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

RISKINESS LEVERAGE MODELS 45This riskiness leverage ratio is zero up to a point, and thenconstant. Here the constant is chosen so as to exactly recreateTVAR, but clearly any constant will give a similar result. In fact,a riskiness leverage ratio that is constant up to a point and thenjumps to another constant will give a similar result.ZC = ¹ + dxf(x)(x ¡ ¹) (x ¡ x q )1 ¡ qZ 1= ¹ + dxf(x) x ¡ ¹x q1 ¡ q= ¹ ¡ ¹Z1 1(1 ¡ q)+ dxf(x)x1 ¡ q 1 ¡ q x q= 1 Z 1dxf(x)x: (4.7)1 ¡ q x qThis is the definition of TVAR, well known to be coherent. 11We see shortly that the allocated capital is just the averagevalue of the variable of interest in the situations where the totalis greater than x q . This is one example of the conditional expectationreferred to earlier.ZC k = ¹ k + dF(x k ¡ ¹ k ) (x ¡ x q )1 ¡ q= ¹ k ¡ ¹ ZRkdFxk (x ¡ x q )dF(x ¡ x1 ¡ qq )+1 ¡ qR dFxk (x ¡ x q )=: (4.8)1 ¡ qThis measure says that only the part of the distribution at thehigh end is relevant.11 [5], Op. cit.