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

Two Indian insurers' annual claim totals X and Y have continuous marginals. Their copula is C(u,v)=uv. Which statement is correct about P(X ≤ x, Y ≤ y) when F1(x)=0.4 and F2(y)=0.5?

The joint probability is 0.20. The copula uv is the independence copula, so the joint cdf is the product of the marginal cdfs, 0.4 times 0.5. The minimum copula would give 0.40, and the countermonotonic copula would give zero.

  1. A0.20, because the copula implies X and Y are independentCorrect
  2. B0.40, because it equals the minimum of the two marginals
  3. C0.90, because the probabilities add
  4. D0.10, because the probability is the product of the marginals squared
  5. 0.00, because the countermonotonic copula applies

Explanation

C(0.4,0.5)=0.4×0.5=0.20. The product copula means independence. The value 0.40 would be the comonotonic copula min(u,v), and 0 would be the countermonotonic max(u+v−1,0)=0 here.

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