Financial Risk Management Professor: Thierry Roncalli Evry University Assistant: Enareta Kurtbegu Evry University Tutorial exercices #4 1 Correlation and copulas 1. The bivariate Gaussian copula is given by the following expression: C (u 1, u ) = Φ ( Φ 1 (u 1 ), Φ 1 (u ) ; ρ ) where Φ (x) is the distribution function of univariate standard normal distribution while Φ (x, y; ρ) is the distribution function of the bivariate standard normal distribution with correlation ρ. Let (X 1, X ) be a random Gaussian vector of distribution Φ. Show that the copula of (X 1, X ) is the same with the one of the random vector (Φ (X 1 ), Φ (X )). Deduce an algorithm to simulate the Gaussian copula of parameter ρ.. Let us consider a random vector (R 1, R ) corresponding to the returns of two assets. We assume that R 1 = σ 1 X 1 and R = σ X with (X 1, X ) the random Gaussian vector of distribution Φ and of parameter ρ. Show that the linear correlation of (R 1, R ) is equal to ρ. Let S 1 and S be the normalized asset prices. We have S 1 = e R 1 and S = e R. Show that the linear correlation of (S 1, S ) is equal to 1 if and only if the linear correlation of (R 1, R ) is equal to 1 and if the volatility σ 1 is equal to the volatility σ. Comment this result in the case of Black-Scholes model. 3. Let it be X 1 N (µ 1, σ 1 ) and X N (µ, σ ). We consider that the copula C X1,X is an ordinal sum of copulas C and C + of parameter θ. (a) We assume that µ 1 = µ = 0 and σ 1 = σ = 1. Specify the copula C X1,X such that the linear correlation of X 1 and X is null. Show that there exists a function f such that X 1 = f (X ). Comment this result. (b) Calculate the linear correlation of X 1 and X as a function of the parameters µ 1, µ, σ 1, σ and θ. (c) Propose a method of moments to estimate θ. The exponential distribution 1. Let it be τ E (λ). Show that: Pr {τ > t τ > s} = Pr {τ > t s} with t > s. Interpret this result. Which is its interest in modeling the credit risk?. Let τ 1,..., τ n n be the random variables of distribution E (λ i ). (a) Calculate the distribution of min (τ 1,..., τ n ) et max (τ 1,..., τ n ) in the independent case. (b) Same question if the random variables τ 1,..., τ n are co-monotone. 1
(c) We place ourselves in the independent case. Let us consider τ = min (τ 1,..., τ n ). Show that: Pr {τ = τ i } = λ i n j=1 λ j 3. Let us consider two exponential default times τ 1 and τ of parameter λ 1 and λ. (a) Show that if the dependence function (τ 1, τ ) is C +, then the following relation is true: τ 1 = λ λ 1 τ (b) Show that there exists a function f such that τ 1 = f (τ ) is the dependence function of (τ 1, τ ) is C. (c) We denote ρ the coefficient of linear correlation of (τ 1, τ ). Show that: 1 < ρ 1 (d) In a more general case, show that the correlation coefficient ρ of a random vector (X 1, X ) can not be equal to 1 if the support of the random variables X 1 and X is [0, + ]. 3 Copula functions 1. Let us consider the function: (a) Show that C is a copula 1 for θ [ 1, 1]. C (u 1, u ) = u 1 u (1 + θ (1 u 1 ) (1 u )) (b) Calculate the tail dependence coefficient λ of this copula, the Kendall tau and the Spearman rho as well. (c) Let us consider X 1 N (µ, σ) and X E (λ). We assume that the copula of the random vector (X 1, X ) is the previous function. Propose an algorithm to simulate the random vector (X 1, X ). { (x ) } i (d) Calculate the log-likelihood of the sample 1, x i i=n. i=1. Let S be the bivariate function defined by: with θ [0, 1], x 1 0 et x 0. S (x 1, x ) = exp (a) Show that S is a survival distribution. (b) Define the survival copula associated tos. ( ( x 1 + x θ x )) 1x x 1 + x 3. We recall that an Archimedean copula has the following expression: where φ is a function called generator. C (u 1, u ) = φ 1 (φ (u 1 ) + φ (u )) (a) Under which conditions for φ should we be, in a way C to be a copula? 1 This copula is known with the name of Farlie-Gumbel-Morgenstern (see Nelsen, 1999, page 68).
