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April  2018, 14(2): 541-559. doi: 10.3934/jimo.2017059

Sparse markowitz portfolio selection by using stochastic linear complementarity approach

Qiyu Wang 1, and  Hailin Sun 2,,

1. 

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

School of Economics and Management, Nanjing University of Science and Technology, Nanjing, 210094, China

* Corresponding author

Received  January 2016 Published  April 2018 Early access  June 2017

We consider the framework of the classical Markowitz mean-variance (MV) model when multiple solutions exist, among which the sparse solutions are stable and cost-efficient. We study a two - phase stochastic linear complementarity approach. This approach stabilizes the optimization problem, finds the sparse asset allocation that saves the transaction cost, and results in the solution set of the Markowitz problem. We apply the sample average approximation (SAA) method to the two - phase optimization approach and give detailed convergence analysis. We implement this methodology on the data sets of Standard and Poor 500 index (S & P 500), real data of Hong Kong and China market stocks (HKCHN) and Fama & French 48 industry sectors (FF48). With mock investment in training data, we construct portfolios, test them in the out-of-sample data and find their Sharpe ratios outperform the $\ell_1$ penalty regularized portfolios, $\ell_p$ penalty regularized portfolios, cardinality constrained portfolios, and $1/N$ investment strategy. Moreover, we show the advantage of our approach in the risk management by using the criteria of standard deviation (STD), Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR).

Keywords: Stochastic programming, sparse linear complimentarily problem, sparse Markowitz portfolio selection, sample average approximation, convergence analysis.
Mathematics Subject Classification: Primary: 90C15, 90C90; Secondary: 91G10.
Citation: Qiyu Wang, Hailin Sun. Sparse markowitz portfolio selection by using stochastic linear complementarity approach. Journal of Industrial & Management Optimization, 2018, 14 (2) : 541-559. doi: 10.3934/jimo.2017059
References:
[1]

D. Bertsimas and R. Shioda, Algorithms for cardinality-constrained quadratic optimization, Computational Optimization and Applications, 43 (2009), 1-22.  doi: 10.1007/s10589-007-9126-9.  Google Scholar

[2]

W. Bian and X. Chen, Neural network for nonsmooth, nonconvex constrained minimization via smooth approximation, IEEE Transactions on Neural Networks and Learning Systems, 25 (2014), 545-556.  doi: 10.1109/TNNLS.2013.2278427.  Google Scholar

[3]

P. Bonami and M. A. Lejeune, An exact solution approach for portfolio optimization problems under stochastic and integer constraints, Operations Research, 57 (2009), 650-670.  doi: 10.1287/opre.1080.0599.  Google Scholar

[4]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1394-9.  Google Scholar

[5]

J. M. Borwein and Q. J. Zhu, A survey of subdifferential calculus with applications, Nonlinear Analysis: Theory, Methods & Applications, 38 (1999), 687-773.  doi: 10.1016/S0362-546X(98)00142-4.  Google Scholar

[6]

J. Brodie, I. Daubechies, C. DeMol, D. Giannone and I. Loris, Sparse and stable markowitz portfolios, Proceedings of the National Academy of Sciences, 106 (2009), 12267-12272.   Google Scholar

[7]

E. J. Candes and T. Tao, Decoding by linear programming, IEEE Transactions on Information Theory, 51 (2005), 4203-4215.  doi: 10.1109/TIT.2005.858979.  Google Scholar

[8]

F. Cesarone, A. Scozzari and F. Tardella, A new method for mean-variance portfolio optimization with cardinality constraints, Annals of Operations Research, 205 (2013), 213-234.  doi: 10.1007/s10479-012-1165-7.  Google Scholar

[9]

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

[10]

X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Mathematical Programming, 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0.  Google Scholar

[11]

X. Chen, L. Guo, Z. Lu and J. Ye, An augmented Lagrangian method for non-Lipschitz nonconvex programming, SIAM Journal on Numerical Analysis, 55 (2017), 168-193.  doi: 10.1137/15M1052834.  Google Scholar

[12]

X. Chen and S. Xiang, Sparse solutions of linear complementarity problems, Mathematical Programming, 159 (2016), 539-556.  doi: 10.1007/s10107-015-0950-x.  Google Scholar

[13]

R. W. Cottle, J. -S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, Boston, MA, 1992.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[16]

G. F. Deng, W. T. Lin and C. C. Lo, Markowitz-based portfolio selection with cardinality constraints using improved particle swarm optimization, Expert Systems with Applications, 39 (2012), 4558-4566.  doi: 10.1016/j.eswa.2011.09.129.  Google Scholar

[17]

D. W. Diamond and R. E. Verrecchia, Constraints on short-selling and asset price adjustment to private information, Journal of Financial Economics, 18 (1987), 277-311.  doi: 10.1016/0304-405X(87)90042-0.  Google Scholar

