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

RISKINESS LEVERAGE MODELS 39Various proposed schemes 7 have utilized the fact that an allocationformula of the formC k = ®¹ k + ¯ Cov(X k ,X) (3.1)will always be additive no matter what the dependency betweenthe X k may be. That is,C ´ ®¹ + ¯ Var(X)= ®E(X)+¯ Cov(X,X)nX nX= ® ¹ k + ¯ Cov(X k ,X)=k=1k=1nXC k : (3.2)k=1A similar result will hold for the sum of any subset of thevariables, thus ensuring the desired properties of the allocation.The sum of covariances of the individual variables with the totalis the covariance of the total with itself. This paper generalizesthis notion.This form can be pushed further by imposing the reasonablerequirement 8 that if a variable has no variation, then the capitalto support it is simply its mean value with no additional capitalrequirement. This requires ® = 1. Then, with capital being thesum of the mean and the risk load,andand so finallyR k = ¯ Cov(X k ,X) (3.3)R = ¯ Var(X) (3.4)R k = R Cov(X k,X): (3.5)Var(X)7 For a sampling, try [6], [2], and [4]. There are no doubt others.8 In [6], since the company can default, a constant value carries a negative risk load. Weare assuming an ongoing company.