A new method for evaluating the distribution of aggregate claims

Abstract

In the present paper, we propose a method of practical utility for calculating the aggregate claims distribution in a discrete framework. It is an approximated method but unlike the other approximated methods proposed in the literature: • the approximation concerns both the counting distribution and the convolution of the severity distributions; • the approximation does not consist in truncating the original distribution up to a given number of terms nor in replacing it with another distribution or a more general function (but simply in considering only the significant numerical realizations and in neglecting the others); • the resulting approximation of the aggregate claims distribution is lower than a prefixed maximum error (10 −6 in our applications). In particular, the probability distribution and also the first three moments are exact with the prefixed maximum error. The proposed method does not require special assumptions on the counting distribution nor the identical distribution of the severity random variables and it does not incur in underflow and overflow computational problems. It proves to be more flexible, easier and cheaper than the (exact and approximated) methods using recursion and Fast Fourier Transform. We show some applications using both a Poisson distribution and a Generalized Pareto mixture of Poisson distributions as counting distribution. In addition to the specific application proposed in this paper, the method can be applied in many other (life and non-life) actuarial fields where the sum of discrete random variables and the calculation of compound distributions are involved. Besides, it can be extended in multivariate cases.