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

WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 109exceeds that charge for a risk whose mean equals the averagemean of the population.We will now study a specific class of risk-size models withparameter uncertainty that are constructed by applying priors tothe risk sizes in a conditionally decomposable model. Under theusual Bayesian construction, the (prior probability) weighted averageof the conditional distributions generates the unconditionaldistribution. If the family of priors itself forms a well-definedrisk-size model, the family of resulting unconditional distributionswill also be a well-defined risk-size model. If the riskmodel of priors is sufficiently well-behaved, we will be ableto derive conclusions about the insurance charges of the unconditionaldistributions. Suppose the priors have charges thatdecrease, not necessarily strictly, with the unconditional risksize. We will then show the resulting unconditional distributionsmust also have charges that decrease with unconditionalrisk size.To begin the mathematical development of this construction,let T() be a non-negative random variable parametrized by such that E[T()] = . Suppose the family fT() j > 0g is differentiablewith respect to and that it has insurance charges thatdecrease with risk size. Now we view the parameter as a randomvariable £ and let H()=H( j ¹) denote its cumulative distribution.Assume H(0) = 0 and that £ has density h()=H 0 (),which is continuously differentiable. Let E[£] befinite.SinceE[T()] = , it follows that the unconditional risk size, E[T(£)],is equal to the mean, E[£], of the parameter distribution. Weoften will use ¹ =E[£] to simplify notation. Finally, we let' T(£) denote the unconditional insurance charge. Given thesedefinitions, the usual Bayesian construction leads to:4.1. Unconditional Insurance Charge Formula' T(£) (r)= 1 ¹Zd h()' T() r¹: (4.1)