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

134 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGESAPPENDIX BCOLLECTIVE RISK MODEL SUMMARYThe quick summary uses notation that is equivalent to, but notalways identical with, the notation used by Heckman and Meyers[4].We start with the claim count model and define the number ofclaims as a counting random variable, N. Let be the conditionalexpected number of claims so that E[Nj ]=. We also write N()to denote the conditional claim count distribution.Let ¹ be the unconditional mean claim count. To introduceparameter uncertainty, we let º be a non-negative random variablewith E[º]=1andVar(º)=c. The parameter c is called thecontagion. To model unconditional claim counts, we first selecta value of º at random and then randomly select a claim count Nfrom the distribution N() where = º¹. Heckman and Meyersassume N is conditionally Poisson, so that it follows thatB.1.Unconditional Claim Count Mean and VarianceE[N] = E[E[N() j = º¹]] = E[º¹]=¹E[º]=¹:(B.1a)Var(N)=E[Var(N() j = º¹)] + Var(E[N() j = º¹])=E[¹º]+Var(¹º)=¹E[º]+¹ 2 Var(º)=¹ + ¹ 2 c:(B.1b)If º is Gamma distributed, the unconditional claim count distributionis Negative Binomial.We now add severity to the model. We let X(¸) be the conditionalclaim severity random variable defined so that E[X(¸)] =¸. Heckman and Meyers model severity parameter uncertainty byassuming the shape of the severity distribution is known but thereis uncertainty about its scale. Let ¯ be a positive random variablesuch that E[1=¯] = 1 and Var(1=¯)=b. Wecallb the mixing pa-