Generalized Pareto distribution
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.
Balkema and de Haan (Annals of Probability, 1974), then Pickands (Annals of Statistics, 1975), established that the excess over a high enough threshold approximately follows a generalized Pareto distribution. It has two parameters, a scale and a shape written xi, and the shape 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 the result is blunt: moments of order equal to or above one over xi are infinite. At xi equal to 0.6 the variance does not exist, and the standard deviation your software prints on a finite sample is an artifact that grows with sample size. From xi equal to 1 the mean itself is infinite: unlimited cover no longer has a finite pure premium. That is the mathematical reason behind policy limits on highly dispersed lines. Estimation is by maximum likelihood on the exceedances. Estimator variance stays large, and the result is reported with its confidence interval, never as 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. At the central estimate the variance does not exist (1 / 0.52 = 1.92, below 2), but the lower end of the interval, under 0.5, leaves it finite: the data cannot settle the question. The estimated 99.5th percentile moves from 48M to 96M EUR depending on which end of the interval you take. The risk committee pack should carry that range rather than the central value.
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