Limit distribution of exceedances above a high threshold, whose shape parameter decides whether the mean and variance exist at all, and therefore whether an unlimited treaty is financeable.
The generalized Pareto distribution has two parameters, a scale and a shape written xi, and it describes the excess above a high threshold. Its value lies in the shape parameter, which reads as a diagnosis. Negative, the tail is bounded and a theoretical maximum loss exists. Zero, the tail is exponential and every moment exists. Positive, the tail is heavy, and a blunt result applies: moments of order above one over xi are infinite. At xi equal to 0.6 the variance does not exist, so the standard deviation your software prints on a finite sample is an artifact that grows with sample size. At xi above 1 the mean itself is infinite, which makes unlimited cover strictly unpriceable, and that is the mathematical reason behind policy limits on highly dispersed lines. Estimation is by maximum likelihood on the exceedances, and estimator variance stays large, hence the duty to return a confidence interval and never a bare value.
Fit on industrial fire losses above 2M EUR in a European book, 2010 to 2025, 84 exceedances. Shape parameter estimated at 0.52 with a 95% confidence interval of 0.31 to 0.73. Variance therefore does not exist, the second moment is infinite, and the estimated 99.5th percentile moves from 48M to 96M EUR depending on which end of the interval you take: it is that range, not the central value, that belongs in the risk committee pack.
GPD, loi GPD, Pareto généralisée, loi des dépassements