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

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.

Figures (2)

• Figure 1: Comparative evolution of $|a_i(\cdot)|$ ($i=1,2$) under the first-order stochastic dominance. The red solid line corresponds to Investor 1 with $\tilde{\mathbb{P}}(\boldsymbol{R}_1=1)=0.9$ and $\tilde{\mathbb{P}}(\boldsymbol{R}_1=3)=0.1$; the blue dashed line corresponds to Investor 2 with $\tilde{\mathbb{P}}(\boldsymbol{R}_2=1)=0.9$ and $\tilde{\mathbb{P}}(\boldsymbol{R}_2=2)=0.1$. The red shaded region indicates where $|a_1(\cdot)| > |a_2(\cdot)|$. Parameters: $\lambda = 0.4$, $T = 20$.
• Figure 2: Comparative evolution of risk exposure magnitudes for individual risk aversions and their convex combination. The black dashed line corresponds to Investor 1 with $\tilde{\mathbb{P}}(\boldsymbol{R}_1 = 0.1) = 0.2$ and $\tilde{\mathbb{P}}(\boldsymbol{R}_1 = 8) = 0.8$; the blue dash-dotted line corresponds to Investor 2 with degenerate distribution $\tilde{\mathbb{P}}(\boldsymbol{R}_2 = 1.5) = 1$; the red solid line corresponds to the aggregated investor with risk aversion $\boldsymbol{R} = 0.5\boldsymbol{R}_1 + 0.5\boldsymbol{R}_2$. The red-shaded Reversal Region indicates where $|a(t)|>\max\{|a_1(t)|,|a_2(t)|\}$. Parameters: $\lambda = 0.5$, $T = 50$.

Theorems & Definitions (63)

• Definition 2.1
• Remark 2.2
• Remark 2.3
• Theorem 2.4
• proof
• Theorem 2.5
• proof
• Example 2.6
• Theorem 3.2
• proof