Three-Layer Problems and the Generalized Pareto Distribution

Abstract

The classical way to get an analytical model for the (supposedly heavy) tail of a loss severity distribution is via parameter inference from empirical large losses. However, in the insurance practice it occurs that one has much less information, but nevertheless needs such a model, say for reinsurance pricing or capital modeling.

We use the Generalized Pareto distribution to build consistent underlying models from very scarce data like: the frequencies at three thresholds, the risk premiums of three layers, or a mixture of both. It turns out that for typical real-world data situations such GPD “fits” exist and are unique.
We also provide a scheme enabling practitioners to construct reasonable models in situations where one has even less, or somewhat more, than three such bits of information.

Finally, we have a look at model risk, by applying some parameter-free inequalities for distribution tails and a particular representation for loss count distributions. It turns out that, in the data situation given above, the uncertainty about the severity can be surprisingly low, such that the overall uncertainty is driven by the loss count.