A Generalized Measure of Risk Aversion, Without the Independence Axiom

Abstract

Risk aversion is currently measured by the Arrow-Pratt index, which requires the existence of the von Neumann -Morgenstern function of utility. In truth a neo-Bernoullian utility is not implied by the notion of risk aversion and, consequently, by its measure. It is sufficient to consider the behavioral interpretations given by Arrow and Pratt for understanding how the notion of risk aversion requires only the existence of a certainty equivalent for any action under examination. For instance, Arrow (1965) refers risk aversion to the difference between 1/2 and the probability p for which Y is the certainty equivalent of the action with probability p for the consequence Y + h and probability 1–p for the consequence Y−h. Such an interpretation does not imply the independence axiom, so that also the measure of risk aversion does not imply the neo-Bernoullian utility. Similarly, Pratt (1964) refers risk aversion to the risk premium, which is the difference between the expected value of an action and its certainty equivalent. Again, only the existence of a certainty equivalent is required, not the independence axiom. Nevertheless, both Arrow and Pratt assume the neo-Bernoullian utility and define the measure of risk aversion in terms of the von Neumann-Morgenstern utility function.