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

108 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGESsuch a model, we use the a priori expected mean to define risksize. If we let represent the true mean of a risk, and ¹ theapriorimean, then M ¹ consists of all the risks with prior mean equal to¹. While M ¹ can therefore contain many risks each having atrue mean, , which is not equal to ¹, we do insist that § ¹ ,themeasure, is defined so that ¹ is the average mean over all risks inM ¹ . Following the usual Bayesian construction and the CRM, wewill restrict our attention to models in which we may represent§ ¹ using a prior distribution, H( j ¹).Before going further, it is instructive to see how such a constructioncan be used to model the combined effects of populationparameter uncertainty and population heterogeneity.EXAMPLE 3: Population Uncertainty and HeterogeneityConsider a model in which the group of risks of size 100 actuallyconsisted of risks whose true means were 90, 100, and 110.Assume we have no way of determining the true mean of anyrisk in advance of an experiment. Suppose there are two possiblestates of the world, “L” and “H,” each of which has an equalrandom chance of occurring. If “L” applies, then we will samplefrom a subgroup of low risks, half of which have a true meanof 90 and half which have a true mean of 100. If “H” applies,the sampling subgroup will consist of an even split of high riskswith true means of 100 and 110. We then take independent sampleswith replacement. In this analogy, the two possible statesof the world correspond to population parameter uncertainty andthe mix of risks in each state corresponds to heterogeneity of thepopulation. To carry the analogy further, suppose the expectedlosses excess of 120 are 4, 10, and 18 for risks whose true meansare 90, 100, and 110 respectively. If we are in state “L,” our samplingwill produce an average excess loss of 7, while the averageexcess loss will be 14 if we are in state “H.” The average over allreplications of this sampling process over all states will be 10.5.With a prior distribution of 25%, 50%, and 25% for risks withtrue means of 90, 100, and 110 respectively, we will duplicatethis result. Note the correct average charge for the population