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

RISKINESS LEVERAGE MODELS 35where f(x 1 ,x 2 ,:::,x n ) is the joint probability density function ofall the variables, the individual means are defined byZ¹ k ´ x k dF, (2.4)and the overall mean isZ " nX¹ ´ x k#dF =k=1nX¹ k : (2.5)k=1Riskiness leverage models have the formZR k ´ dF(x k ¡ ¹ k )L(x) with x ´ThenZR =ZdF(x ¡ ¹)L(x)=nXx k : (2.6)k=1f(x)(x ¡ ¹)L(x)dx: (2.7)The essential key to this formulation is that the riskiness leverageL depends only on the sum of the individual variables. Inthesecond form of Equation (2.7), f(x) is the density function forX, the sum of random variables.It follows directly from their definitions that R = P nk=1 R k andC = P nk=1 C k , no matter what the joint dependence of the variablesmay be.In analogy with the relation of covariance to variance, theR k will be referred to as co-measures of risk for the measureR. On occasion, the C k will also be referred to as co-measureswhen the context is clear. Since additivity is automatic with theseco-measures, what remains is to find appropriate forms for theriskiness leverage L(x).The form can be thought of as the risk load being aprobability-weighted average of risk loads over outcomes of thetotal net loss:ZR = dxf(x)r(x) where r(x)=(x ¡ ¹)L(x): (2.8)