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

WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 91FIGURE 1Actuarial Intuition About Charges and Risk Sizeis a business analyst who merits honorable mention as a closetactuary whose examples, thought experiments, and practical observationshelped motivate this work.1. INTRODUCTIONOur broad objective is to study the dependence of insurancechargefunctions on risk size. We would like to arrive at somesufficiently general conditions that will force charges to obeythe actuarial intuition that larger risks ought to have smaller insurancecharges. More precisely, we would like to show thatthe assumptions of the Collective Risk Model (CRM) [4] leadto decreasing charges by size of risk. In keeping with standardactuarial terminology, the insurance charge refers to the excessratio, not the absolute dollar amount, and the charge is viewed asa function of the entry ratio. When we say the charge is smaller,we mean that the excess ratio is less than or equal to its initialvalue at every entry ratio (see Figure 1).Before proving this holds under certain conditions, we shouldnote that no one has published any article disputing it. Neitherdoes the literature contain any example with actual data for whichit fails to hold. In practice, it is implicitly assumed to be true or