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

WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 1195.2. Fixed Independent SeverityA compound risk-size model has independent fixed severityif:i) all risks share a common severity distribution, X.ii) fX 1 ,X 2 ,:::,X N g is an independent set.iii) X i is independent of N.iv) X i is independent of , where isthetruemeanof N for a risk.(5.2)Given these severity assumptions and a decomposable claimcount model, we can show the aggregate loss model is also decomposable.5.3. Aggregate Loss Model Inherits Decomposability fromClaim Count Model Assuming Fixed Independent SeverityIf M N is a decomposable claim count model and thecompound model, M T(N,X) = fT(N,X) j N 2 M N ghas fixed independent severity, then M T(N,X) =fT(N,X) j N 2 M N g is also decomposable. (5.3)Proof Recall we have assumed without loss of generality thatE[X] = 1. Thus E[T(N,X)] = E[N]E[X]=E[N]. Given > 0,completeness of M N implies there exists a unique N() 2 M Nsuch that E[N()] = . ItfollowsthatE[T(N(),X)] = . Thus,M T(N,X) is complete. Now let T(N( 1 ),X) andT(N( 2 ),X) bein M T(N,X) . Then using our severity assumptions we can showT(N( 1 ),X)+T(N( 2 ),X)=T(N( 1 )+N( 2 ),X). Since M N isclosed under independent summation, it follows that N( 1 )+N( 2 )=N( 1 + 2 )andN( 1 + 2 ) 2 M N . Therefore, T(N( 1 )+N( 2 ),X) 2 M T(N,X) , proving that M T(N,X) . is closed under independentsummation. Now we apply Equation (3.2) to finish theproof.