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

An actuary says a Gaussian copula with correlation 0.6 is applied to two continuous loss variables with different marginal distributions. Which statement is correct?

The dependence structure is unchanged by strictly increasing transformations of the losses. A copula captures only dependence, separate from the marginals, so rank-based measures stay the same, while Pearson correlation of the losses generally differs from 0.6.

  1. AThe Pearson correlation of the two losses must equal 0.6
  2. BThe copula, and hence Kendall's tau, changes if each loss is transformed by a strictly increasing function
  3. CThe dependence structure is unchanged by applying strictly increasing transformations to the lossesCorrect
  4. DThe joint distribution is bivariate normal whatever the marginals
  5. Marginal distributions must be normal for the copula to be valid

Explanation

Copulas are invariant under strictly increasing transformations of the margins, so rank-based measures such as Kendall's tau are unchanged. The Pearson correlation of the losses generally differs from 0.6 when the marginals are non-normal, and the joint need not be normal.

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