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

48 RISKINESS LEVERAGE MODELSandZ¯R k =dF(x1 ¡ F(¹) k ¡ ¹ k )(x ¡ ¹): (4.17)In some sense this may be the most natural naive measure, asit simply assigns capital for bad outcomes in proportion to howbad they are. Both this measure and the preceding one could beused for risks such as not achieving plan, even though ruin isnot in question.In fact, there is a heuristic argument suggesting that ¯ ¼ 2. Itruns as follows: suppose the underlying distribution is uniformin the interval ¹ ¡ ¢ · x · ¹ + ¢. Then in the cases where thehalf-width ¢ is small compared to ¹, the natural risk load is ¢.For example, if the liability is $95M to $105M, then the naturalrisk load is $5M. So from Equation (4.17)¢ = R(X)= ¯0:5Z ¹+¢¹dx2¢ (x ¡ ¹)=¯¢ 2 : (4.18)However, for a distribution that is not uniform or tightly gatheredaround the mean, if one decided to use this measure, themultiplier would probably be chosen by some other test such asthe probability of seriously weakening surplus.Proportional Excess 15Take the riskiness leverageh(x)[x ¡ (¹ + ¢)]L(x)= , (4.19)x ¡ ¹where to maintain the integrability of R k either h(¹)=0or¢>0.ThenZR = f(x)h(x)[x ¡ (¹ + ¢)]dx, (4.20)andZR k =dF x k ¡ ¹ kh(x)[x ¡ (¹ + ¢)]: (4.21)x ¡ ¹15 Another contribution from Gary Venter.