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

WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 125to aggregate loss models in which severity is subject to scaleparameter risk. Though our final model is based on conditionallydecomposable claim counts, the parameter risk on both countsand severity produce a model that is not decomposable.The CRM is based on conditional Poisson counts and hasseverities that satisfy our independence assumptions. The parameterscale uncertainty in the CRM is the same as in our model.Thus the CRM satisfies the assumptions of our key result inStatement 5.8 and therefore it will generate charges that declineby size of risk. This is what we set out to prove.The latest NCCI Table M was produced with the Gamma-Poisson claim-count model, where, to fit the data, the contagiondeclines with risk size [3]. This does not imply that the latestTable M is based on a decomposable model, but rather that thestraight CRM model with constant contagion by size may leadto an overstatement of the charges for large risks.In our unconditional model, we found charges were not forcedto asymptotically approach the lowest possible charge function,' 0 (r) = max(0,1 ¡ r), as risk size tends to infinity. While we haveshown ' 0 (r) is indeed the charge function for “an infinitely largerisk” in a decomposable model, in our unconditional count modelthe charge for a very large risk approaches the charge for theprior of that risk.Though severity increases insurance charges, the introductionof severity did not cause our size versus charge relation to fail.Intuitively this is because risk size is driven by the expected claimcount. In short, severity does increase the insurance charge, butit does not change the relation between charge and size in themodels we have developed here. While in the actual derivation ofthe latest Table M, severity did vary a bit by size in order to reconcileagainst the expected losses and fitted frequencies [3], thevariation was not sufficient to cause inversions of the decliningcharge by size of risk relation. It is a topic of future research to