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

118 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGEShas finite variance and write ¿ 2 =Var(X). Let X i be the ith ina sequence of trials of X. Define the aggregate loss randomvariable for the risk via T(N,X)=X 1 + X 2 + ¢¢¢+ X N .Noteinthis process of generating results, we are generating losses for aparticular risk. When no confusion should result, we will writeT instead of T(N,X). Using Equation (A.6), we know the randomvariable, (T=¹ X ), has the same insurance charge as T. Thismeans we can assume ¹ X = 1 for the purpose at hand withoutloss of generality. When there are exactly n claims, defineT(n,X)=X 1 + X 2 + ¢¢¢+ X n . Assuming ¹ X = 1, it follows thatE[T(n,X)] = n. To simplify notation, we may write ' n=X or even' n in place of ' T(n,X) .The insurance charge for T(N,X) can be decomposed as aweighted sum of the insurance charges for T(n,X), each evaluatedat an appropriately scaled entry ratio.5.1. Count Decomposition of the Insurance Charge forAggregate Loss' T(N,X) (r)= 1 X r¹Np¹ N (n) ¢ n ¢ ' T(n,X) : (5.1)N nn=1ProofLeft as an exercise for the reader.While one could get some general results by working withthe claim count decomposition and using discrete distributionanalogues of integration by parts, the proofs are a bit messy. Instead,we will employ the simpler strategy of deriving propertiesof compound distributions from their claim count models. Thenwe can use the results of Chapters 3 and 4 to arrive at relativelypainless conclusions about charges for the aggregate loss models.We start by proving that if the claim count model is decomposable,then so is the resulting compound distribution model. Todo this, we will first make the following severity assumptions: