Comparison of Approximations for Compound Poisson Processes

The aim of this paper is to provide a comparison of the error in several approximation methods for the cumulative aggregate claim distribution customarily used in the collective model of insurance theory. In this theory it is usually supposed that a portfolio of clients is at risk for a time period of length t. The claims take place according to a Poisson process, so that the number of claims is a Poisson random variable N. Each single claim is an independent replication of the random variable X, representing the claim severity. The object of study is the cumulative distribution function of the random sum of N independent replications of X, i.e. a compound Poisson process representing the aggregate claim or total claim amount process in a period of length t. Due to the complexity of its computation, several approximation methods for this cdf have been proposed in the literature. In this paper, we only consider approximations using the lower order moments of the involved distributions. This requirement rules out the Esscher approximation as well as methods of exact computation based on a preliminary discretization of the claim distribution (Panjer recursion, FFT). Therefore, we consider the Normal, Edgeworth, NP2, NP2a, Adjusted NP2, NP3, Wilson-Hilferty, Haldane A and B, Lognormal, Gamma, Translated Gamma, Bowers Gamma, Inverse Gaussian and Gamma-IG approximations. For these fifteen approximations put forward in the literature, we consider the difference between the true distribution and the approximating one and we propose to use expansions of this difference related to Edgeworth series to measure their accuracy as the observation period diverges to infinity. Using these expansions, several statements concerning the quality of these approximations can find theoretical support, while other statements can be disproved on the same grounds. At last we investigate numerically the accuracy of the proposed formulas.