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

Two random variables have Kendall's tau of 0.5 and are linked by a Gumbel copula with parameter alpha, where tau = 1 - 1/alpha. Which statement is correct?

Solving 0.5 = 1 - 1/alpha gives alpha = 2. The Gumbel copula has upper tail dependence 2 - 2^(1/alpha), which is 2 - sqrt(2), about 0.586, and no lower tail dependence. Independence would need alpha = 1.

  1. Aalpha = 1.5 and the variables are independent
  2. Balpha = 2 and the upper tail dependence coefficient is 2 - 2^(1/2)Correct
  3. Calpha = 2 and the lower tail dependence coefficient is 2 - 2^(1/2)
  4. Dalpha = 0.5 and the upper tail dependence coefficient is 0
  5. alpha = 2 and the upper tail dependence coefficient is 0.5

Explanation

From tau = 1 - 1/alpha, 0.5 = 1 - 1/alpha gives alpha = 2. Gumbel upper tail dependence is 2 - 2^(1/alpha) = 2 - sqrt(2), about 0.586. Lower tail dependence is zero for the Gumbel, and independence corresponds to alpha = 1, not 2.

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