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

WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 135rameter. Suppose ° is the unconditional expected severity and letVar(X(°)) = ¿ 2 To model severity, we first take a sample from ¯and then sample the scaled severity distribution Y = X(°)=¯.To generate aggregate losses T, we first independently samplethe number of claims N, and the scaling parameter, ¯. Thenweindependently draw N samples from the severity random variableY = X(°)=¯. The aggregate loss T is the sum of these N severitysamples.Formulas for the unconditional mean and variance of the aggregateloss are derived as followsB.2.Unconditional Aggregate Loss Mean and VarianceE[T]=E[N]E[Y]=E[º¹]E[X(°)=¯]=¹E[º]°E[1=¯]=¹°:(B.2a)Var(T)=E[Var(Tj º ,¯)] + Var(E[T j º,¯])=E[E[N j ¹º]Var(Y j ¯)+Var(N j ¹º)E[Y j ¯] 2 ]+Var(¹º°=¯)= ¹E[º]¿ 2 E[(1=¯) 2 ]+¹E[º]° 2 E[(1=¯) 2 ]+(¹°) 2 Var(º=¯)= ¹¿ 2 (1 + b)+¹° 2 (1 + b)+(¹°) 2 Var(º=¯)= ¹(¿ 2 + ° 2 )(1 + b)+¹ 2 ° 2 (b + c + bc): (B.2b)Heckman and Meyers assume that ¯ has a Gamma distribution.