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

112 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGESThen change variables using ´ =(=1+c) toget' T(£2 ) (r)= 1 Z 1 r¹1d´h¹ 1 (´) ¢ ´ ¢ ' (T(´(1+c)) :1 0´Because we assumed the conditional insurance charges were decreasingwith risk size, it follows that' T(´(1+c)) · ' T(´) :This leads to' T(£2 ) (r) · 1 Z 1 r¹1d´h¹ 1 (´) ¢ ´ ¢ ' T(´) = '1 ´ T(£1 ) (r):0Why does this result make intuitive sense, despite the factthat scaling up the prior doesn’t change its insurance charge?The answer is that scaling up does raise the mean of the prior sothat larger conditional risks have more weight in the weightedaverage signified by the integral. Since the large conditional riskshave smaller charges, the net effect of scaling up the prior is toreduce the charge of the unconditional distribution. Note thatthis result did not depend on decomposability of the conditionalmodel, merely that the conditional risk-size model had chargesthat decrease with size.For an example, consider the following:EXAMPLE 4: Gamma Contagion on Conditional PoissonsLet T() be conditionally Poisson with parameter, . Suppose = ¹ ¢ º where º is Gamma distributed with shape parameter ®,and scale parameter ¸ = ®, sothatE[º] = 1. The variable, º, introducesparameter uncertainty and Var(º)=(1=®)=c is calledthe contagion [4] parameter for claim counts. It follows that£ ¹ , the random variable for , is Gamma distributed with shapeparameter ®, and scale parameter ¸ = ®=¹, sothatE[£ ¹ ]=¹.The unconditional distribution T(£ ¹ ), is Negative Binomial withfailure rate probability parameter q =(1+¸) ¡1 = ¹=(¹ + ®). IfQ = f£ ¹ j ¹>0g and M(Q) is the resulting set of Negative Bino-