A Globally Optimal Portfolio for m-Sparse Sharpe Ratio Maximization

TL;DR

This paper introduces a novel method to find the globally optimal m-sparse portfolio maximizing the Sharpe ratio, overcoming nonconvexity challenges with a new quadratic programming approach and convergence guarantees.

Contribution

It converts the m-sparse Sharpe ratio optimization into a quadratic problem and develops an efficient algorithm with theoretical convergence guarantees for the first time.

Findings

Achieves globally optimal m-sparse Sharpe ratio under certain conditions.

Develops an efficient Proximal Gradient Algorithm with proven convergence rates.

First method with theoretical guarantees for m-sparse Sharpe ratio maximization.

Abstract

The Sharpe ratio is an important and widely-used risk-adjusted return in financial engineering. In modern portfolio management, one may require an m-sparse (no more than m active assets) portfolio to save managerial and financial costs. However, few existing methods can optimize the Sharpe ratio with the m-sparse constraint, due to the nonconvexity and the complexity of this constraint. We propose to convert the m-sparse fractional optimization problem into an equivalent m-sparse quadratic programming problem. The semi-algebraic property of the resulting objective function allows us to exploit the Kurdyka-Lojasiewicz property to develop an efficient Proximal Gradient Algorithm (PGA) that leads to a portfolio which achieves the globally optimal m-sparse Sharpe ratio under certain conditions. The convergence rates of PGA are also provided. To the best of our knowledge, this is the first proposal that achieves a globally optimal m-sparse Sharpe ratio with a theoretically-sound guarantee.