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doi: 10.3934/jimo.2021111
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Sparse minimax portfolio and Sharpe ratio models

Chenchen Zu 1,, , Xiaoqi Yang 1, and  Carisa Kwok Wai Yu 2,

1. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China
2. 

Department of Mathematics, Statistics and Insurance, The Hang Seng University of Hong Kong, Shatin, Hong Kong, China

* Corresponding author: Chenchen Zu

Received  October 2020 Revised  March 2021 Early access June 2021

In this paper, we investigate sparse portfolio selection models with a regularized $ l_p $-norm term $ (0<p\leq 1) $ and negatively bounded shorting constraints. We obtain some basic properties of several linear $ l_p $-sparse minimax portfolio models in terms of the regularization parameter. In particular, we introduce an $ l_1 $-sparse minimax Sharpe ratio model by guaranteeing a positive denominator with a pre-selected parameter and design a parametric algorithm for finding its global solution. We carry out numerical experiments of linear $ l_p $-sparse minimax portfolio models with 1200 stocks from Hang Seng Index, Shanghai Securities Composite Index, and NASDAQ Index and compare their performance with $ l_p $-sparse mean-variance models. We test the effect of the regularization parameter and the negatively bounded shorting parameter on the level of sparsity, risk, and rate of return respectively and find that portfolios including fewer stocks of the linear $ l_p $-sparse minimax models tend to have lower risks and lower rates of return. However, for the $ l_p $-sparse mean-variance models, the corresponding changes are not so significant.

Keywords: Sparse minimax portfolio selection model, Sharpe ratio, short selling, sparse mean-variance model, $ l_p $ regularization.
Mathematics Subject Classification: Primary: 91G10, 90C90, 90-10.
Citation: Chenchen Zu, Xiaoqi Yang, Carisa Kwok Wai Yu. Sparse minimax portfolio and Sharpe ratio models. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021111
References:
[1]

C. R. Bacon, Practical Portfolio Performance Measurement and Attribution, 2$^{nd}$ edition, John Wiley & Sons, New York, 2008. doi: 10.1002/9781119206309.

[2]

S. Benninga, Financial Modeling, 4$^{th}$ edition, The MIT Press, London, 2014.

[3]

A. B. Berkelaar, K. Roos and T. Terlaky, The optimal set and optimal partition approach to linear and quadratic programming, in Advances in Sensitivity Analysis and Parametic Programming (eds. T. Gal and H.J. Greenberg), Springer, (1997), 159–202.

[4]

J. Brodie, I. Daubechies, C. De Mol, D. Giannone and I. Loris, Sparse and stable Markowitz portfolios, PNAS, 106 (2009), 12267-12272.  doi: 10.1073/pnas.0904287106.

[5]

G. Burke, A sharper Sharpe ratio, Futures, 23 (1994), 56.

[6]

X. Cai, K.-L. Teo, X. Yang and X. Zhou, Portfolio optimization under a minimax rule, Management Science, 46 (2000), 957-972.  doi: 10.1287/mnsc.46.7.957.12039.

[7]

T.-J. Chang, N. Meade, J. E. Beasley and Y. M. Sharaiha, Heuristics for cardinality constrained portfolio optimisation, Computers & Operations Research, 27 (2000), 1271-1302.  doi: 10.1016/S0305-0548(99)00074-X.

[8]

R. Chartrand, Exact reconstruction of sparse signals via nonconvex minimization, IEEE Signal Processing Letters, 14 (2007), 707-710. 

[9]

S. S. Chen, D. L. Donoho and M. A. Saunders, Atomic decomposition by basis pursuit, SIAM Rev., 43 (2001), 129-159.  doi: 10.1137/S003614450037906X.

[10]

C. Chen, X. Li, C. Tolman, S. Wang and Y. Ye, Sparse portfolio selection via quasi-norm regularization, preprint, arXiv: 1312.6350.

