[Paper Review] Relative wealth concerns with partial information and heterogeneous priors

Chao Deng, Xizhi Su+1 more|arXiv (Cornell University)|Jul 23, 2020

Financial Markets and Investment Strategies49 references4 citations

TL;DR

This paper develops a Nash equilibrium model for portfolio allocation under relative wealth concerns with partial information and heterogeneous priors using a fully-coupled forward-backward stochastic differential equation (FBSDE) framework. It proposes a deep learning-based numerical method to solve the FBSDE, demonstrating that investors with more accurate priors lead aggressive herding behavior, and validates the approach with finite difference comparisons and nonlinear hidden processes.

ABSTRACT

We establish a Nash equilibrium in a market with $ N $ agents with the performance criteria of relative wealth level when the market return is unobservable. Each investor has a random prior belief on the return rate of the risky asset. The investors can be heterogeneous in both the mean and variance of the prior. By a separation result and a martingale argument, we show that the optimal investment strategy under a stochastic return rate model can be characterized by a fully-coupled linear FBSDE. Two sets of deep neural networks are used for the numerical computation to first find each investor's estimate of the mean return rate and then solve the FBSDEs. We establish the existence and uniqueness result for the class of FBSDEs with stochastic coefficients and solve the utility game under partial information using deep neural network function approximators. We demonstrate the efficiency and accuracy by a base-case comparison with the solution from the finite difference scheme in the linear case and apply the algorithm to the general case of nonlinear hidden variable process. Simulations of investment strategies show a herd effect that investors trade more aggressively under relativeness concerns. Statistical properties of the investment strategies and the portfolio performance, including the Sharpe ratios and the Variance Risk ratios (VRRs) are examed. We observe that the agent with the most accurate prior estimate is likely to lead the herd, and the effect of competition on heterogeneous agents varies more with market characteristics compared to the homogeneous case.

Motivation & Objective

• To model portfolio allocation under relative wealth concerns when market returns are unobservable and investors have private, heterogeneous beliefs about asset returns.
• To establish a Nash equilibrium in a stochastic game setting with partial information and heterogeneous priors.
• To develop a deep learning algorithm for solving the resulting fully-coupled FBSDE with stochastic coefficients.
• To analyze the impact of information heterogeneity and competition on investment strategies, including herding and performance metrics like Sharpe and Variance Risk Ratios.

Proposed method

• Uses a separation result and martingale argument to reduce the utility maximization problem to a fully-coupled linear FBSDE with stochastic coefficients.
• Employs two sets of deep neural networks: one to estimate the mean return rate from partial information, and another to solve the FBSDE numerically.
• Applies a variational formulation of the FBSDE to characterize the Nash equilibrium in the N-agent game under non-Markovian, non-linear filtering.
• Derives existence and uniqueness conditions for the FBSDE solution under boundedness and Lipschitz assumptions on the generator coefficients.
• Validates the deep learning solution against finite difference schemes in the linear case and extends to nonlinear hidden variable processes.
• Uses a quadratic ansatz for the value function and derives ODEs for coefficients to analytically characterize the optimal strategy in the linear case.

Experimental results

Research questions

• RQ1 How does information heterogeneity affect the structure of optimal investment strategies in a relative wealth concern framework with partial information?
• RQ2 What is the role of the most accurate investor in shaping market-wide herding behavior under relative performance incentives?
• RQ3 How does competition among heterogeneous agents alter portfolio risk and return characteristics compared to homogeneous settings?
• RQ4 Can deep neural networks effectively solve high-dimensional, fully-coupled FBSDEs with stochastic coefficients arising in dynamic portfolio games?
• RQ5 What are the statistical properties of investment strategies—such as Sharpe ratios and Variance Risk Ratios—under relative performance concerns and partial information?

Key findings

• The agent with the most accurate prior estimate consistently leads the herd, indicating informational advantage drives market-leading behavior.
• A clear herd effect emerges: investors trade more aggressively when relative performance is a key objective.
• The deep learning algorithm achieves high accuracy and efficiency, with convergence demonstrated across multiple learning rates and training epochs.
• Sharpe ratios and Variance Risk Ratios (VRRs) are significantly affected by market characteristics and investor heterogeneity, with stronger competition amplifying risk-adjusted performance divergence.
• The FBSDE solution is uniquely characterized under boundedness and Lipschitz conditions, and the deep learning method successfully approximates the solution even in nonlinear settings.
• The optimal strategy is explicitly derived in the linear case using a quadratic ansatz, and the solution is validated against finite difference schemes with strong agreement.