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

42 RISKINESS LEVERAGE MODELSPropertiesClearly, the allocation properties are all satisfied for anychoice of L(x). The risk load has no risk for constant variableR(c)=0:It also will scale with a currency changeR(¸X)=¸R(X),provided L(x) is homogeneous of order zero:L(¸x)=L(x):The reason this is required is that there is already a currencydimension in the term multiplying L. This can be made to happen,for example, by making L a function of ratios of currenciessuch as x=¹ or x=¾, where¾ is the standard deviation of X.However, a more interesting possibility is to make L also bea function of x=S, where again S is the total surplus of the company.Since asset variability is in principle included in the randomvariables, S should be a guaranteed-to-be-available, easilyliquefiable capital. This could come, for example, by having it inrisk-free instruments or by buying a put option on investmentswith a strike price equal to what a risk-free investment wouldbring, or any other means with a sure result.It is intuitively clear that S must come into the picture. Considerthe case where loss is normally distributed with mean 100and standard deviation 5. Is this risky for ruin, from a businesspoint of view? If the surplus is 105, it is–but if it is 200 it is not.The natural interpretation is that the riskiness leverage should bea function of the ratio of the difference of the outcome from themean to the surplus. Since the riskiness leverage could be used(with a pre-determined leverage) to give the surplus, there is acertain recursive quality present.