  In this paper we consider the problem of computing tail probabilities of the
distribution of a random sum of positive random variables. We assume that the
individual variables follow a reproducible natural exponential family (NEF)
distribution, and that the random number has a NEF counting distribution with a
cubic variance function. This specific modelling is supported by data of the
aggregated claim distribution of an insurance company. Large tail probabilities
are important as they reflect the risk of large losses, however, analytic or
numerical expressions are not available. We propose several simulation
algorithms which are based on an asymptotic analysis of the distribution of the
counting variable and on the reproducibility property of the claim
distribution. The aggregated sum is simulated efficiently by importancesampling
using an exponential cahnge of measure. We conclude by numerical experiments of
these algorithms.

Probability