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

An actuary measures the dependence between two insurance loss variables using Pearson's linear correlation coefficient. Which statement about this measure is correct?

Pearson correlation measures only linear dependence, so it can equal zero even when the variables are strongly dependent, such as X and X squared with X symmetric about zero. Zero correlation implies independence only for special joint distributions like the bivariate normal.

  1. AIt is unchanged by any strictly increasing transformation of either variable
  2. BIt measures only the strength of linear dependence and can be zero when the variables are strongly dependentCorrect
  3. CA value of zero implies that the variables are independent for any joint distribution
  4. DIt is always defined, even when the variances are infinite
  5. Its value depends only on the copula and not on the marginal distributions

Explanation

Pearson correlation captures linear association only. For example, if X is symmetric about zero and Y = X^2, then Y is fully determined by X yet Cov(X,Y)=0. Zero correlation implies independence only in special cases such as the bivariate normal. It also depends on the marginals and is undefined with infinite variance.

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