Abstract
There has been much research about regularizing optimal portfolio selections through \(\ell _1\) norm and/or \(\ell _2\)-norm squared. The common consensuses are (i) \(\ell _1\) leads to sparse portfolios and there exists a theoretical bound that limits extreme shorting of assets; (ii) \(\ell _2\) (norm-squared) stabilizes the computation by improving the condition number of the problem resulting in strong out-of-sample performance; and (iii) there exist efficient numerical algorithms for those regularized portfolios with closed-form solutions each step. When combined such as in the well-known elastic net regularization, theoretical bounds are difficult to derive so as to limit extreme shorting of assets. In this paper, we propose a minimum variance portfolio with the regularization of \(\ell _1\) and \(\ell _2\) norm combined (namely \(\ell _{1, 2}\)-norm). The new regularization enjoys the best of the two regularizations of \(\ell _1\) norm and \(\ell _2\)-norm squared. In particular, we derive a theoretical bound that limits short-sells and develop a closed-form formula for the proximal term of the \(\ell _{1,2}\) norm. A fast proximal augmented Lagrange method is applied to solve the \(\ell _{1,2}\)-norm regularized problem. Extensive numerical experiments confirm that the new model often results in high Sharpe ratio, low turnover and small amount of short sells when compared with several existing models on six datasets.
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Acknowledgements
We thank the editor and two referees for their detailed comments that have improved the quality of the paper. In particular, one referee points us to the much relevant paper [35], which leads to the comment Remark 3 in the paper.
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The paper was supported in part by 111 Project of China (B16002), IEC/NSFC/191543 and NSFC(12071022)
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Zhao, H., Kong, L. & Qi, HD. Optimal portfolio selections via \(\ell _{1, 2}\)-norm regularization. Comput Optim Appl 80, 853–881 (2021). https://doi.org/10.1007/s10589-021-00312-4
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DOI: https://doi.org/10.1007/s10589-021-00312-4
Keywords
- Portfolio optimization
- Minimum variance portfolio
- \(\ell _{1, 2}\)-norm regularization
- Proximal augmented Lagrange method
- Out-of-sample performance