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

Paper: arXiv 2512.00830

Authors: Weilun Cheng, Zongxia Liang, Sheng Wang, Jianming Xia

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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly rigorous mathematical analysis of continuous-time portfolio selection with random risk aversion, establishing closed-form characterizations of equilibria and addressing time-inconsistency. However, it is entirely theoretical with no empirical backtesting or real-world financial data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Characterize portfolio equilibria under Random Risk Aversion RRA] --> B[Input: Continuous-time model with general RRA distribution]
    B --> C[Derive Deterministic Equilibrium Strategies in Closed Form]
    C --> D{Expectation of RRA}
    D -- Finite E[RRA] --> E[Unique Non-Trivial Equilibrium]
    D -- Infinite E[RRA] --> F[Multiple Equilibria or Unique Trivial Risk-Free]
    F --> G[Select Optimal Equilibrium via Initial Functional Maximization]
    E & G --> H[Comparative Statics: Apply Reverse Hazard Rate Order for Risk Monotonicity]

Cited References (152)

1. Equilibrium Portfolio Selection Under Beliefs-Dependent Utilities (2025)
2. Stochastic Finance: An Introduction in Discrete Time (2016)
3. Time-Inconsistent Optimal Control Problems and Related Issues (2019)
4. Essays in the Theory of Risk Bearing (1970)
5. Risk aversion in the small and in the large (1964)
6. Stochastic Orders (2007)
7. MONOTONICITY PROPERTIES OF OPTIMAL INVESTMENT STRATEGIES FOR LOG- B ROWNIAN ASSET PRICES (2007)
8. Consistent investment of sophisticated rank-dependent utility agents in continuous time (2021)

7 of 152 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.