Relative portfolio optimization via a value at risk based constraint

ArXiv ID: 2503.20340 “View on arXiv”

Authors: Unknown

Abstract

In this paper, we consider $n$ agents who invest in a general financial market that is free of arbitrage and complete. The aim of each investor is to maximize her expected utility while ensuring, with a specified probability, that her terminal wealth exceeds a benchmark defined by her competitors’ performance. This setup introduces an interdependence between agents, leading to a search for Nash equilibria. In the case of two agents and CRRA utility, we are able to derive all Nash equilibria in terms of terminal wealth. For $n>2$ agents and logarithmic utility we distinguish two cases. In the first case, the probabilities in the constraint are small and we can characterize all Nash equilibria. In the second case, the probabilities are larger and we look for Nash equilibria in a certain set. We also discuss the impact of the competition using some numerical examples. As a by-product, we solve some portfolio optimization problems with probability constraints.

Keywords: Nash equilibrium, CRRA utility, Portfolio optimization, Probability constraints, Competitive finance, Multi-asset general financial market

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is heavily theoretical, deriving Nash equilibria using advanced stochastic control and game theory in complete markets, with extensive formal proofs and no empirical data or backtesting.

  flowchart TD
    A["Research Goal:<br>Find Nash Equilibria in<br>Relative Portfolio Optimization<br>with Value at Risk Constraints"] --> B["Methodology: Model Setup<br>Agents: n investors<br>Market: Arbitrage-free, Complete<br>Utility: CRRA or Logarithmic<br>Constraint: Terminal Wealth > Benchmark VaR"]
    B --> C{"Number of Agents n?"}
    C -->|n = 2| D["CRRA Utility Analysis<br>Analytical derivation of<br>all Nash equilibria in terminal wealth"]
    C -->|n > 2| E["Logarithmic Utility Analysis<br>Case 1: Small probabilities<br>Characterize all equilibria"]
    C -->|n > 2| F["Logarithmic Utility Analysis<br>Case 2: Large probabilities<br>Search equilibria in restricted set"]
    D --> G["Key Findings & Outcomes<br>- Full characterization of equilibria for 2 agents<br>- Distinct equilibria for n>2 based on risk levels<br>- Impact of competition on portfolio weights<br>- Solved new portfolio optimization problems with VaR constraints"]
    E --> G
    F --> G
    style A fill:#e1f5fe
    style B fill:#fff3e0
    style G fill:#e8f5e8