On Risk and Price: Stochastic Orderings and Measures

Following the axiomatic approach to measures of statistical quantities initiated by van Zwet (1964) and developed by several other authors, we present a general axiomatic system for the measure of the quantities risk and price. We argue that risk and insurance price are closely related through the notion of risk loading, viewed as function of the measure of risk, and that risk should be closely related to the measures of scale, skewness and kurtosis. We consider "universal" measures of scale and risk, which can be adjusted for skewness and kurtosis. Concerning the measure of price, the distortion pricing principle introduced by Denneberg (1990), studied further by Wang (1996a/b), and justified axiomatically as insurance price in a competitive market setting by Wang et al.(1997), is a measure of price for our more general axiomatic system. Our presentation includes numerous examples, some of which have so far not been encountered in actuarial science.

Keywords : axiomatic approach, measure of risk, measure of price, distortion pricing, stop-loss order, relative inverse convex order, scale, skewness, kurtosis