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A Globally Optimal Portfolio for m-Sparse Sharpe Ratio Maximization

Yizun Lin, Zhao-Rong Lai, Cheng Li

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.

A Globally Optimal Portfolio 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.

Paper Structure

This paper contains 25 sections, 21 theorems, 109 equations, 2 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works and Existing Problems
  3. Ordinary Portfolio Optimization
  4. Sharpe Ratio Optimization
  5. PGA for m-Sparse Sharpe Ratio Maximization
  6. m-Sparse Sharpe Ratio Maximization Model
  7. Proximal Gradient Algorithm
  8. Convergence Analysis of PGA
  9. Experimental results
  10. Results for Sharpe ratios
  11. Results for Cumulative Wealths
  12. Results for Transaction Costs
  13. Sparsity for mSSRM-PGA
  14. Concluding Remarks
  15. Appendix
  16. ...and 10 more sections

Key Result

Theorem 1

Suppose that there exists some $\tilde{\bm{w}}\in\Omega_1$ such that $\bm{p}^\top\tilde{\bm{w}}>0$. If $\hat{\bm{v}}$ is an optimal solution of model model:Sharpeiota, then $\frac{\bm{p}^\top\hat{\bm{v}}}{\hat{\bm{v}}^\top\bm{Q}_{\epsilon}\hat{\bm{v}}}\hat{\bm{v}}$ is an optimal solution of model sp

Figures (2)

  • Figure 1: Simulation results of PGA for model \ref{['targaltermod']}. Left: normalized error of the iterative sequence versus number of iterations. Right: normalized error of function value versus number of iterations.
  • Figure 2: Final cumulative wealths of portfolio optimization methods w.r.t. transaction cost rate $\nu$ on $6$ benchmark data sets.

Theorems & Definitions (37)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Proposition 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Lemma 8
  • proof
  • Proposition 9
  • ...and 27 more