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

A bivariate copula C(u,v) is a joint distribution function on the unit square. Which property must every such copula satisfy?

A copula is a joint distribution function on the unit square whose marginal distributions are uniform on [0,1]. It need not be symmetric, need not equal the product copula, and need not have a density.

  1. AIts marginal distributions are uniform on [0,1]Correct
  2. BIts marginal distributions are standard normal
  3. CIt must equal the product uv for all u and v
  4. DIt must be symmetric, so that C(u,v) = C(v,u)
  5. It must have a density that is positive everywhere on the unit square

Explanation

A copula is a joint cdf of two random variables with U(0,1) marginals. Independence (uv) is only one special copula, symmetry is not required, and a density need not exist or be positive everywhere (for example the comonotonic copula has none).

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