Figure 1.
The convergence of the SAA problem in Example 4.1
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Integrated recycling-integrated production - distribution planning for decentralized closed-loop supply chain
April
2018, 14(2): 541-559.
doi: 10.3934/jimo.2017059
Sparse markowitz portfolio selection by using stochastic linear complementarity approach
| 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 |
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 and Management Optimization,
2018, 14
(2)
: 541-559.
doi: 10.3934/jimo.2017059
![]()
References:
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D. Bertsimas and R. Shioda,
Algorithms for cardinality-constrained quadratic optimization, Computational Optimization and Applications, 43 (2009), 1-22.
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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. |
| [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. |
| [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. |
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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. |
| [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.
|
| [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. |
| [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. |
| [9] |
C. Chen, X. Li, C. Tolman, S. Wang and Y. Ye, Sparse portfolio selection via quasi-norm regularization, preprint, arXiv: 1312.6350. |
| [10] |
X. Chen,
Smoothing methods for nonsmooth, nonconvex minimization, Mathematical Programming, 134 (2012), 71-99.
doi: 10.1007/s10107-012-0569-0. |
| [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. |
| [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. |
| [13] |
R. W. Cottle, J. -S. Pang and R. E. Stone,
The Linear Complementarity Problem, Academic Press, Boston, MA, 1992. |
| [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] |
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. |
| [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. |
| [18] |
J. Gao and D. Li,
Optimal cardinality constrained portfolio selection, Operations Research, 61 (2013), 745-761.
doi: 10.1287/opre.2013.1170. |
| [19] |
H. Markowitz,
Portfolio selection, The Journal of Finance, 7 (1952), 77-91.
doi: 10.1111/j.1540-6261.1952.tb01525.x. |
| [20] |
A. J McNeil, R. Frey and P. Embrechts,
Quantitative Risk Management: Concepts, Techniques and Tools, Princeton University Press, 2015. |
| [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. |
| [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. |
| [23] |
B. K. Natarajan,
Sparse approximate solutions to linear systems, SIAM Journal on Computing, 24 (1995), 227-234.
doi: 10.1137/S0097539792240406. |
| [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. |
| [25] |
W. F. Sharpe,
The Sharpe ratio, The Journal of Portfolio Management, 21 (1994), 49-58.
doi: 10.3905/jpm.1994.409501. |
| [26] |
A. Shleifer and R. W. Vishny,
The limits of arbitrage, The Journal of Finance, 52 (1997), 35-55.
|
| [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. |
| [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. |
| [29] |
F. Xu, G. Wang and Y. Gao,
Nonconvex L1/2 regularization for sparse portfolio selection, Pacific Journal of Optimization, 10 (2014), 163-176.
|
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [9] |
C. Chen, X. Li, C. Tolman, S. Wang and Y. Ye, Sparse portfolio selection via quasi-norm regularization, preprint, arXiv: 1312.6350. |
| [10] |
X. Chen,
Smoothing methods for nonsmooth, nonconvex minimization, Mathematical Programming, 134 (2012), 71-99.
doi: 10.1007/s10107-012-0569-0. |
| [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. |
| [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. |
| [13] |
R. W. Cottle, J. -S. Pang and R. E. Stone,
The Linear Complementarity Problem, Academic Press, Boston, MA, 1992. |
| [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] |
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. |
| [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. |
| [18] |
J. Gao and D. Li,
Optimal cardinality constrained portfolio selection, Operations Research, 61 (2013), 745-761.
doi: 10.1287/opre.2013.1170. |
| [19] |
H. Markowitz,
Portfolio selection, The Journal of Finance, 7 (1952), 77-91.
doi: 10.1111/j.1540-6261.1952.tb01525.x. |
| [20] |
A. J McNeil, R. Frey and P. Embrechts,
Quantitative Risk Management: Concepts, Techniques and Tools, Princeton University Press, 2015. |
| [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. |
| [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. |
| [23] |
B. K. Natarajan,
Sparse approximate solutions to linear systems, SIAM Journal on Computing, 24 (1995), 227-234.
doi: 10.1137/S0097539792240406. |
| [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. |
| [25] |
W. F. Sharpe,
The Sharpe ratio, The Journal of Portfolio Management, 21 (1994), 49-58.
doi: 10.3905/jpm.1994.409501. |
| [26] |
A. Shleifer and R. W. Vishny,
The limits of arbitrage, The Journal of Finance, 52 (1997), 35-55.
|
| [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. |
| [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. |
| [29] |
F. Xu, G. Wang and Y. Gao,
Nonconvex L1/2 regularization for sparse portfolio selection, Pacific Journal of Optimization, 10 (2014), 163-176.
|
| [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. |
| [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. |
| [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. |
| [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. |
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
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
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
Table 1.
Convergence analysis of SAA sparse portfolio optimal value (STD) for Example 4.1
| N | true | 500 | 1500 | 3000 | 4500 | 6000 | 7500 | 9000 | 10000 | dmissing |
| Val | 2.489 | 2.490 | 2.487 | 2.487 | 2.487 | 2.489 | 2.488 | 2.490 | 2.486 | 2.496 |
| N | true | 500 | 1500 | 3000 | 4500 | 6000 | 7500 | 9000 | 10000 | dmissing |
| Val | 2.489 | 2.490 | 2.487 | 2.487 | 2.487 | 2.489 | 2.488 | 2.490 | 2.486 | 2.496 |
Table 2.
