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

106 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES3.5. Charges Decrease with Size in Poisson, Negative Binomial,and Gamma ModelsThe insurance charge decreases by size of risk in each of thefollowing models:Poisson:M = fN » Poisson(¹) j ¹>0g:(3.5a)Negative Binomial with common q:M = fN » Negative Binomial(®,q) j q is fixed and ®>0g:(3.5b)Gamma with common scale parameter ¸:M = fT » Gamma(®,¸) j ¸ is fixed and ®>0g:(3.5c)Proof With the given restrictions, it can be easily shown thateach of the families is a unique size model that has well-definedcharges. It is also readily seen that each is decomposable. Theresults then follow from Equation (3.4).Exhibit 4 shows columns of charges for risks of different sizesfor Poisson random variables, Negative Binomials with commonfailure rate parameter, and Gammas with common scale parameter.In a decomposable model, the charge decreases to the smallestpossible charge as risk size goes to infinity.3.6. Charge for an Infinitely Large Risk Equals SmallestPossible Charge in Decomposable ModelSuppose M is a differentiable decomposable model. Then' T¹! ' 0 as ¹ !1 where ' 0 (r) = max(0,1 ¡ r):(3.6)