A Method for Efficient Simulation of the Collective Risk Model

david l. homer and richard a. rosengarten The Collective Risk Model (CRM) constructs aggregate losses from a claim count distribution and a claim size distribution. The aggregate losses are Z = X1 +...+XN, where the Xi are independent and identically distributed as well as independent from the claim counts N. Simulating individual claims can be a lengthy process when the expected number of claims is large. Often it is sufficient to collect only individual claims greater than some threshold τ together with the aggregate smaller claims. This is the case when modeling the effects of excess of loss reinsurance. The simulation run time can be significantly reduced, therefore, by simulating large losses individually and small losses in aggregate. The challenge in doing this is to preserve the risk characteristics of the original CRM, because the small losses and the large losses are not generally independent. This paper shows how to do this by first simulating the total claim counts and then conditionally simulating both the individual large losses and an approximation to the aggregate small losses. In the case where the claim count distribution is a mixed Poisson, it is shown that the distribution of losses simulated from this method converges to the CRM distribution. This result is a generalization of the principle that the limiting behavior of a mixed Poisson CRM is controlled by the mixing distribution.