Evolution of the Arrow-Pratt Measure of Risk-Tolerance for Predictable Forward Utility Processes

Abstract

We study the evolution of the Arrow-Pratt measure of risk-tolerance in the framework of discrete-time predictable forward utility (or performance) processes. An agent starts with an initial utility function, which is then sequentially updated forward in time under the guidance of the martingale optimality principle. We first characterize completely the class of forward utility functions that have a time-constant measure of risk-tolerance and thus a preservation of preferences. We then show that, in general, preferences vary over time and whether the agent becomes more or less tolerant to risk is related to the curvature of the measure of risk-tolerance. An example where the initial utility function belongs to the SAHARA class, which is found to be analytically tractable and stable in the sense that all the subsequent utility functions belong to the same class as the initial one, illustrates the obtained results.