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

RISKINESS LEVERAGE MODELS 41Then, following the same argument as in Equation (3.9), forany mnXE m (X)= E m (X k ): (3.11)k=1Notice that the moment expectation for m = 1 is just the covarianceof X k with the total.The individual risk load may now be formulated as1XR k = ¯mE m (X k ), (3.12)m=1and there are now an infinite number of arbitrary constants toplay with. Since there are so many independent constants, essentiallyany form can be approximated arbitrarily well.For any choice of the constants ¯m, the total risk load is thesum of the individual risk loads:R =1X¯mE m (X)=m=11X¯mm=1 k=1nXE m (X k )=nXR k : (3.13)This risk load can be put into a more transparent form by writingit as1XZ1XR k = ¯mE m (X k )= dF(x k ¡ ¹ k ) ¯m(x ¡ ¹) m :m=1m=1k=1(3.14)Since the term with m = 0 integrates to 0 (that being the definitionof the mean), what is present is a Taylor series expansionof a function of the total losses about ¹. Thus, Equation (3.14)may be written asZR k = dF(x k ¡ ¹ k )L(x): (3.15)This is the framework described earlier.