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

WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 113mials. It follows as a consequence of Equation (4.2) that chargesdecrease with risk size in M(Q).Note M(Q) consists of Negative Binomials with a commonshape but different failure rate parameters. This is different fromour previous decomposable Negative Binomial risk-size model,from Equation (3.5b), in which all the variables had a commonfailure rate parameter. Our result is that charges declinewith size in the standard Gamma-Poisson claim-count contagionmodel. Note this model is not closed under independent summation.Further, observe that the square of the coefficient of variation,CV 2 =Var(£ ¹ )=¹ 2 =1=(®q)=c +(1=¹), decreases towardthe contagion, and not zero, as the risk size grows infinite. SeeExhibit 5 for tables of charges for Negative Binomials as definedin Example 3.Next we will consider two priors with the same mean. Assumethese priors are acting on a continuously differentiabledecomposable conditional risk-size model. We will show that,under certain conditions, if one prior has a smaller insurancecharge, then its resulting unconditional random variable also hasa smaller insurance charge. In order to prove this, we will firstuse integration by parts to express the unconditional insurancecharge in terms of an integral of a product of the risk partialsof the limited expected values of the conditional model and theprior.4.3. Unconditional Charge Formula' T (£)(r)=1¡ 1 ¹Z 10d @E[£;]@¢ @E[T();r¹] : (4.3)@ProofWe write' T(£) (r)=1¡ 1 ¹Z 10d h() ¢ E[T();r¹]: