Long-Term Mean-Variance Optimization Under Mean-Reverting Equity Returns

TL;DR

This paper derives explicit solutions for long-term mean-variance portfolio optimization in markets with mean-reverting risk-free rates and equity premiums, showing long horizons improve risk-return trade-offs.

Contribution

It introduces a novel spectral approach to solve the mean-variance optimization problem with mean-reverting factors, providing explicit solutions and boundary conditions.

Findings

Long-term investors achieve better risk-return trade-offs.

Optimal policies are characterized by a matrix differential equation.

Eigenvalues of the lambda-matrix determine the solution.

Abstract

This paper studies the mean-variance optimal portfolio choice of an investor pre-committed to a deterministic investment policy in continuous time in a market with mean-reversion in the risk-free rate and the equity risk-premium. In the tradition of Markowitz, optimal policies are restricted to a subclass of factor exposures in which losses cannot exceed initial capital and it is shown that the optimal policy is characterized by an Euler-Lagrange equation derived by the method of Calculus of Variations. It is a main result, that the Euler-Lagrange equation can be recast into a matrix differential equation by an integral transformation of the factor exposure and that the solution to the characteristic equation can be parametrized by the eigenvalues of the associated lambda-matrix, hence, the optimization problem is equivalent to a spectral problem. Finally, explicit solutions to the optimal policy are provided by application of suitable boundary conditions and it is demonstrated that - if in fact the equity risk-premium is slowly mean-reverting - then investors committing to long investment horizons realize better risk-return trade-offs than investors with shorter investment horizons.