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

46 RISKINESS LEVERAGE MODELSVAR (Value At Risk)Take the riskiness leverageL(x)= ±(x ¡ x q ) : (4.9)f(x q )In Equation (4.9) ±(x) is the Dirac delta function. 12 Its salientfeatures are that it is zero everywhere except at (well, arbitrarilyclose to) zero and integrates to one. 13 See Appendix A forremarks about this very useful function. Here the riskiness leverageratio is all concentrated at one point. The constant factor hasbeen chosen to reproduce VAR exactly, but clearly could havebeen anything.ZC = ¹ +dxf(x)(x ¡ ¹) ±(x ¡ x q )f(x q )= ¹ + x q ¡ ¹= x q : (4.10)This gives value at risk, known not to be coherent. 14 This measuresays that only the value x q is relevant; the shape of the lossdistribution does not matter except to determine that value.The capital co-measure is the mean of the variable over thehyperplane where the total is constant at x q :ZC k = ¹ i + dF(x k ¡ ¹ k ) ±(x ¡ x q)= 1 Zf(x q )f(x q )0nXdFx k ± @j=1x j ¡ x q1A: (4.11)In a simulation environment one would have to take a smallregion rather than a plane. This could most easily be done as the12 Introduced in 1926.13 This implies that R dxf(x)±(x ¡ a)=f(a). See Appendix 1.14 [5], Op. cit.