[18]

J. Gao and D. Li, Optimal cardinality constrained portfolio selection, Operations Research, 61 (2013), 745-761.  doi: 10.1287/opre.2013.1170.  Google Scholar

[19]

H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x.  Google Scholar

[20]

A. J McNeil, R. Frey and P. Embrechts, Quantitative Risk Management: Concepts, Techniques and Tools, Princeton University Press, 2015.  Google Scholar

[21]

R.C. Merton, On estimating the expected return on the market: An exploratory investigation, Journal of Financial Economics, 8 (1980), 323-361.  doi: 10.3386/w0444.  Google Scholar

[22]

H. Qi and D. Sun, A quadratically convergent Newton method for computing the nearest correlation matrix, SIAM Journal on Matrix Analysis and Applications, 28 (2006), 360-385.  doi: 10.1137/050624509.  Google Scholar

[23]

B. K. Natarajan, Sparse approximate solutions to linear systems, SIAM Journal on Computing, 24 (1995), 227-234.  doi: 10.1137/S0097539792240406.  Google Scholar

[24]

R.T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-42.  doi: 10.21314/JOR.2000.038.  Google Scholar

[25]

W. F. Sharpe, The Sharpe ratio, The Journal of Portfolio Management, 21 (1994), 49-58.  doi: 10.3905/jpm.1994.409501.  Google Scholar

[26]

A. Shleifer and R. W. Vishny, The limits of arbitrage, The Journal of Finance, 52 (1997), 35-55.   Google Scholar

[27]

Y. Tian, S. Fang, Z. Deng and Q. Jin, Cardinality constrained portfolio selection problem: A completely positive programming approach, Journal of Indstrial and Management Optimization, 12 (2016), 1041-1056.  doi: 10.3934/jimo.2016.12.1041.  Google Scholar

[28]

F. Xu, Z. Lv and Z. Xu, An efficient optimization approach for a cardinality-constrained index tracking problem, Optimization Methods and Software, 31 (2016), 258-271.  doi: 10.1080/10556788.2015.1062891.  Google Scholar

[29]

F. Xu, G. Wang and Y. Gao, Nonconvex L1/2 regularization for sparse portfolio selection, Pacific Journal of Optimization, 10 (2014), 163-176.   Google Scholar

[30]

H. Xu and D. Zhang, Monte Carlo methods for mean-risk optimization and portfolio selection, Computational Management Science, 9 (2012), 3-29.  doi: 10.1007/s10287-010-0123-6.  Google Scholar

[31]

L. Xue, S. Ma and H. Zou, Positive-definite $\ell_{1}$-penalized estimation of large covariance matrices, Journal of the American Statistical Association, 107 (2012), 1480-1491.  doi: 10.1080/01621459.2012.725386.  Google Scholar

[32]

J. Ye, Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints, SIAM Journal on Optimization, 10 (2000), 943-962.  doi: 10.1137/S105262349834847X.  Google Scholar

[33]

X. Zheng, X. Sun and D. Li, Improving the performance of MIQP solvers for quadratic programs with cardinality and minimum threshold constraints: A semidefinite program approach, INFORMS Journal on Computing, 26 (2014), 690-703.  doi: 10.1287/ijoc.2014.0592.  Google Scholar

show all references

References:
[1]

D. Bertsimas and R. Shioda, Algorithms for cardinality-constrained quadratic optimization, Computational Optimization and Applications, 43 (2009), 1-22.  doi: 10.1007/s10589-007-9126-9.  Google Scholar

[2]

W. Bian and X. Chen, Neural network for nonsmooth, nonconvex constrained minimization via smooth approximation, IEEE Transactions on Neural Networks and Learning Systems, 25 (2014), 545-556.  doi: 10.1109/TNNLS.2013.2278427.  Google Scholar

[3]

P. Bonami and M. A. Lejeune, An exact solution approach for portfolio optimization problems under stochastic and integer constraints, Operations Research, 57 (2009), 650-670.  doi: 10.1287/opre.1080.0599.  Google Scholar

[4]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1394-9.  Google Scholar

[5]

J. M. Borwein and Q. J. Zhu, A survey of subdifferential calculus with applications, Nonlinear Analysis: Theory, Methods & Applications, 38 (1999), 687-773.  doi: 10.1016/S0362-546X(98)00142-4.  Google Scholar

[6]

J. Brodie, I. Daubechies, C. DeMol, D. Giannone and I. Loris, Sparse and stable markowitz portfolios, Proceedings of the National Academy of Sciences, 106 (2009), 12267-12272.   Google Scholar

[7]

