TOWARD A CERTAIN EQUIVALENT DISCOUNTED CASH FLOW MODEL

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. Introduction In many situations we are asked to analyze the loss distribution of an excess policy which includes coverage for retained losses that exceed some aggregate accumulation. While it is possible to determine the distributions ol both the retained losses and the excess losses, it is not readily apparent how to combine the distributions because the distributions are not independent, since a claim could contribute to each distribution. The procedure described in this note decomposes the problem in such a way as to make the. two distributions independent. The key to this decomposition is to determine the excess and retained loss distributions for a given and fixed number of excess claims. The first step is to determine the claim frequency distribution of non-excess claims conditioned on the number of excess claims. This problem is solved under the assumption of a negative binomial claim distribution using Bayes Theorem. The next step is to determine the aggregate loss distributions for the excess and retained losses. For a given number of excess claims, say Nx, the distribution of excess losses is given by the Nx-fold convolution of the excess loss distribution, which will be denoted fxL(X)Nx. The distribution of losses below the underlying per occurrence loss limit, L. is given by. the Compound Distribution of the non-excess claims plus NxL. the amount contributed by the excess claims. From this. the distribution of underlying losses over an annual aggregate, Agg. can be determined. I will call this fAgg(XINx). The key observation is that in the subspacc defined by the number of excess claims, the aggregate distribution of the excess claims and the aggregate distribution of the retained losses are independent. The argument is as follows: The excess distribution is determined by the Nxfold convolution of the excess claim distribution, The distribution of the aggregate retained losses is made up of two components. The tirst is the aggregate distribution of the non-excess claims. Under the usual assumptions the individual claims are independent, thus the size of the non-excess claims is not influenced by the size of the excess claims and vise versa. The other component is the retained portion of the excess claims, In the entire probability space, this isn’t independent of the number of excess claims, but for a given number of excess claims it is a ftxed amount. Thus, in each subspace defined by the number of excess claims, the distribution of the retained amounts is independent of the distribution of amount of the excess losses. Finally, since the excess losses and the retained losses are independent. so are the excess losses and the excess of the retained losses over the annual aggregate. This is discussed in more detail in Appendix A. Thus for any number of excess claims, say Nx. the total loss distribution is given by the convolution of fxL(X)Nx and fAgg(XINx).