[11]

Z. Dai and F. Wen, A generalized approach to sparse and stable portfolio optimization problem, J. Ind. Manag. Optim., 14 (2018), 1651-1666.  doi: 10.3934/jimo.2018025.

[12]

S. R. Das, H. M. Markowitz, J. Scheid and M. Statman, Portfolios for investors who want to reach their goals while staying on the mean-variance efficient frontier, The Journal of Wealth Management, 14 (2011), 25-31. 

[13]

I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), 1413-1457.  doi: 10.1002/cpa.20042.

[14]

V. DeMiguel, L. Garlappi, F. J. Nogales and R. Uppal, A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms, Management Science, 55 (2009), 798-812. 

[15]

V. DeMiguel, L. Garlappi and R. Uppal, Optimal versus naive diversification: how inefficient is the $1/N$ portfolio strategy?, Review of Financial Studies, 22 (2009), 1915-1953.  doi: 10.1093/acprof:oso/9780199744282.003.0034.

[16]

E. J. Elton, M. J. Gruber, S. J. Brown and W. N. Goetzmann, Modern Portfolio Theory and Investment Analysis, 9$^{th}$ edition, John Wiley & Sons, New York, 2014.

[17]

B. Fastrich, S. Paterlini and P. Winker, Constructing optimal sparse portfolios using regularization methods, Comput. Manag. Sci., 12 (2015), 417-434.  doi: 10.1007/s10287-014-0227-5.

[18]

C. J. Goh and X. Q. Yang., Analytic efficient solution set for multi-criteria quadratic programs, Nonlinear Anal., 30 (1997), 4309-4316.  doi: 10.1016/S0362-546X(97)00130-2.

[19]

M. Grant and S. Boyd, CVX: Matlab Software for Disciplined Convex Programming, Version 2.2, 2020. Available from: http://cvxr.com/cvx.

[20]

Y. Hu, C. Li, K. Meng, J. Qin and X. Yang, Group sparse optimization via $l_{p, q}$ regularization, J. Mach. Learn. Res., 18 (2017), 960-1011. 

[21]

R. Jagannathan and T. Ma, Risk reduction in large portfolios: why imposing the wrong constraints helps?, The Journal of Finance, 58 (2003), 1651-1683.  doi: 10.3386/w8922.

[22]

M. C. Jensen, The performance of mutual funds in the period 1945-1964, The Journal of Finance, 23 (1968), 389-416. 

[23]

H. Konno and T. Kuno, Generalized linear multiplicative and fractional programming, Ann. Oper. Res., 25 (1990), 147-161.  doi: 10.1007/BF02283691.

[24]

H. Konno and H. Yamazaki, Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Management Science, 37 (1991), 519-531.  doi: 10.1287/mnsc.37.5.519.

[25]

Z. Lu, Iterative reweighted minimization methods for $l_p$ regularized unconstrained nonlinear programming, Math. Program., 147 (2014), 277-307.  doi: 10.1007/s10107-013-0722-4.

[26] D. G. Luenberger, Investment Science, Oxford University Press, New York, 1997. 
[27]

H. M. Markowitz, How to represent mark-to-market possibilities with the general portfolio selection model, Journal of Portfolio Management, 39 (2013), 1-3. 

[28]

H. M. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91. 

[29]

H. M. Markowitz and G. P. Todd, Mean-Variance Analysis in Portfolio Choice and Capital Markets, John Wiley & Sons, New York, 2000.

[30]

H. M. Markowitz and E. L. van Dijk, Single-period mean-variance analysis in a changing world, Financ. Anal. J., 59 (2003), 30-44.  doi: 10.1007/978-1-4419-1642-6_10.

[31]

J. Matousek and B. Gartner, Understanding and Using Linear Programming, Springer-Verlag, Heidelberg, 2007.

[32]

T. A. McCafferty, The Market is Always Right., McGraw Hill, New York, 2003.

[33]

A. Niedermayer and D. Niedermayer, Applying Markowitz's critical line algorithm, in Handbook of Portfolio Construction (eds. J. B. Guerard, Jr.), Springer, (2010), 383–400. doi: 10.1007/978-0-387-77439-8_12.

