October  2018, 14(4): 1651-1666. doi: 10.3934/jimo.2018025

A generalized approach to sparse and stable portfolio optimization problem

1. 

College of Mathematics and Statistics, Changsha University of Science and Technology, Hunan 410114, China

2. 

College of business, Central South University, Hunan 410083, China

3. 

Supply Chain and Logistics Optimization Research Centre, Faculty of Engineering, University of Windsor, Windsor, ON, Canada

* Corresponding author: Fenghua Wen

Received  August 2017 Published  October 2018 Early access  January 2018

In this paper, we firstly examine the relation between the portfolio weights norm constraints method and the objective function regularization method in portfolio selection problems. We find that the portfolio weights norm constrained method mainly tries to obtain stable portfolios, however, the objective function regularization method mainly aims at obtaining sparse portfolios. Then, we propose some general sparse and stable portfolio models by imposing both portfolio weights norm constraints and objective function $L_{1}$ regularization term. Finally, three empirical studies show that the proposed strategies have better out-of-sample performance and lower turnover than many other strategies for tested datasets.

Citation: Zhifeng Dai, Fenghua Wen. A generalized approach to sparse and stable portfolio optimization problem. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1651-1666. doi: 10.3934/jimo.2018025
References:
[1]

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

[2]

F. Black and R. Litterman, Global portfolio optimization, Journal of Financial and Analysis, 48 (1992), 28-43.  doi: 10.2469/faj.v48.n5.28.

[3]

J. BrodieI. DaubechiesC. De MolD. Giannone and I. Loris, Sparse and stable markowitz portfolios, Proceedings of the National Academy of Sciences of the United States of America, 106 (2009), 12267-12272.  doi: 10.1073/pnas.0904287106.

[4]

P. BehrA. Guettler and F. Miebs, On portfolio optimization: Imposing the right constraints, Journal of Banking and Finance, 37 (2013), 1232-1242. 

[5]

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

[6]

V. K. Chopra and W. T. Ziemba, The effect of errors in means, variance and covariances on optimal portfolio choice, Journal of Portfolio Management, 19 (1993), 6-11. 

[7]

C. H. Chen and Y. Y. Ye, Sparse portfolio selection via quasi-norm regularization, preprint, arXiv: 1312.6350, 2015.

[8]

X. T. CuiX. J. ZhengS. S. Zhu and X. L. Sun, Convex relaxations and MIQCQP reformulations for a class of cardinality-constrained portfolio selection problems, Journal of Global Optimization, 56 (2013), 1409-1423.  doi: 10.1007/s10898-012-9842-2.

[9]

Z. F. DaiD. H. Li and F. H. Wen, Worse-case conditional value-at-risk for asymmetrically distributed asset scenarios returns, Journal of Computational Analysis and Application, 20 (2016), 237-251. 

[10]

Z. F. DaiX. H. Chen and F. H. Wen, A modified Perry's conjugate gradient method-based derivative-free method for solving large-scale nonlinear monotone equations, Applied Mathematics and Computation, 270 (2015), 378-386.  doi: 10.1016/j.amc.2015.08.014.

[11]

V. DeMiguelL. Garlappi and R. Uppal, Optimal versus naive diversification: How ineffecient is the 1/n portfolio strategy?, Review of Financial Studies, 22 (2009), 1915-1953.  doi: 10.1093/acprof:oso/9780199744282.003.0034.

[12]

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

[13]

J. FanJ. Zhang and K. Yu, Vast portfolio selection with gross-exposure constraints, Journal of the American Statistical Association, 107 (2012), 592-606.  doi: 10.1080/01621459.2012.682825.

[14]

B. FastrichS. Paterlini and P. Winker, Constructing optimal sparse portfolios using regularization methods, Computational Management Science, 12 (2015), 417-434.  doi: 10.1007/s10287-014-0227-5.

[15]

A. Frangioni and C. Gentile, Perspective cuts for a class of convex 0-1 mixed integer programs, Mathematical Programming, 106 (2006), 225-236.  doi: 10.1007/s10107-005-0594-3.

[16]

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

[17]

R. Green and B. Hollifield, When will mean-variance efficient portfolios be well diversified?, Journal of Finance, 47 (1992), 1785-1809.  doi: 10.1111/j.1540-6261.1992.tb04683.x.

