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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Introduction to copulas

A Gumbel copula C(u,v) = exp(-[(-ln u)^θ + (-ln v)^θ]^(1/θ)) with θ = 2 is used for two Indian insurer's catastrophe losses. The upper tail dependence coefficient of a Gumbel copula is 2 - 2^(1/θ). What is the value here, and which tail shows dependence?

The upper tail dependence coefficient is 2 - √2 ≈ 0.586, and the Gumbel copula has upper tail dependence only, with none in the lower tail. This makes it suitable for joint extreme losses such as catastrophes.

  1. A0.586, in the upper tail, with no lower tail dependence
  2. B0.586, in the lower tail, with no upper tail dependence
  3. C0.414, in the upper tail, with no lower tail dependenceCorrect
  4. D0.414, in both tails equally
  5. 0.000, in neither tail

Explanation

With θ = 2, 2^(1/2) = 1.4142, so the coefficient is 2 - 1.4142 = 0.5858. Check: 2 - 1.4142 = 0.5858, not 0.414. So the value is 0.586 and it is upper tail only; the Gumbel has no lower tail dependence.

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