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

Losses X and Y are independent with continuous marginals. A modeller writes their joint distribution function using Sklar's theorem. Which copula results, and what is P(X ≤ x, Y ≤ y) when F_X(x)=0.4 and F_Y(y)=0.5?

Independence corresponds to the product copula C(u,v)=uv. The joint probability is therefore 0.4 multiplied by 0.5, which equals 0.20. Other copulas such as the comonotonic one would give a different value.

  1. AComonotonic copula; 0.4
  2. BIndependence copula; 0.20Correct
  3. CIndependence copula; 0.90
  4. DCountermonotonic copula; 0.00
  5. Independence copula; 0.10

Explanation

Independence gives F(x,y)=F_X(x)F_Y(y), so C(u,v)=uv. Then 0.4 x 0.5 = 0.20. The comonotonic copula would give min(0.4,0.5)=0.4, and the countermonotonic copula would give max(0.4+0.5-1,0)=0, so both are wrong here. The value 0.90 is the sum and 0.10 is an arithmetic slip.

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