Bayesian credibility model with heavy tail random variables: calibration of the prior and application to natural disasters and cyber insurance

Abstract

The Bayesian credibility approach is a method for evaluating a certain risk of a segment of a portfolio (such as policyholder or category of policyholders) by compensating for the lack of historical data through the use of a prior distribution. This prior distribution can be thought as a preliminary expertise, that gathers information on the target distribution. This paper describes a particular Bayesian credibility model that is well-suited for situations where collective data are available to compute the prior, and when the distribution of the variables are heavy-tailed. The credibility model we consider aims to obtain a heavy-tailed distribution (namely a Generalized Pareto distribution) at a collective level and provides a closed formula to compute the severity part of the credibility premium at an individual level. Two cases of application are presented: one related to natural disasters and the other to cyber insurance. In the former, a large database on flood events is used as the collective information to define the prior, which is then combined with individual observations at a city level. In the latter, a classical database on data leaks is used to fit a model for the volume of data exposed during a cyber incident, while the historical data on a given firm is taken into account to consider individual experience.