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

WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 93FIGURE 2Random Variable with Smaller CV Has Larger Chargeat some Entry Ratiosextend our result to models in which independent decomposabilityis only conditionally true. To do this, we will follow theusual Bayesian construction and view the true mean of a risk asa random variable having a prior distribution. A family of priorswill then be used to define a family of unconditional distributions.This introduces parameter risk. Assuming an arbitrarilydecomposable conditional model and priors having charges thatdecline with size, we will show the charges for the unconditionalmodel decline with the unconditional risk size. However, the unconditionalrisk-size model will not be decomposable, and, dueto the parameter uncertainty introduced via the priors, the CVfor an unconditional risk will not tend towards zero as risk sizebecomes infinite.We then turn to aggregate loss distributions that are generatedby sampling claim counts and sampling claim severities in themanner described by the CRM. We will first show that an aggregateloss model inherits decomposability from its underlyingclaim count model, assuming severities are independently sampledfrom a fixed severity distribution. This leads to the conclusionthat decomposable counts and independent fixed severitiesproduce a model in which charges decrease by size of risk. We