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

120 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGESUsing this, it follows as a direct application of Equation (3.4)that the aggregate loss model has charges that decrease with risksize.5.4. Decomposable Claim Counts Imply Aggregate Loss ModelHas Charges that Decrease with Risk Size Assuming FixedIndependent SeverityIf M N is a differentiable decomposable risk-sizemodel for claim counts and the compound model,M T (N,X)=fT(N,X) j N 2 M N g has fixed independentseverity, then M T (N,X) has charges that decreasewith risk size. (5.4)Proof Via Equation (5.2), M T(N,X) is decomposable and theintroduction of fixed independent severity does not affect differentiabilitywith respect to risk size. The result then follows fromEquation (3.4).Note in this result that all risks share a common severity distributionand there is no parameter uncertainty regarding thisseverity distribution. While adding severity to the model doeslead to larger charges for all risks, the result says that underthese assumptions charges for aggregate loss still decline by sizeof risk.We now apply Statement 5.4 to prove that insurance chargesdecrease with risk size for several classes of distributions commonlyused in insurance models.5.5. Charges for Aggregate Loss Decrease with Risk Size WhenCounts Are Poisson or Negative Binomial (Fixed Failure Rate)Assuming claim sizes are independently and identically distributedand independent of the claim count, the insurance chargedecreases as the size of a risk is increased in each of the follow-