A Class of Distortion Operators for Pricing Financial and Insurance Risks

This article introduces a class of distortion operators, ga(t) = D[44-(u) + a], where D is the standard normal cumulative distribution. For any loss (or asset) variable X with a probability distribution Sx(x) = 1Fx(x), ga [Sx(x)] defines a distorted probability distribution whose mean value yields a risk-adjusted premium (or an asset price). The distortion operator ga can be applied to both assets and liabilities, with opposite signs in the parameter a. Based on CAPM, the author establishes that the parameter ca should correspond to the systematic risk of X. For a normal (L,aU2) distribution, the distorted distribution is also normal with '= u + aa and a5' = a. For a lognormal distribution, the distorted dis