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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)^k + (-ln v)^k]^(1/k)) with k >= 1, is fitted to the joint claims of two lines of business. Which feature describes it correctly?

The Gumbel copula exhibits upper tail dependence with coefficient 2 - 2^(1/k), positive when k exceeds 1. It has no lower tail dependence, and at k = 1 it reduces to independence, making it suitable for joint large losses.

  1. AIt has lower tail dependence only, with coefficient 2^(-1/k)
  2. BIt has no tail dependence for any k
  3. CIt has upper tail dependence with coefficient 2 - 2^(1/k), which is positive for k > 1Correct
  4. DIt has symmetric tail dependence equal to that of a t copula
  5. It reduces to the comonotonic copula when k = 1

Explanation

The Gumbel copula has upper tail dependence 2 - 2^(1/k), positive when k > 1 and zero at k = 1. At k = 1 it becomes the independence copula uv, not the comonotonic one, and its lower tail dependence is zero.

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