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

WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 121ing models, M T(N,X) ,whereM N is:Poisson:M = fN 2 Poisson(¹) j ¹ ¸ 0g(5.5a)Negative Binomial with common q:M = fN 2 Negative Binomial(®,q) j q is fixed and ® ¸ 0g:(5.5b)Proof Apply Statement 5.4.We now introduce parameter uncertainty regarding the meanseverity for a risk. We follow the CRM [4] as shown in AppendixB and assume severity may vary from risk to risk only due toa scale factor. Let ¯ be a positive continuous random variablewith density, w(¯), such that E[1=¯] = 1 and Var(1=¯)=b. Theparameter b is called the mixing parameter.We let X be a fixed severity distribution that does not changeby risk. The severity distribution for a particular risk Y is obtainedby first randomly selecting a ¯ and then using the formulaY = X=¯. Under this construction, each risk has a particular¯ that does not change from claim to claim. We will provethat when the selection of ¯ is independent of , it follows thatthe compound model on decomposable counts has charges thatdecrease with size. To ensure clarity, we first define the notionof independent severity with scale parameter uncertainty by risk.This provides with us a terminology for describing the severitymodel just presented.5.6. Independent Severity with Scale Parameter UncertaintyA compound risk-size model has independent severity withscale parameter uncertainty ifi) each particular risk has a particular ¯ and associatedseverity distribution, Y = X=¯,whereX is fixed for