E. J. Candes and T. Tao, Decoding by linear programming, IEEE Transactions on Information Theory, 51 (2005), 4203-4215.  doi: 10.1109/TIT.2005.858979.  Google Scholar

[8]

F. Cesarone, A. Scozzari and F. Tardella, A new method for mean-variance portfolio optimization with cardinality constraints, Annals of Operations Research, 205 (2013), 213-234.  doi: 10.1007/s10479-012-1165-7.  Google Scholar

[9]

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

[10]

X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Mathematical Programming, 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0.  Google Scholar

[11]

X. Chen, L. Guo, Z. Lu and J. Ye, An augmented Lagrangian method for non-Lipschitz nonconvex programming, SIAM Journal on Numerical Analysis, 55 (2017), 168-193.  doi: 10.1137/15M1052834.  Google Scholar

[12]

X. Chen and S. Xiang, Sparse solutions of linear complementarity problems, Mathematical Programming, 159 (2016), 539-556.  doi: 10.1007/s10107-015-0950-x.  Google Scholar

[13]

R. W. Cottle, J. -S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, Boston, MA, 1992.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[16]

G. F. Deng, W. T. Lin and C. C. Lo, Markowitz-based portfolio selection with cardinality constraints using improved particle swarm optimization, Expert Systems with Applications, 39 (2012), 4558-4566.  doi: 10.1016/j.eswa.2011.09.129.  Google Scholar

[17]

D. W. Diamond and R. E. Verrecchia, Constraints on short-selling and asset price adjustment to private information, Journal of Financial Economics, 18 (1987), 277-311.  doi: 10.1016/0304-405X(87)90042-0.  Google Scholar

[18]

J. Gao and D. Li, Optimal cardinality constrained portfolio selection, Operations Research, 61 (2013), 745-761.  doi: 10.1287/opre.2013.1170.  Google Scholar

[19]

H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x.  Google Scholar

[20]

A. J McNeil, R. Frey and P. Embrechts, Quantitative Risk Management: Concepts, Techniques and Tools, Princeton University Press, 2015.  Google Scholar

[21]

R.C. Merton, On estimating the expected return on the market: An exploratory investigation, Journal of Financial Economics, 8 (1980), 323-361.  doi: 10.3386/w0444.  Google Scholar

[22]

H. Qi and D. Sun, A quadratically convergent Newton method for computing the nearest correlation matrix, SIAM Journal on Matrix Analysis and Applications, 28 (2006), 360-385.  doi: 10.1137/050624509.  Google Scholar

[23]

B. K. Natarajan, Sparse approximate solutions to linear systems, SIAM Journal on Computing, 24 (1995), 227-234.  doi: 10.1137/S0097539792240406.  Google Scholar

[24]

R.T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-42.  doi: 10.21314/JOR.2000.038.  Google Scholar

[25]

W. F. Sharpe, The Sharpe ratio, The Journal of Portfolio Management, 21 (1994), 49-58.  doi: 10.3905/jpm.1994.409501.  Google Scholar

[26]

A. Shleifer and R. W. Vishny, The limits of arbitrage, The Journal of Finance, 52 (1997), 35-55.   Google Scholar

[27]

Y. Tian, S. Fang, Z. Deng and Q. Jin, Cardinality constrained portfolio selection problem: A completely positive programming approach, Journal of Indstrial and Management Optimization, 12 (2016), 1041-1056.  doi: 10.3934/jimo.2016.12.1041.  Google Scholar

[28]

F. Xu, Z. Lv and Z. Xu, An efficient optimization approach for a cardinality-constrained index tracking problem, Optimization Methods and Software, 31 (2016), 258-271.  doi: 10.1080/10556788.2015.1062891.  Google Scholar

[29]

F. Xu, G. Wang and Y. Gao, Nonconvex L1/2 regularization for sparse portfolio selection, Pacific Journal of Optimization, 10 (2014), 163-176.   Google Scholar

[30]

H. Xu and D. Zhang, Monte Carlo methods for mean-risk optimization and portfolio selection, Computational Management Science, 9 (2012), 3-29.  doi: 10.1007/s10287-010-0123-6.  Google Scholar

[31]

L. Xue, S. Ma and H. Zou, Positive-definite $\ell_{1}$-penalized estimation of large covariance matrices, Journal of the American Statistical Association, 107 (2012), 1480-1491.  doi: 10.1080/01621459.2012.725386.  Google Scholar

[32]

J. Ye, Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints, SIAM Journal on Optimization, 10 (2000), 943-962.  doi: 10.1137/S105262349834847X.  Google Scholar

[33]

X. Zheng, X. Sun and D. Li, Improving the performance of MIQP solvers for quadratic programs with cardinality and minimum threshold constraints: A semidefinite program approach, INFORMS Journal on Computing, 26 (2014), 690-703.  doi: 10.1287/ijoc.2014.0592.  Google Scholar

