Loss Models: From Data to Decisions, 4th Edition

9.5 Computing the aggregate claims distribution

The computation of the compound distribution function

(9.17)

or the corresponding probability (density) function is generally not an easy task, even in the simplest of cases. In this section we discuss several approaches to numerical evaluation of (9.17) for specific choices of the frequency and severity distributions as well as for arbitrary choices of one or both distributions.

One approach is to use an approximating distribution to avoid direct calculation of (9.17). This approach is used in Example 9.4 where the method of moments is used to estimate the parameters of the approximating distribution. The advantage of this method is that it is simple and easy to apply. However, the disadvantages are significant. First, there is no way of knowing how good the approximation is. Choosing different approximating distributions can result in very different results, particularly in the right-hand tail of the distribution. Of course, the approximation should improve as more moments are used; but after four moments, one quickly runs out of distributions!

The approximating distribution may also fail to accommodate special features of the true distribution. For example, when the loss distribution is of the continuous type and there is a maximum possible claim (e.g., when there is a policy limit), the severity distribution may have a point mass ...