Equilibrium Investment with Random Risk Aversion: (Non-)uniqueness, Optimality, and Comparative Statics

TL;DR

This paper characterizes equilibrium investment strategies under random risk aversion, revealing how the distribution of risk aversion influences the existence, uniqueness, and optimality of equilibria, with implications for investment behavior and comparative statics.

Contribution

It provides a complete closed-form characterization of equilibrium strategies considering general RRA distributions and establishes conditions for their uniqueness and optimality.

Findings

Equilibrium is unique if the expectation of RRA is finite.

Infinite RRA expectation can lead to multiple or trivial equilibria.

Larger risk aversion under reverse hazard rate order reduces investment risk.

Abstract

This paper studies a continuous-time portfolio selection problem under a general distribution of random risk aversion (RRA). We provide a complete characterization of all deterministic equilibrium strategies in closed form. Our results show that the structure of the solution depends crucially on the distribution of RRA: the equilibrium is unique (if exits) when the expectation of RRA is finite, whereas an infinite expectation leads either to infinitely many equilibria or to a unique trivial one (i.e. risk-free investment). To resolve this multiplicity of equilibria, we select, among all deterministic equilibria, the one that maximizes the objective functional at the initial time. We establish a necessary and sufficient condition for the existence of such an optimal equilibrium, which is then shown to be unique and uniformly optimal. Finally, we conduct a comparative statics. Using counterexamples based on two-point distributed RRA, we demonstrate that a larger risk aversion in the sense of first-order stochastic dominance does not necessarily lead to less risky investment. Within the two-point distribution framework, we further examine the single-crossing property of equilibrium strategies and the monotonicity of the crossing time. We show that a larger risk aversion under a stronger stochastic order -- the reverse hazard rate order -- always leads to less risky investment. In addition, we analyze how the convex combination of independent and identically distributed RRAs influences investment.