(b) Are the following copulas C, C and C + Archimedean? If yes, give the corresponding generators. (c) Which is the copula associated to generator φ (u) = ( ln u) θ where θ 1? Show that thos copula is a copula of extreme values. (d) Calculate the conditional distribution C 1 associated to the Archimedean copula. Deduce an algorithm to simulate the Archimedean copula. 4. Let (X 1, X ) be a standard Gaussian vector of correlation ρ. (a) Calculate the distribution of X knowing that X 1 = x 1. (b) Deduce an expression of the copula C associated to (X 1, X ). Calculate the conditional distribution C 1. Deduce an algorithm to simulate the copula C. 5. Which is the property of a copula of extreme values? Show that C and C + are copulas of extreme values, while that is not the case for C. 4 Calculation of the upper/lower bounds of correlation 1. Give the mathematical definitions of the copulas C, C and C +.. Define the Normal bivariate copula C (ρ) of correlation ρ. 3. Which are the probabilistic interpretations of the three copulas defined in question 1? Deduce that C (ρ= 1) = C, C (ρ=0) = C and C (ρ=1) = C +. 4. Let us consider the random vector (τ, LGD) which model the joint law of default τ and the loss given default LGD of a counterpart. We assume that τ E λ and LGD U [0,1]. (a) Show that the dependence of (τ, LGD) is maximal while: LGD +e λτ 1 = 0 (b) Show that the correlation ρ (τ, LGD) verifies the following inequality: (c) Comment these results. ρ (τ, LGD) 3 5 The generalized exponential model 1. We denote F and S the distribution function and the survival function of the random variable τ. Define the function S (t) and deduce the expression of the associated density function f (t).. Give the definition of hazard rate λ (t). Deduce that the exponential model corresponds to a particular case λ (t) = λ. 3. How could we simulate the random variable τ in the case when λ (t) = λ. 4. Let us assume now that: λ 1 if t 3 λ (t) = λ if 3 < t 5 λ 3 if t > 5 Give the expression of the survival distribution S (t). Deduce the expression of the density function f (t). Verify that: f (t) S (t) = constant 3
5. We assume that the interest rate is constant r. We recall that in the case of CDS whose margin is payed in ongoing basis, the spread of CDS is equal to: s = (1 R) T 0 e rt f (t) dt T 0 e rt S (t) dt where S and f are the survival and density functions of default time τ associated to CDS, R is the recovery rate and T is the CDS maturity. Give the triangular equality while τ E λ. 6. Let us consider that τ is the generalized exponential model of question 4. We assume that there exist three market spreads CDS s 1, s and s 3 whose respective maturity are 3 years, 5 years and 7 years. Show that the calibration of the generalized exponential model implies to solve a system of 3 equations and 3 unknowns λ 1, λ and λ 3. Which is the name of this calibration method? 6 Risk contributions We denote L the loss of a portfolio of n credits, and x i the exposure to default of the i-th credit. We have: n L = x e = x i e i with e i the unitary loss of the i-th credit. We denote F the distribution function of L. 1. Let us assume that e = (e 1,..., e n ) N (0, Σ). Calculate the value at risk of confidence level α. i=1. Deduce the marginal value-at-risk of the i-th credit. Define the risk contribution of the i-th credit. 3. Verify that the value at risk is equal to: Interpret this result. VaR x i = E [ e i L = F 1 (α) ] 4. Let us consider the Bâle II model of credit risk.. We have: e i = LGD i D i where D i = 1 {τ i < M i } is the default indicator and the M i is the maturity of the i-th credit. Which are the necessary conditions to be satisfied to obtain the following result: E [ e i L = F 1 (α) ] = E [LGD i ] E [ D i L = F 1 (α) ] 5. We assume that the default occurs before the maturity M i if the latent variable Z i falls below a certain level B i : τ i M i Z i B i We model Z i = ρx + 1 ρε i with Z i, X and ε i three independent standard normal random variables. X is the factor (or the systemic risk) and ε i is the individual. Calculate the conditional probability of default. 6. Show that in the Bâle II model, we have: E [ e i L = F 1 (α) ] = E [LGD i ] E [ D i X = Φ 1 (1 α) ] 7. Deduce the expression of the risk contribution of the i-th credit in the Bâle II model. 4
8. We assume that the portfolio is homogeneous, which means that the credits have the same exposure to default, the same distribution of loss given default and the same default probability. Using the following result: c Φ(a + bx)ϕ(x) dx = Φ (c, a 1 + b ; ) b 1 + b with Φ (x, y; ρ) the distribution function of the bivariate Gaussian distribution of correlation ρ in the domain [, x] [, y], calculate the expected shortfall in the case of Bâle II model. Comment this result. 5