A Globally Optimal Portfolio for m-Sparse Sharpe Ratio Maximization
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| Format: | Preprint |
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2024
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| _version_ | 1866914993783439360 |
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| author | Lin, Yizun Lai, Zhao-Rong Li, Cheng |
| author_facet | Lin, Yizun Lai, Zhao-Rong Li, Cheng |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_21100 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Globally Optimal Portfolio for m-Sparse Sharpe Ratio Maximization Lin, Yizun Lai, Zhao-Rong Li, Cheng Optimization and Control 90C26, 90C32, 65K05 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. |
| title | A Globally Optimal Portfolio for m-Sparse Sharpe Ratio Maximization |
| topic | Optimization and Control 90C26, 90C32, 65K05 |
| url | https://arxiv.org/abs/2410.21100 |