A Class of Distortion Operators for Pricing Financial and Insurance Risks

TLDR

The paper introduces a class of distortion operators, ga(t)=D[44-(u)+a], to price financial and insurance risks. These operators distort the probability distribution of a loss or asset variable X, yielding a risk‑adjusted premium or asset price, and preserve normality by shifting the mean while applying opposite signs for assets and liabilities. Using CAPM, the authors show that the distortion parameter a is proportional to the systematic risk of X.

Abstract

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