Risk Aversion: Differential Conditions for the Iso-Utility Curves with Positive Slope in Transformed Two Parameter Distributions

Abstract

The condition of Risk Aversion implies that the Utility Function must be concave. We take into account the dependence of the Utility Function on the return that has any type of two-parameter distribution; it is possible to define Risk and Target, the former may be the Standard Deviation of the return, and the latter is usually the Expected value of the return, as a generic function of these two parameters. Considering the 3D space of Risk, Target and Expected Utility, this paper determines the Differential Conditions for these three functions so that the Expected Utility Function depends decreasingly on Risk and increasingly on Target, that means the iso-utility curves have positive slope in the plane of Risk and Target. As a specific case, we discuss these conditions in the case of the CRRA Utility Function and the Truncated Normal distribution. Furthermore, different measures of Risk are chosen, such as Value at Risk (VaR) and Expected Shortfall (ES), to verify if these measures maintain a positive slope of the iso-utility curves in the Risk-Target plane.