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(θ) with a fixed parameter (vector) θ. If the undertaking does not know θ, it faces parameter uncertainty (see e.g. [1,2,4,5,10]). It is well-known that neglecting parameter uncertainty can lead to an underestimation of the true risk capital requirement.In this contribution we address some practical questions. A risk capital requirement not taking into account parameter uncertainty can imply a probability of solvency significantly below the required confidence level. However, the underestimation of the confidence level depends on the distribution, the size of the sample and, in general, on the true parameters of the distribution. We determine the probability of solvency for different distributions and samples sizes. We then follow the "inversion method" introduced in [4], which is 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 but are frequently used to model the losses of an insurance undertaking: the lognormal distribution together with the method of moments and the two-parameter gamma distribution. Experimental data demonstrate that the inversion method also succeeds for these cases in modelling a risk capital requirement that achieves the required probability of solvency in good approximation.