The Calculation of Aggregate Loss Distributions from Claim Severity and Claim Count Distributions [Discussion]

Aggregate losses are easily defined as the sum of individual claims, but the distribution of aggregate losses has not been easy to calculate. In fact, this has been a central, and perhaps the central, problem of collective risk theory. The mean of the aggregate loss distribution can be calculated as the product of the means of the underlying frequency and severity distributions; similarly, there are well known formulas for the higher moments of the aggregate distribution in terms of the corresponding frequency and severity moments (e.g., see [5] Appendix C). However the aggregate distribution function, and thus the all important excess pure premium ratio, has been awkward to calculate from the distribution functions of frequency and severity. It is this calculation problem that is addressed and solved in this important paper. The result is generalized somewhat to the case where the severity distribution is known only up to a scale multiplicative factor, which itself follows a specific distribution (inverse gamma). In this review the approach in the paper is abstracted somewhat in an attempt to focus on the areas where the specific assumptions come into play.