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

110 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGESProofOmitted.The Bayesian construction implies there is some parameteruncertainty about the true mean of any risk. Under one interpretation,the final charge value is the (prior) probability-weightedaverage of the dollar charge over all values of the true mean,divided by the expected value of the true mean. Under anotherinterpretation, we are dealing with a population of risks whosetrue overall mean we know, even though there is some parameteruncertainty regarding the mean of any particular risk in thepopulation. The prior then represents the spread in the populationand the formula arrives at the correct average charge forthe population. It is also important to note that, as in our example,the charge for an average risk is not the same as (and isusually lower than) the weighted average charge for the populationof risks. These two interpretations correspond to two typesof parameter risk. The first expresses our uncertainty about theoverall mean of a population, while the second expresses ouruncertainty about the parameter dispersion or heterogeneity of apopulation. While a hierarchical, “double-integral” model couldbe used to separately delineate their effects, Example 3 showsthat with respect to insurance charges, Equation (4.1) can beused to model both types of parameter risk together. For otherapplications such as in credibility theory, it may be importantto maintain a distinction between these sources of parameterrisk.Now let Q be a family of £ random variables and M(Q)the associated set of unconditional random variables. One quick,but important, result can be obtained assuming all the priors arescaled versions of a single distribution. Thus all the priors havethe same insurance-charge function. Assume we have a conditionalmodel in which charges decline with size. Then we showthat applying the scaled priors to this model generates an unconditionalmodel in which the insurance charge declines as afunction of the unconditional risk size.