Extreme Value Theory: GEV and GPD for Insurance

GEV Distribution

The Generalized Extreme Value distribution unifies the three extreme value types. The CDF is G(x) = exp(-(1 + xi*(x-mu)/sigma)^{-1/xi}) for xi not equal to 0, and exp(-exp(-(x-mu)/sigma)) for xi = 0 (Gumbel). Parameters: location mu, scale sigma > 0, and shape xi. When xi > 0 (Frechet), the distribution has a heavy right tail with finite moments only up to order 1/xi. When xi < 0 (Weibull type), the distribution has a finite upper endpoint at mu minus sigma/xi. Maximum likelihood estimates GEV parameters from block maxima data. Return levels (quantiles) answer questions like "what is the 100-year loss level?"

GPD for Exceedances

The Generalized Pareto Distribution models threshold exceedances: H(y) = 1 minus (1 + xi*y/sigma_u)^{-1/xi} for y > 0. The expected exceedance is sigma_u/(1-xi) for xi < 1. The GPD shape xi matches the GEV shape, connecting block maxima and POT approaches. Threshold selection balances bias (too low) against variance (too high). The mean residual life plot and parameter stability plots guide threshold choice. For insurance, GPD models large claims above a retention, pricing excess layers and estimating tail risk measures. Exam MAS-II tests GEV and GPD fitting, return level estimation, and threshold diagnostics.