Excess Loss Distributions over an Underlying Annual Aggregate

The purpose of the paper is IO develop a method of calculating the aggregate loss distribution for a policy covering excess claims over occurrence limit plus claims arising from the primary losses over an underlying annual aggregate. Usually. when working with losses from more than one source you would determine the aggregate distributions of each component and convolute the result IO get the overall distribution. The problem is that the two distributions-the excess over occurrence limits and the excess over the retained annual aggregate are not independent. Using results developed in my earlier note, I develop the conditional probability distribution of the number of non-excess claims based on the number of excess claims. It is argued that, in the probability subspace defined by a particular number of excess claims, the random 'variables describing the distributions of excess and retained losses are independent and thus so arc the distributions of the excess losses and the excess of the retained losses over an annual aggregate. Thus the distribution of their sum can be determined by convoluting the respective distributions. The conditional results for zero, one, two etc. excess claims arc then summed using the probabilities of that number of excess claims. Finally I outline a computer implementation of the process. I have created a simple demonstration version in Turbo Pascal for the Macintosh. It is limited in that I used a simple loss distribution to limit the number of points required for the calculations. While not developed explicitly in this paper. this approach could also bc used to determine increased limits factors as a function of the expected number of claims when an underlying aggregate is involved.