S & P 500 Portfolio return, STD, Sharpe Ratio, sparsity, VaR, CVaR and distance
| S & P 500 | LCPSP | | CCPS100 | 1/N |
| return | 0.001 | 0.000823 | 0.0014 | -0.00003 |
| STD | 0.0024 | 0.002287 | 0.0084 | 0.0074 |
| Sharpe | 0.3989 | 0.359736 | 0.1694 | -0.0045 |
| VaR | 0.0041 | 0.004289 | 0.0115 | 0.0131 |
| CVaR | 0.0046 | 0.004292 | 0.0147 | 0.0131 |
| sparsity | 89(406) | 66.25 | 58.6 | 500 |
| distance | 1.00E-05 | 3.50E-07 |
| S & P 500 | LCPSP | | CCPS100 | 1/N |
| return | 0.001 | 0.000823 | 0.0014 | -0.00003 |
| STD | 0.0024 | 0.002287 | 0.0084 | 0.0074 |
| Sharpe | 0.3989 | 0.359736 | 0.1694 | -0.0045 |
| VaR | 0.0041 | 0.004289 | 0.0115 | 0.0131 |
| CVaR | 0.0046 | 0.004292 | 0.0147 | 0.0131 |
| sparsity | 89(406) | 66.25 | 58.6 | 500 |
| distance | 1.00E-05 | 3.50E-07 |
Table 3.
Hong Kong and Mainland China Cross Market Portfolio return, STD, Sharpe Ratio, sparsity, VaR and CVaR and distance
| HKCHN | LCPSP | | CCPS 20 | CCPS25 | | 1/N |
| return | 0.001296 | 0.00044 | 0.001348 | 0.001583 | 0.000537 | -0.00171 |
| STD | 0.007345 | 0.007115 | 0.013617 | 0.012368 | 0.010248 | 0.006009 |
| Sharpe | 0.1764 | 0.061903 | 0.099 | 0.128 | 0.0524 | -0.2841 |
| VaR | 0.013248 | 0.011514 | 0.023592 | 0.024182 | 0.014756 | 0.010944 |
| CVaR | 0.01408 | 0.011807 | 0.026692 | 0.024382 | 0.015791 | 0.010944 |
| sparsity | 27(49) | 14 | 19.45 | 23.1 | 24.1 | |
| distance | 0.0024 | 0.003406 | 0.00012 | 0.144709 |
| HKCHN | LCPSP | | CCPS 20 | CCPS25 | | 1/N |
| return | 0.001296 | 0.00044 | 0.001348 | 0.001583 | 0.000537 | -0.00171 |
| STD | 0.007345 | 0.007115 | 0.013617 | 0.012368 | 0.010248 | 0.006009 |
| Sharpe | 0.1764 | 0.061903 | 0.099 | 0.128 | 0.0524 | -0.2841 |
| VaR | 0.013248 | 0.011514 | 0.023592 | 0.024182 | 0.014756 | 0.010944 |
| CVaR | 0.01408 | 0.011807 | 0.026692 | 0.024382 | 0.015791 | 0.010944 |
| sparsity | 27(49) | 14 | 19.45 | 23.1 | 24.1 | |
| distance | 0.0024 | 0.003406 | 0.00012 | 0.144709 |
Table 4.
FF48 Portfolio return, STD, Sharpe Ratio, VaR, CVaR, sparsity and distance
| FF48 | LCPSP | | CCPS18 | CCPS 24 | | 1/N |
| return | -0.1201 | -0.13259 | -0.6413 | -0.3736 | -0.3334 | -0.7838 |
| STD | 5.9265 | 5.917113 | 8.5639 | 7.5703 | 5.3545 | 8.002 |
| Sharpe | -0.0203 | -0.0224 | -0.0749 | -0.0493 | -0.0623 | -0.098 |
| VaR | 8.5242 | 14.59896 | 12.8905 | 9.8635 | 10.0079 | 7.827 |
| CVaR | 10.3835 | 14.86594 | 15.514 | 12.8947 | 13.5395 | 10.0896 |
| sparsity | 29(48) | 24.35 | 21.45 | 7.2 | 31.2 | 48 |
| distance | 0.0044 | 0.2232 | 0.5608 | 0.0938 |
| FF48 | LCPSP | | CCPS18 | CCPS 24 | | 1/N |
| return | -0.1201 | -0.13259 | -0.6413 | -0.3736 | -0.3334 | -0.7838 |
| STD | 5.9265 | 5.917113 | 8.5639 | 7.5703 | 5.3545 | 8.002 |
| Sharpe | -0.0203 | -0.0224 | -0.0749 | -0.0493 | -0.0623 | -0.098 |
| VaR | 8.5242 | 14.59896 | 12.8905 | 9.8635 | 10.0079 | 7.827 |
| CVaR | 10.3835 | 14.86594 | 15.514 | 12.8947 | 13.5395 | 10.0896 |
| sparsity | 29(48) | 24.35 | 21.45 | 7.2 | 31.2 | 48 |
| distance | 0.0044 | 0.2232 | 0.5608 | 0.0938 |
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