Copulas: Measuring Dependence Beyond Correlation

Copula Fundamentals

Sklar's theorem states that any multivariate distribution can be decomposed into its marginals and a copula that captures the dependence structure. For bivariate (X,Y), the joint CDF is F(x,y) = C(F_X(x), F_Y(y)) where C is the copula function. This separation allows modeling marginal distributions and dependence independently. The copula C: [0,1]^2 to [0,1] is itself a bivariate distribution on the unit square with uniform marginals. Frechet-Hoeffding bounds constrain all copulas: max(u+v-1, 0) <= C(u,v) <= min(u,v).

Common Copulas

The independence copula C(u,v) = u*v corresponds to independent marginals. The Gaussian copula is derived from the bivariate normal and depends on correlation rho. The Clayton copula C(u,v) = (u^{-theta} + v^{-theta} - 1)^{-1/theta} captures lower tail dependence (joint extreme lows). The Gumbel copula captures upper tail dependence. Kendall's tau and Spearman's rho are rank correlations that measure concordance and relate directly to copula properties. For insurance, copulas model dependent risks like correlated lines of business or joint mortality. Exam MAS-I covers copula theory, tail dependence concepts, and basic applications.