Practical aspects of modelling parameter uncertainty for risk capital calculation

Abstract

We assume that an insurance undertaking models its risk by a random variable X=X(¿0) with a fixed parameter (vector) ¿0. If the undertaking does not know ¿0 and can only estimate it from historical data, it faces parameter uncertainty. Neglecting parameter uncertainty can lead to an underestimation of the true risk capital requirement (see e.g. Gerrard and Tsanakas 2011; Fröhlich and Weng 2015). In this contribution we address some practical questions. To illustrate the relevance of the parameter risk we determine the probability of solvency for a risk capital model not taking parameter uncertainty into account for different distributions and samples sizes. We then follow the “inversion method” introduced in Fröhlich and Weng (2015) known to model an appropriate risk capital requirement respecting parameter uncertainty for a wide class of distributions and common estimation methods. We extend the idea to distribution families and estimation methods that have not been considered so far in this context but are frequently used to model the losses of an insurance undertaking