[34]

Y. Qi, Parametrically computing efficient frontiers of portfolio selection and reporting and utilizing the piecewise-segment structure, Journal of the Operational Research Society, 71 (2020), 1675-1690.  doi: 10.1080/01605682.2019.1623477.

[35]

Y. Qi, Y. Zhang and S. Ma, Parametrically computing efficient frontiers and reanalyzing efficiency-diversification discrepancies and naive diversification, INFOR Inf. Syst. Oper. Res., 57 (2019), 430-453.  doi: 10.1080/03155986.2018.1533207.

[36]

W. F. Sharpe, A linear programming algorithm for mutual fund portfolio selection, Management Science, 13 (1967), 499-510.  doi: 10.1287/mnsc.13.7.499.

[37]

W. F. Sharpe, Mutual fund performance, Journal of Business, 39 (1966), 119-138.  doi: 10.1086/294846.

[38]

F. A. Sortino and R. van der Meer, Downside risk, The Journal of Portfolio Management, 17 (1991), 27-31.  doi: 10.3905/jpm.1991.409343.

[39]

R. Tibshirani, Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. Ser. B, 58 (1996), 267-288.  doi: 10.1111/j.2517-6161.1996.tb02080.x.

[40]

J. L. Treynor, How to rate management of investment funds, Harvard Business Review, 43 (1965), 63-75.  doi: 10.1002/9781119196679.ch10.

[41]

M. Woodside-Oriakhi, C. Lucas and J. E. Beasley, Heuristic algorithms for the cardinality constrained efficient frontier, European J. Oper. Res., 213 (2011), 538-550.  doi: 10.1016/j.ejor.2011.03.030.

[42]

Z. Xu, H. Zhang, Y. Wang, X. Chang and Y. Liang, $L_{1/2}$ regularization, Sci. China Inf. Sci., 53 (2010), 1159-1169.  doi: 10.1007/s11432-010-0090-0.

[43]

M. R. Young, A minimax portfolio selection rule with linear programming solution, Management Science, 44 (1998), 673-683.  doi: 10.1287/mnsc.44.5.673.

[44]

T. W. Young, Calmar ratio: A smoother tool, Futures, 20 (1991), 40-40. 

show all references

References:
[1]

C. R. Bacon, Practical Portfolio Performance Measurement and Attribution, 2$^{nd}$ edition, John Wiley & Sons, New York, 2008. doi: 10.1002/9781119206309.

[2]

S. Benninga, Financial Modeling, 4$^{th}$ edition, The MIT Press, London, 2014.

[3]

A. B. Berkelaar, K. Roos and T. Terlaky, The optimal set and optimal partition approach to linear and quadratic programming, in Advances in Sensitivity Analysis and Parametic Programming (eds. T. Gal and H.J. Greenberg), Springer, (1997), 159–202.

[4]

J. Brodie, I. Daubechies, C. De Mol, D. Giannone and I. Loris, Sparse and stable Markowitz portfolios, PNAS, 106 (2009), 12267-12272.  doi: 10.1073/pnas.0904287106.

[5]

G. Burke, A sharper Sharpe ratio, Futures, 23 (1994), 56.

[6]

X. Cai, K.-L. Teo, X. Yang and X. Zhou, Portfolio optimization under a minimax rule, Management Science, 46 (2000), 957-972.  doi: 10.1287/mnsc.46.7.957.12039.

[7]

T.-J. Chang, N. Meade, J. E. Beasley and Y. M. Sharaiha, Heuristics for cardinality constrained portfolio optimisation, Computers & Operations Research, 27 (2000), 1271-1302.  doi: 10.1016/S0305-0548(99)00074-X.

