second model we still attribute 80% of the default intensities λ j to idiosyncratic shocks, but we now have 20% in common shocks. In the third model we have 40% in idiosyncratic and 60% in common shocks. In the final model we have only common shocks. c 2000 (F. Lindskog, RiskLab) 31
0 1000 2000 3000 4000 0 200 600 1000 1400 0 1000 2000 3000 4000 0 1000 2000 3000 4000 0 100 300 500 0 100 200 300 0 1000 2000 3000 4000 0 1000 2000 3000 4000 The effect of increasing the intensities of common shocks while decreasing the intensities of idiosyncratic shocks when the univariate conditional default probabilities are held fixed. c 2000 (F. Lindskog, RiskLab) 32