Dynamic portfolio selection without risk-free assets

TL;DR

This paper develops a semi-explicit solution for mean-variance portfolio optimization without risk-free assets using a game theoretic approach, revealing wealth-dependent investment strategies and superior performance in bull markets.

Contribution

It introduces a novel framework for portfolio selection without risk-free assets, deriving a semi-explicit solution via extended Hamilton-Jacobi-Bellman equations.

Findings

Optimal investment depends on current wealth.

Value function is quadratic in wealth.

Model outperforms classical and variance models in bull markets.

Abstract

We consider the mean--variance portfolio optimization problem under the game theoretic framework and without risk-free assets. The problem is solved semi-explicitly by applying the extended Hamilton--Jacobi--Bellman equation. Although the coefficient of risk aversion in our model is a constant, the optimal amounts of money invested in each stock still depend on the current wealth in general. The optimal solution is obtained by solving a system of ordinary differential equations whose existence and uniqueness are proved and a numerical algorithm as well as its convergence speed are provided. Different from portfolio selection with risk-free assets, our value function is quadratic in the current wealth, and the equilibrium allocation is linearly sensitive to the initial wealth. Numerical results show that this model performs better than both the classical one and the variance model in a bull market.