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

An insurer models the losses from its motor and home portfolios with fixed marginal distributions and wants to change only the way the two lines move together in the extreme tail. According to Sklar's theorem, which statement describes what the copula contributes?

The copula captures only the dependence structure between the variables, separate from the marginal distributions. By Sklar's theorem, the joint distribution is the copula applied to the marginals, so dependence in the tail can be altered without changing either line's marginal loss distribution.

  1. AIt specifies the dependence structure linking the marginals, separate from the marginal distributions themselvesCorrect
  2. BIt determines the mean and variance of each marginal loss distribution
  3. CIt replaces the marginal distributions with a single joint normal distribution
  4. DIt guarantees that the correlation between the losses is zero
  5. It fixes the aggregate claim amount for the combined portfolio

Explanation

Sklar's theorem says a joint distribution can be written as a copula applied to the marginals. The copula carries only the dependence structure, so the marginals can be kept fixed while the copula is changed. The other options attribute marginal features or zero correlation to the copula, which is wrong.

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