[8]

R. Chartrand, Exact reconstruction of sparse signals via nonconvex minimization, IEEE Signal Processing Letters, 14 (2007), 707-710. 

[9]

S. S. Chen, D. L. Donoho and M. A. Saunders, Atomic decomposition by basis pursuit, SIAM Rev., 43 (2001), 129-159.  doi: 10.1137/S003614450037906X.

[10]

C. Chen, X. Li, C. Tolman, S. Wang and Y. Ye, Sparse portfolio selection via quasi-norm regularization, preprint, arXiv: 1312.6350.

[11]

Z. Dai and F. Wen, A generalized approach to sparse and stable portfolio optimization problem, J. Ind. Manag. Optim., 14 (2018), 1651-1666.  doi: 10.3934/jimo.2018025.

[12]

S. R. Das, H. M. Markowitz, J. Scheid and M. Statman, Portfolios for investors who want to reach their goals while staying on the mean-variance efficient frontier, The Journal of Wealth Management, 14 (2011), 25-31. 

[13]

I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), 1413-1457.  doi: 10.1002/cpa.20042.

[14]

V. DeMiguel, L. Garlappi, F. J. Nogales and R. Uppal, A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms, Management Science, 55 (2009), 798-812. 

[15]

V. DeMiguel, L. Garlappi and R. Uppal, Optimal versus naive diversification: how inefficient is the $1/N$ portfolio strategy?, Review of Financial Studies, 22 (2009), 1915-1953.  doi: 10.1093/acprof:oso/9780199744282.003.0034.

[16]

E. J. Elton, M. J. Gruber, S. J. Brown and W. N. Goetzmann, Modern Portfolio Theory and Investment Analysis, 9$^{th}$ edition, John Wiley & Sons, New York, 2014.

[17]

B. Fastrich, S. Paterlini and P. Winker, Constructing optimal sparse portfolios using regularization methods, Comput. Manag. Sci., 12 (2015), 417-434.  doi: 10.1007/s10287-014-0227-5.

[18]

C. J. Goh and X. Q. Yang., Analytic efficient solution set for multi-criteria quadratic programs, Nonlinear Anal., 30 (1997), 4309-4316.  doi: 10.1016/S0362-546X(97)00130-2.

[19]

M. Grant and S. Boyd, CVX: Matlab Software for Disciplined Convex Programming, Version 2.2, 2020. Available from: http://cvxr.com/cvx.

[20]

Y. Hu, C. Li, K. Meng, J. Qin and X. Yang, Group sparse optimization via $l_{p, q}$ regularization, J. Mach. Learn. Res., 18 (2017), 960-1011. 

[21]

R. Jagannathan and T. Ma, Risk reduction in large portfolios: why imposing the wrong constraints helps?, The Journal of Finance, 58 (2003), 1651-1683.  doi: 10.3386/w8922.

[22]

M. C. Jensen, The performance of mutual funds in the period 1945-1964, The Journal of Finance, 23 (1968), 389-416. 

[23]

H. Konno and T. Kuno, Generalized linear multiplicative and fractional programming, Ann. Oper. Res., 25 (1990), 147-161.  doi: 10.1007/BF02283691.

[24]

H. Konno and H. Yamazaki, Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Management Science, 37 (1991), 519-531.  doi: 10.1287/mnsc.37.5.519.

[25]

Z. Lu, Iterative reweighted minimization methods for $l_p$ regularized unconstrained nonlinear programming, Math. Program., 147 (2014), 277-307.  doi: 10.1007/s10107-013-0722-4.

[26] D. G. Luenberger, Investment Science, Oxford University Press, New York, 1997. 
[27]

H. M. Markowitz, How to represent mark-to-market possibilities with the general portfolio selection model, Journal of Portfolio Management, 39 (2013), 1-3. 

