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

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”

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

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.

Archimedean Copulas Charateristics of the generator φ(t): –  (1) = 0 – is monotonically decreasing; and – is convex (  ’ exists and  ’  0). If  ’’ exists, then  ’’  0 C(u 1,…,u n ) =  -1 (  (u 1 ) + … +  (u n ))

Archimedean Copulas - Examples Gumbel Copula –  (t) = (-log(u)) 1/  –  -1 (t) = exp(-u  ) Frank Copula –  (t) = - log((e -  t – 1)/(e -  – 1) –  -1 (t) = - log(1 – (1 - e -  )e -t ) /log(  )

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 ) =  - 1 (  (u 1 ) +  (u 2 )) for some generator . Define the random variables S =  (u 1 )/(  (u 1 ) +  (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 –  (t)/  ’(t).

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 C -1 (w) where K C (t) = t –  (t)/  ’(t). Set u 1 =  -1 (s  (t)) and u 2 =  -1 ((1-s)  (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).

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 ) =  -1 (  (u 1 ) + … +  (u n )) or some generator . Define the n tranformed random variables S 1,…,S n-1 and T, where S k = (  (u 1 ) + … +  (u k )) / (  (u 1 ) + … +  (u k+1 )) T = C(u 1,…u n ) =  -1 (  (u 1 ) + … +  (u n ))

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 1 0 s 2 1 s 3 2 …. s n-1 n-2  -1(n) (t)[  (t)] /  ’ (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).

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 S k (s) = ks k-1, for s  (0,1) The distribution functions for S k can be written as: F S k (s) = s k, for s  (0,1) Corollary: The marginal density for T is given by: f T (t) =  -1(n) (t)[  (t)] n-1  ’(t) for t  (0,1)

Algorithm for simulating multi- dimensional 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 =  -1 (s 1 …s n-1  (t)), u n =  -1 ((1-s n-1 )  (t)) and for k = 2,…,n, u k =  -1 ((1-s k-1 )  s j  (t). 5. x k = F k -1 (u k ) for k = 1,…,n.

Example: Multivariate Gumbel Copula Gumbel Copulas –  (u) = (-log(u)) 1/  –  -1 (u) = exp(-u  ) –  -1(k) = (-1) k  exp(-u  )u -(k+1)/   k-1 (u  ) –  k (x) = [  (x-1) + k]  k-1 (x) -   ’ k-1 (x) Recursive with  0 (x) = 1.

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

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

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))

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

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

Application: VaR and TailVaR (4) Frank Gumbel

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.

Application: VaR and TailVaR (6) Gumbel Frank

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

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