Simulating Exchangeable Multivariate Archimedean Copulas and its Applications. Authors: Florence Wu Emiliano A. Valdez Michael Sherris

Transcription

1 Simulating Exchangeable Multivariate Archimedean Copulas and its Applications Authors: Florence Wu Emiliano A. Valdez Michael Sherris

2 Literatures Frees and Valdez (1999) Understanding Relationships Using Copulas Whelan, N. (2004) Sampling from Archimedean Copulas Embrechts, P., Lindskog, and A. McNeil (2001) Modelling Dependence with Copulas and Applications to Risk Management

3 This paper: Extending Theorem in Nelson (1999) to multi-dimensional copulas Presenting an algorithm for generating Exchangeable Multivariate Archimedean Copulas based on the multi-dimensional version of theorem Demonstrating the application of the algorithm

4 Exchangeable Archimedean Copulas One parameter Archimedean copulas Archimedean copulas a well known and often used class characterised by a generator, φ(t) Copula C is exchangeable if it is associative C(u,v,w) = C(C(u,v),w) = C(u, C(v,w)) for all u,v,w in I.

5 Archimedean Copulas Charateristics of the generator φ(t): ϕ(1) = 0 is monotonically decreasing; and is convex (ϕ exists and ϕ 0). If ϕ exists, then ϕ 0 C(u 1,,u n ) = ϕ -1 (ϕ(u 1 ) + + ϕ(u n ))

6 Archimedean Copulas - Examples Gumbel Copula ϕ(t) = (-log(u)) 1/θ ϕ -1 (t) = exp(-u θ ) Frank Copula ϕ(t) = - log((e -θt 1)/(e -θ 1) ϕ -1 (t) = - log(1 (1 - e -θ )e -t ) /log(θ)

7 Theorem Let (U 1,U 2 ) be a bivariate random vector with uniform marginals and joint distribution function defined by Archimedean copula C(u 1,u 2 ) = ϕ - 1 (ϕ(u 1 ) + ϕ(u 2 )) for some generator ϕ. Define the random variables S = ϕ(u 1 )/(ϕ(u 1 ) + ϕ(u 2 )) and T = C(u 1,u 2 ). The joint distribution function of (S,T) is characterized by H(s,t) = P(S s, T t) = s K C (t) where K C (t) = t ϕ(t)/ ϕ (t).

8 Simulating Bivariate Copulas Algorithm for generating bivariate Archimedean copulas (refer Embrecht et al (2001): Simulate two independent U(0,1) random variables, s and w. Set t = K -1 C (w) where K C (t) = t ϕ(t)/ ϕ (t). Set u 1 = ϕ -1 (s ϕ(t)) and u 2 = ϕ -1 ((1-s) ϕ(t)). x 1 = F 1-1 (u 1 ) and x 2 = F 2-1 (u 2 ) if inverses exist. (F 1 and F 2 are the marginals).

9 Theorem for Multi-dimensional Archimedean Copulas (1) Let (U 1,,U n ) be an n-dimensional random vector with uniform marginals and joint distribution function defined by the Archimedean copula C(u 1,,u n ) = ϕ -1 (ϕ(u 1 ) + + ϕ(u n )) or some generator ϕ. Define the n tranformed random variables S 1,,S n-1 and T, where S k = (ϕ(u 1 ) + + ϕ(u k )) / (ϕ(u 1 ) + + ϕ(u k+1 )) T = C(u 1, u n ) = ϕ -1 (ϕ(u 1 ) + + ϕ(u n ))

10 Theorem for Multi-dimensional Archimedean Copulas (2) The joint density distribution for S 1,,S n-1 and T can be defined as follows. h(s 1,s 2,,s n,t) = J c(u 1, u n ) or h(s 1,s 2,,s n-1,t) = s 10 s 21 s 32. s n-2 n-1 ϕ -1(n) (t)[ϕ(t)] /ϕ (t) Hence S 1,,S n-1 and T are independent, and 1. S 1 and T are uniform; and 2. S 2,,S n-1 each have support in (0,1).

11 Theorem for Multi-dimensional Archimedean Copulas (3) Distribution functions for S k: Corollary: The density for S k for k = 1,2, n-1 is given by f Sk (s) = ks k-1, for s (0,1) The distribution functions for S k can be written as: F Sk (s) = s k, for s (0,1) Corollary: The marginal density for T is given by: f T (t) = ϕ -1(n) (t)[ϕ(t)] n-1 ϕ (t) for t (0,1)

12 Algorithm for simulating multidimensional Archimedean Copulas 1. Simulate n independent U(0,1) random variables, w 1, w n. 2. For k = 1,2,, n-1, set s k =w k 1/k 3. Set t = F T -1 (w n ) 4. Set u 1 = ϕ -1 (s 1 s n-1 ϕ(t)), u n = ϕ -1 ((1-s n-1 ) ϕ(t)) and for k = 2,,n, u k = ϕ -1 ((1-s k-1 )Πs j ϕ(t). 5. x k = F -1 k (u k ) for k = 1,,n.

13 Example: Multivariate Gumbel Copula Gumbel Copulas ϕ(u) = (-log(u)) 1/θ ϕ -1 (u) = exp(-u θ ) ϕ -1(k) = (-1) k θexp(-u θ )u -(k+1)/ θ Ψ k-1 (u θ ) Ψ k (x) = [θ(x-1) + k] Ψ k-1 (x) - θ Ψ k-1 (x) Recursive with Ψ 0 (x) = 1.

14 Example: Gumbel Copula (Kendall Tau 0.5, Theta =2)

15 Example: Gumbel Copula (3) Normal vs Lognormal vs Gamma

16 Application: VaR and TailVaR (1) Insurance portfolio Contains multiple lines of business, with tail dependence Copulas Gumbel copula distributions have heavy right tails Frank copula lower tail dependence than Gumbel at the same level of dependence Economic Capital: VaR/TailVaR VaR: the k-th percentile of the total loss TailVaR: the conditional expectation of the total loss at a given level of VaR (or E(X X VaR))

17 Application: VaR and TailVaR (2) Density of Gumbel Copulas Density of Frank Copulas

18 Application: VaR and TailVaR (3) Assumptions: Lines of business: 4 Kendall s tau = 0.5 (linear correlation = 0.7) theta = 2 for Gumbel copula theta = 5.75 for Frank copula Mean and variance of marginals are the same

19 Application: VaR and TailVaR (4) Gumbel Frank

20 Application: VaR and TailVaR (5) Gumbel copula has higher TailVaR s than Frank copula for Lognormal and Gamma marginals Lognormal has the highest TailVaR and VaR at both 95% and 99% confidence level.

21 Application: VaR and TailVaR (6) Gumbel 1.4 Frank The Case of LogNormal Distribution The Case of LogNormal Distribution The Case of Gamma Distribution The Case of Gamma Distribution

22 Application: VaR and TailVaR (7) Impact of the choice of Kendall s correlation on VaR and TailVaR

23 Conclusion Derived an algorithm for simulating multidimensional Archimedean copula. Applied the algorithm to assess risk measures for marginals and copulas often used in insurance risk models. Copula and marginals have a significant effect on economic capital