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

Two loss variables X and Y have a Gaussian copula. A modeller replaces both marginals by new continuous distributions, each via a strictly increasing transformation of the original variables, keeping the copula parameters the same. Which statement is correct?

Spearman's rank correlation is unchanged. Strictly increasing transformations of the marginals leave the copula invariant, and rank correlations depend only on the copula. Pearson's linear correlation depends on the marginals and will generally change, and the Gaussian copula stays tail independent.

  1. ASpearman's rank correlation of the pair is unchangedCorrect
  2. BPearson's linear correlation is unchanged
  3. CThe coefficient of tail dependence becomes positive
  4. DThe copula changes to a Student t copula
  5. Kendall's tau must change to retain the same copula

Explanation

Copulas are invariant under strictly increasing transformations of the marginals. Rank-based measures such as Spearman's rho and Kendall's tau depend only on the copula, so they are unchanged. Pearson's correlation depends on the marginals and generally changes.

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