Figure 1.  The convergence of the SAA problem in Example 4.1
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Figure 2.  S & P 500 Portfolio (a)Sharpe Ratio, (b)Sparsity. The bar from the left to the right in each test stands for LCP sparse portfolio, $\ell_1$ 0.1, CCPS100 and 1/N
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Figure 3.  Hong Kong and Mainland China Cross Market Portfolio (a)Sharpe Ratio, (b)Sparsity. The bar from the left to the right in each test stands for LCP sparse portfolio, $\ell_1$ 0.1, CCPS20, CCPS25, $\ell_p$ 0.015 and 1/N
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Figure 4.  FF48 Portfolio (a)Sharpe Ratio, (b)Sparsity. The bar from the left to the right in each test stands for LCP sparse portfolio, $\ell_1$ 0.1, CCPS18, CCPS24, $\ell_p$ 0.015 and 1/N
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Table 1.  Convergence analysis of SAA sparse portfolio optimal value (STD) for Example 4.1
Ntrue50015003000450060007500900010000dmissing
Val2.4892.4902.4872.4872.4872.4892.4882.4902.4862.496
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Ntrue50015003000450060007500900010000dmissing
Val2.4892.4902.4872.4872.4872.4892.4882.4902.4862.496
Table 2.  S & P 500 Portfolio return, STD, Sharpe Ratio, sparsity, VaR, CVaR and distance
S & P 500LCPSP $\ell_1$ $0.1$CCPS1001/N
return0.0010.0008230.0014-0.00003
STD0.00240.0022870.00840.0074
Sharpe0.39890.3597360.1694-0.0045
VaR0.00410.0042890.01150.0131
CVaR0.00460.0042920.01470.0131
sparsity89(406)66.2558.6500
distance1.00E-053.50E-07
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S & P 500LCPSP $\ell_1$ $0.1$CCPS1001/N
return0.0010.0008230.0014-0.00003
STD0.00240.0022870.00840.0074
Sharpe0.39890.3597360.1694-0.0045
VaR0.00410.0042890.01150.0131
CVaR0.00460.0042920.01470.0131
sparsity89(406)66.2558.6500
distance1.00E-053.50E-07
Table 3.  Hong Kong and Mainland China Cross Market Portfolio return, STD, Sharpe Ratio, sparsity, VaR and CVaR and distance
HKCHNLCPSP $\ell_1$ $0.1$CCPS 20CCPS25 $\ell_p$ 0.0151/N
return0.0012960.000440.0013480.0015830.000537-0.00171
STD0.0073450.0071150.0136170.0123680.0102480.006009
Sharpe0.17640.0619030.0990.1280.0524-0.2841
VaR0.0132480.0115140.0235920.0241820.0147560.010944
CVaR0.014080.0118070.0266920.0243820.0157910.010944
sparsity27(49)1419.4523.124.1
distance0.00240.0034060.000120.144709
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HKCHNLCPSP $\ell_1$ $0.1$CCPS 20CCPS25 $\ell_p$ 0.0151/N
return0.0012960.000440.0013480.0015830.000537-0.00171
STD0.0073450.0071150.0136170.0123680.0102480.006009
Sharpe0.17640.0619030.0990.1280.0524-0.2841
VaR0.0132480.0115140.0235920.0241820.0147560.010944
CVaR0.014080.0118070.0266920.0243820.0157910.010944
sparsity27(49)1419.4523.124.1
distance0.00240.0034060.000120.144709
Table 4.  FF48 Portfolio return, STD, Sharpe Ratio, VaR, CVaR, sparsity and distance
FF48LCPSP $\ell_1$ $0.1$CCPS18CCPS 24 $\ell_p$ 0.0151/N
return-0.1201-0.13259-0.6413-0.3736-0.3334-0.7838
STD5.92655.9171138.56397.57035.35458.002
Sharpe-0.0203-0.0224-0.0749-0.0493-0.0623-0.098
VaR8.524214.5989612.89059.863510.00797.827
CVaR10.383514.8659415.51412.894713.539510.0896
sparsity29(48)24.3521.457.231.248
distance0.00440.22320.56080.0938
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FF48LCPSP $\ell_1$ $0.1$CCPS18CCPS 24 $\ell_p$ 0.0151/N
return-0.1201-0.13259-0.6413-0.3736-0.3334-0.7838
STD5.92655.9171138.56397.57035.35458.002
Sharpe-0.0203-0.0224-0.0749-0.0493-0.0623-0.098
VaR8.524214.5989612.89059.863510.00797.827
CVaR10.383514.8659415.51412.894713.539510.0896
sparsity29(48)24.3521.457.231.248
distance0.00440.22320.56080.0938
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