PROCEEDINGS May 15, 16, 17, 18, 2005 - Casualty Actuarial Society

WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 123aggregate model is not even a unique model as the introductionof the scale parameter leads to an infinite number of risks withthe same a priori expected aggregate loss and thus the same size.We also need to be careful in interpreting the order in which thescaling parameter is averaged over the population. Suppose eachrisk in a decomposable count model has an exponential severitydistribution and the prior on severity is a Gamma. It follows thatthe unconditional severity over all risks is Pareto distributed. Ifwe then construct a model where each risk has this Pareto asits severity, we will have a decomposable model that is differentand has different charges than the one in Statement 5.7.Next we reprise the work done in Chapter 4 and extend ourresult to aggregate loss models in which the counts are subject toparameter uncertainty. We start with decomposable counts andthen introduce a family of prior distributions on the mean claimcounts, such that the priors constitute a risk-size model. If thepriors have charges that decrease, not necessarily strictly, withrisk size and if the compound model has independent severitywith scaling parameter uncertainty, then the aggregate model hascharges that decline with risk size. This is the key result of thepaper.5.8. Unconditional Aggregate Loss Model Charges Decreasewith Risk Size Assuming Counts Based on DecomposableConditionals with Priors that Decrease by Size and IndependentSeverity with Scaling Parameter UncertaintyAssume M N is a differentiable decomposableclaim-count model parameterized by such thatE[N()] = . LetQ = f£ ¹ g be a complete set ofpriors on having charges that decrease with risksize. Let Y denote risk severity and suppose theaggregate loss model, M T , has independent severitywith scale parameter uncertainty. Then M T hascharges that decrease with risk size. (5.8)