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

122 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGESall risks and b is a positive continuous random variablewith E[1=¯] = 1 and Var(1=¯)=b.ii) fX 1 ,X 2 ,:::,X N g is an independent set.iii) X i is independent of N.iv) The selection of ¯ for a risk is independent of ,where isthetruemeanofN for a risk.v) The selection of ¯ for a risk is independent of ¹,where ¹ istheapriorimeanofN for a risk. (5.6)We now show5.7. Decomposable Claim Counts Imply Aggregate Loss ModelHas Charges that Decrease with Risk Size Assuming IndependentSeverity with Scale Parameter UncertaintyIf M N is a differentiable decomposable risk-sizemodel for claim counts and the compound model,M T(N,Y) = fT(N,Y) j N 2 M N ,Y = X=¯g, hasindependent severity with scale parameter uncertainty,then M T(N,Y) has charges that decrease with risk size.(5.7)Proof By the usual Bayesian conditioning and using the independentseverity assumptions, we can show the charge for themodel at risk size is given via:' T(N(),Y) (r)=Z 10d¯ w(¯)(1=¯)' T(N(),X) (r¯):Using Statement 5.5, we know the integrands decrease by sizeof risk, and the result follows using the independence of ¯and .Even though the aggregate model in Statement 5.7 is based ondecomposable counts, it will not be decomposable. Indeed, the