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

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.

💡 Research Summary

This paper investigates a continuous‑time portfolio selection problem in which the investor’s relative risk aversion (RRA) is modeled as a random variable with an arbitrary distribution. The authors adopt the intra‑personal equilibrium framework to handle the time‑inconsistency that arises when the objective functional is defined as a weighted average of certainty equivalents over the RRA distribution.

The market consists of a risk‑free asset with zero interest rate and d risky assets following Black‑Scholes dynamics. The investor’s utility is of the CRRA type, and the objective functional is
(J(\pi; t,w)=\int_{0}^{\infty}(u_{\gamma})^{-1}\mathbb{E}_{t}