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

WHY LARGER RISKS HAVE SMALLER INSURANCECHARGESIRA ROBBINAbstractThe insurance-charge function is defined as the excessratio (the ratio of expected loss excess of an attachmentpoint to the expected total loss) and is expressedas a function of the entry ratio (the ratio of theattachment to the total loss expectation). Actuaries useinsurance-charge algorithms to price retrospective ratingmaximums and excess of aggregate coverages. Manyof these algorithms are based on models that can beviewed as particular applications of the Collective RiskModel (CRM) developed by Heckman and Meyers [4].If we examine the insurance-charge functions for risksof different sizes produced by these models, we will findinvariably that the insurance charge for a large risk isless than or equal to the charge for a small risk at everyentry ratio. The specific purpose of this paper isto prove that this must be so. In other words, we willshow the assumptions of the CRM force charge functionsto decline by size of risk. We will take a fairlygeneral approach to the problem, develop some theory,and prove several results along the way that apply beyondthe CRM.We will first prove that the charge for a sum of twonon-negative random variables is less than or equal tothe weighted average of their charges. We will extendthat result to show that under certain conditions, thecharge for a sum of identically distributed, but not necessarilyindependent, samples declines with the samplesize. The extension is not entirely straightforward, as thedesired result cannot be directly derived using simple in-89