[28]

H. M. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91. 

[29]

H. M. Markowitz and G. P. Todd, Mean-Variance Analysis in Portfolio Choice and Capital Markets, John Wiley & Sons, New York, 2000.

[30]

H. M. Markowitz and E. L. van Dijk, Single-period mean-variance analysis in a changing world, Financ. Anal. J., 59 (2003), 30-44.  doi: 10.1007/978-1-4419-1642-6_10.

[31]

J. Matousek and B. Gartner, Understanding and Using Linear Programming, Springer-Verlag, Heidelberg, 2007.

[32]

T. A. McCafferty, The Market is Always Right., McGraw Hill, New York, 2003.

[33]

A. Niedermayer and D. Niedermayer, Applying Markowitz's critical line algorithm, in Handbook of Portfolio Construction (eds. J. B. Guerard, Jr.), Springer, (2010), 383–400. doi: 10.1007/978-0-387-77439-8_12.

[34]

Y. Qi, Parametrically computing efficient frontiers of portfolio selection and reporting and utilizing the piecewise-segment structure, Journal of the Operational Research Society, 71 (2020), 1675-1690.  doi: 10.1080/01605682.2019.1623477.

[35]

Y. Qi, Y. Zhang and S. Ma, Parametrically computing efficient frontiers and reanalyzing efficiency-diversification discrepancies and naive diversification, INFOR Inf. Syst. Oper. Res., 57 (2019), 430-453.  doi: 10.1080/03155986.2018.1533207.

[36]

W. F. Sharpe, A linear programming algorithm for mutual fund portfolio selection, Management Science, 13 (1967), 499-510.  doi: 10.1287/mnsc.13.7.499.

[37]

W. F. Sharpe, Mutual fund performance, Journal of Business, 39 (1966), 119-138.  doi: 10.1086/294846.

[38]

F. A. Sortino and R. van der Meer, Downside risk, The Journal of Portfolio Management, 17 (1991), 27-31.  doi: 10.3905/jpm.1991.409343.

[39]

R. Tibshirani, Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. Ser. B, 58 (1996), 267-288.  doi: 10.1111/j.2517-6161.1996.tb02080.x.

[40]

J. L. Treynor, How to rate management of investment funds, Harvard Business Review, 43 (1965), 63-75.  doi: 10.1002/9781119196679.ch10.

[41]

M. Woodside-Oriakhi, C. Lucas and J. E. Beasley, Heuristic algorithms for the cardinality constrained efficient frontier, European J. Oper. Res., 213 (2011), 538-550.  doi: 10.1016/j.ejor.2011.03.030.

[42]

Z. Xu, H. Zhang, Y. Wang, X. Chang and Y. Liang, $L_{1/2}$ regularization, Sci. China Inf. Sci., 53 (2010), 1159-1169.  doi: 10.1007/s11432-010-0090-0.

[43]

M. R. Young, A minimax portfolio selection rule with linear programming solution, Management Science, 44 (1998), 673-683.  doi: 10.1287/mnsc.44.5.673.

[44]

T. W. Young, Calmar ratio: A smoother tool, Futures, 20 (1991), 40-40. 

Figure 1.  Different $ \lambda $ of (generalized) Sharpe ratio
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Figure 2.  Rates of return with different $ \tau $
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Figure 3.  Performances with $ \alpha = -0.2 $
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Figure 4.  Performances with $ \alpha = -0.5 $
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Table 1.  Computational time
$ l_1 $-MM $ l_{\frac{1}{2}} $-MM $ l_1 $-MV $ l_{\frac{1}{2}} $-MV $ l_1 $-SR
0.56s 2.90s 24.35s 85.91s 225.33s
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$ l_1 $-MM $ l_{\frac{1}{2}} $-MM $ l_1 $-MV $ l_{\frac{1}{2}} $-MV $ l_1 $-SR
0.56s 2.90s 24.35s 85.91s 225.33s
Table 2.  Performances of different sparse models
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Table 3.  Performances with different $\alpha$
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