Simulating Exchangeable Multivariate Archimedean Copulas and its Applications

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

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This paper

• Extending Theorem 4.3.7 in Nelson (1999) to
  multi-dimensional copulas
• Presenting an algorithm for generating
  Exchangeable Multivariate Archimedean Copulas
  based on the multi-dimensional version of theorem
  4.3.7
• Demonstrating the application of the algorithm

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Exchangeable Archimedean Copulas

• One parameter Archimedean copulas
• Archimedean copulas a well known and often used
  class characterised by a generator, f(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.

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Archimedean Copulas

• Charateristics of the generator f(t)
• ?(1) 0
• is monotonically decreasing and
• is convex (? exists and ?? 0). If ? exists,
  then ? ? 0
• C(u1,,un) ?-1(?(u1) ?(un))

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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(?)

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Theorem 4.3.7

• Let (U1,U2) be a bivariate random vector with
  uniform marginals and joint distribution function
  defined by Archimedean copula C(u1,u2) ?
  -1(?(u1) ?(u2)) for some generator ?. Define
  the random variables S ?(u1)/(?(u1) ?(u2))
  and T C(u1,u2). The joint distribution function
  of (S,T) is characterized by
• H(s,t) P(S ? s, T ? t) s KC(t)
• where KC(t) t ?(t)/ ?(t).

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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 KC-1 (w) where KC(t) t ?(t)/ ?(t).
• Set u1 ? -1(s ?(t)) and u2 ? -1((1-s) ?(t)).
• x1 F1-1(u1) and x2 F2-1(u2) if inverses
  exist. (F1 and F2 are the marginals).

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Theorem for Multi-dimensional Archimedean Copulas
(1)

• Let (U1,,Un) be an n-dimensional random vector
  with uniform marginals and joint distribution
  function defined by the Archimedean copula
• C(u1,,un) ? -1(?(u1) ?(un)) or some
  generator ?.
• Define the n tranformed random variables
  S1,,Sn-1 and T, where
• Sk (?(u1) ?(uk)) / (?(u1) ?(uk1))
• T C(u1,un) ? -1(?(u1) ?(un))

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Theorem for Multi-dimensional Archimedean Copulas
(2)

• The joint density distribution for S1,,Sn-1 and
  T can be defined as follows.
• h(s1,s2,,sn,t) J c(u1,un)
• or
• h(s1,s2,,sn-1,t) s10s21s32. sn-1n-2 ?
  -1(n)(t)?(t) /?(t)
• Hence S1,,Sn-1 and T are independent, and
• S1and T are uniform and
• S2,,Sn-1 each have support in (0,1).

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Theorem for Multi-dimensional Archimedean Copulas
(3)

• Distribution functions for Sk
• Corollary The density for Sk for k 1,2,n-1 is
  given by
• fSk(s) ksk-1, for s ? (0,1)
• The distribution functions for Sk can be written
  as
• FSk(s) sk , for s ? (0,1)
• Corollary The marginal density for T is given
  by
• fT(t) ?-1(n)(t)?(t)n-1 ?(t) for t ? (0,1)

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Algorithm for simulating multi-dimensional
Archimedean Copulas

• Simulate n independent U(0,1) random variables,
  w1,wn.
• For k 1,2,, n-1, set skwk1/k
• Set t FT-1 (wn)
• Set u1 ? -1(s1sn-1?(t)), un ? -1((1-sn-1)
  ?(t)) and for k 2,,n , uk ?
  -1((1-sk-1)?sj?(t).
• xk Fk-1(uk) for k 1,,n.

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Example Multivariate Gumbel Copula

• Gumbel Copulas
• ?(u) (-log(u))1/?
• ?-1(u) exp(-u?)
• ?-1(k) (-1)k ?exp(-u?)u-(k1)/ ? ?k-1(u?)
• ?k (x) ?(x-1) k ?k-1 (x) - ? ?k-1 (x)
• Recursive with ?0 (x) 1.

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Example Gumbel Copula (Kendall Tau 0.5, Theta 2)
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Example Gumbel Copula (3)

• Normal vs Lognormal vs Gamma

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ApplicationVaR 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))

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Application VaR and TailVaR (2)

• Density of Gumbel Copulas

• Density of Frank Copulas

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Application VaR and TailVaR (3)

• Assumptions
• Lines of business 4
• Kendalls 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

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Application VaR and TailVaR (4)

• Frank

• Gumbel

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Application VaR and TailVaR (5)

• Gumbel copula has higher TailVaRs than Frank
  copula for Lognormal and Gamma marginals
• Lognormal has the highest TailVaR and VaR at both
  95 and 99 confidence level.

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Application VaR and TailVaR (6)

• Gumbel

• Frank

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Application VaR and TailVaR (7)

• Impact of the choice of Kendalls correlation on
  VaR and TailVaR

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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