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

RISKINESS LEVERAGE MODELS 47difference of two closely neighboring TVAR regions. This wasdone using the formulation of the exemplar spreadsheet and a1% width of the region.SVAR (Semi-Variance)Take the riskiness leverageL(x)=¯(x ¡ ¹)(x ¡ ¹): (4.12)SThe risk load is the semi-variance–the “downside” of the variance:R = ¯ Z 1dxf(x)(x ¡ ¹) 2 , (4.13)S ¹andR k = ¯ ZdF(xS k ¡ ¹ k )(x ¡ ¹)(x ¡ ¹): (4.14)This measure says that risk loads are only non-zero for resultsworse (greater) than the mean. This accords with the usual accountant’sview that risk is only relevant for bad results, not forgood ones. Further, this says the load should be quadratic toinfinity.Mean Downside DeviationTake the riskiness leverage(x ¡ ¹)L(x)=¯1 ¡ F(¹) : (4.15)F(x) is the cumulative distribution function for X, the total. Thisrisk load is a multiple of the mean downside deviation, whichis also TVAR with x q = ¹. This riskiness leverage ratio is zerobelow the mean, and constant above it. ThenR(X)=¯1 ¡ F(¹)Z 1¹dxf(x)(x ¡ ¹), (4.16)