Copula Models for Dependent Risks in Insurance

Archimedean Copulas

Archimedean copulas have the form C(u,v) = psi^{-1}(psi(u) + psi(v)) where psi is a generator function. Clayton (psi(t) = t^{-theta} minus 1) captures lower tail dependence: lambda_L = 2^{-1/theta}. Gumbel (psi(t) = (-ln(t))^theta) captures upper tail dependence: lambda_U = 2 minus 2^{1/theta}. Frank (psi(t) = -ln((exp(-theta*t)-1)/(exp(-theta)-1))) has no tail dependence and allows negative dependence. The tail dependence coefficient lambda = lim P(Y > F_Y^{-1}(u) | X > F_X^{-1}(u)) as u approaches 1 measures the probability of joint extreme events.

Fitting and Selection

Copula fitting uses the two-stage inference for margins (IFM) method: first estimate marginal parameters, then estimate copula parameters from pseudo-observations u_i = F-hat_X(x_i), v_i = F-hat_Y(y_i). Goodness-of-fit uses the Cramer-von Mises statistic comparing the empirical copula to the fitted parametric copula. AIC and BIC guide copula family selection. For insurance, copulas model dependencies between correlated lines, joint mortality, and aggregate risk. Exam MAS-II tests copula selection, tail dependence analysis, and the impact of dependence assumptions on aggregate risk measures and reinsurance pricing.