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

90 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGESduction or straightforward analysis of the coefficient ofvariation (CV).To explore size and charge in some generality, wewill define the construct of a risk-size model. A risksizemodel may initially be viewed as a collection ofnon-negative random variables whose sizes are definedby their expectation values. Given an appropriate measureon the risks of a particular size, we will be ableto regard the cumulative distribution and the charge aswell-defined functions of risk size. In a complete andcontinuous model, there are risks of every size and thecumulative distribution is a continuous function of risksize. We will first show that the charge declines withsize if any risk can be decomposed into the independentsum of smaller risks in the model. Then we will employthe usual Bayesian construction to introduce parameterrisk and extend the result to models that are notdecomposable. This is an important extension, becauseactuaries have long known from study of Table M that alarge risk is not the independent sum of smaller ones. Inparticular, our result implies that charges decrease withsize in the standard contagion model of the NegativeBinomial used in the CRM. Finally, we will introduceseverity, prove our result assuming a fixed severity distribution,and then extend it to cover the type of parameteruncertainty in risk severity modeled in the CRM.Thus we will arrive at the conclusion that the assumptionsof the CRM force charges to decline by size ofrisk.ACKNOWLEDGEMENTThe author gratefully acknowledges the valuable contributionsmade by Michael Singer, Dave Clark, and Jim Adams. Dave wasan excellent sounding board for ideas and his insightful responsesgreatly aided this work. Michael served admirably as proofreaderand editor. His suggestions significantly improved the paper. Jim