Deductibles and the Inverse Gaussian Distribution

Abstract

The calculation of mean claim sizes, in the presence of a deductible, is usually achieved through numerical integration. In case of a Lognormal or Gamma distribution, the quantities of interest can easily be expressed as functions of the cumulative distribution function, with modified parameters. This also applies to the F -distribution, where the incomplete Beta function enters the scene; see for instance the appendix in Hogg and Klugman (1984). The purpose of this paper is to derive an explicit formula for the first two moments of the Inverse Gaussian distribution, in the presence of censoring. For reasons of completeness we also consider truncation of the Inverse Gaussian distribution by an upper limit. The tractability of the derivation depends in a crucial way on two properties of the Inverse Gaussian distribution. Firstly, the cumulative distribution function of the Inverse Gaussian can be written as a simple function using the Normal probability integral. Secondly, the moment generating function of a censorized Inverse Gaussian distribution boils down to an expression containing the cumulative Inverse Gaussian distribution. This manifests itself most clearly in case of life insurance where the quantity of interest is the expectation of a present value. In case of non-life insurance, where the dimension of the Inverse Gaussian random variable is money instead of time, a further step is required: differentiation of the moment generating function. So, a natural order of this paper is to address ourselves first to the derivation for the life case and afterwards tackling the more laborious derivation for the non-life case.