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

92 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGESturns out to be true under the assumptions made for a particularmodel. Under the procedure promulgated by the NationalCouncil on Compensation Insurance (NCCI) [6], a column ofinsurance charges is selected for a given risk based on its expectedlosses. The columns of charges have been constructed toeffectively guarantee that a large risk will always be assignedsmaller insurance charge values than a small one [3].But why should this property hold? The basic intuition isthat the excess ratio is related to the propensity of a distributionto take on relatively extreme values. When a large risk canbe viewed as the independent sum of smaller risks, the Law ofLarge Numbers will apply and the likelihood of relatively extremeoutcomes will decline.Looking at the coefficient of variation (CV), the ratio of thestandard deviation to the mean, supports these intuitions. Whena large risk is the sum of independent, identically distributedsmaller risks, the CV declines as risk size increases. Since thesquare of the CV is directly related to the integral of the insurancecharge [8], these arguments suggest the insurance charge shouldalso decline with risk size. However, this does not constitute astrict proof. The counterexample in Exhibit 1 shows that the CVdoes not uniquely determine the charge at every entry ratio. Inthat example, the risk with the smaller CV has a larger chargeat some entry ratios (see Figure 2). In conclusion, because thearrow goes the wrong way, we cannot use a CV argument toarrive at a relatively trivial proof.Instead we will use some numeric tricks and inequalities relatinglimited expected values to rigorously show that insurancecharges do indeed decline as risk size increases when a largerisk can be decomposed into a sum of independent smaller ones.This is a useful result, but alone it is insufficient for our largerpurpose. Actuaries have long known that a large risk in practicehas a distribution different than that resulting from the independentsummation of smaller risks [5]. So we will go further and