Equilibrium investment with random risk aversion

Abstract

We solve the problem of an investor who maximizes utility but faces random preferences. We propose a problem formulation based on expected certainty equivalents. We tackle the time-consistency issues arising from that formulation by applying the equilibrium theory approach. To this end, we provide the proper definitions and prove a rigorous verification theorem.We complete the calculations for the cases of power and exponential utility. For power utility, we illustrate in a numerical example that the equilibrium stock proportion is independent of wealth, but decreasing in time, which we also supplement by a theoretical discussion. For exponential utility, the usual constant absolute risk aversion is replaced by its expectation. K E Y W O R D Scertainty equivalents, equilibrium approach, power and exponential utility, random risk aversion, time-inconsistency INTRODUCTIONWe formulate and solve a dynamic portfolio optimization problem under random preferences by taking a distribution over certainty equivalents. This forms a new approach to optimization under random preferences. The problem becomes time-inconsistent and we deal with this via the equilibrium approach. The elegance of the solution is demonstrated for standard choices of preferences like power (constant relative risk aversion) and exponential (constant absolute riskThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.