[18]

M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1. 21., http://cvxr.com/cvx. 2010.

[19]

W. James and C. Stein, Estimation with quadratic loss, Proc. 4th Berkeley Sympos. Probab. Statist., University of California Press, Berkeley, 1 (1961), 361--379.

[20]

R. Jagannathan and T. Ma, Risk reduction in large portfolios: Why imposing the wrong constraints helps, Journal of Finance, 58 (2003), 1651-1684. 

[21]

O. Ledoit and M. Wolf, Improved estimation of the covariance matrix of stock returns with an application to portfolio selection, Journal of Empirical Finance, 10 (2003), 603-621.  doi: 10.1016/S0927-5398(03)00007-0.

[22]

O. Ledoit and M. Wolf, A well-conditioned estimator for large-dimensional covariance matrices, Journal of Multivariate Analysis, 88 (2004), 365-411.  doi: 10.1016/S0047-259X(03)00096-4.

[23]

D. LiX. L. Sun and J. Wang, Optimal lot solution to cardinality constrained mean-variance formulation for portfolio selection, Mathematical Finance, 16 (2006), 83-101.  doi: 10.1111/j.1467-9965.2006.00262.x.

[24]

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

[25]

D. Maringer and H. Kellerer, Optimization of cardinality constrained portfolios with a hybrid local search algorithm, OR Spectrum, 25 (2003), 481-495.  doi: 10.1007/s00291-003-0139-1.

[26]

D. X. ShawS. Liu and L. Kopman, Lagrangian relaxation procedure for cardinality-constrained portfolio optimization, Optimization Methods and Software, 23 (2008), 411-420.  doi: 10.1080/10556780701722542.

[27]

F. H. WenZ. He and Z. Dai, Characteristics of Investors' Risk Preference for Stock Markets, Economic Computation and Economic Cybernetics Studies and Research, 48 (2014), 235-254. 

[28]

F. H. WenX. Gong and S. Cai, Forecasting the volatility of crude oil futures using HAR-type models with structural breaks, Energy Economics, 59 (2016), 400-413.  doi: 10.1016/j.eneco.2016.07.014.

[29]

F. H. WenJ. Xiao and C. Huang, Interaction between oil and US dollar exchange rate: Nonlinear causality, time-varying influence and structural breaks in volatility, Applied Economics, 50 (2018), 319-334.  doi: 10.1080/00036846.2017.1321838.

[30]

J. XieS. He and S. Zhang, Randomized portfolio selection, with constraints, Pacific Journal of Optimization, 4 (2008), 87-112. 

[31]

X. XingJ. J. Hub and Y. Yang, Robust minimum variance portfolio with L-infinity constraints, Journal of Banking and Finance, 46 (2014), 107-117.  doi: 10.1016/j.jbankfin.2014.05.004.

[32]

F. M. XuG. Wang and Y. L. Gao, Nonconvex $L_{1/2}$ regularization for sparse portfolio selection, Pacific Journal of Optimization, 10 (2014), 163-176. 

[33]

X. J. ZhengX. L. 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 of Computing, 26 (2014), 690-703.  doi: 10.1287/ijoc.2014.0592.

show all references

References:
[1]

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

[2]

F. Black and R. Litterman, Global portfolio optimization, Journal of Financial and Analysis, 48 (1992), 28-43.  doi: 10.2469/faj.v48.n5.28.

[3]

J. BrodieI. DaubechiesC. De MolD. Giannone and I. Loris, Sparse and stable markowitz portfolios, Proceedings of the National Academy of Sciences of the United States of America, 106 (2009), 12267-12272.  doi: 10.1073/pnas.0904287106.

[4]

P. BehrA. Guettler and F. Miebs, On portfolio optimization: Imposing the right constraints, Journal of Banking and Finance, 37 (2013), 1232-1242. 

[5]

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

[6]

V. K. Chopra and W. T. Ziemba, The effect of errors in means, variance and covariances on optimal portfolio choice, Journal of Portfolio Management, 19 (1993), 6-11. 

[7]

C. H. Chen and Y. Y. Ye, Sparse portfolio selection via quasi-norm regularization, preprint, arXiv: 1312.6350, 2015.

[8]

X. T. CuiX. J. ZhengS. S. Zhu and X. L. Sun, Convex relaxations and MIQCQP reformulations for a class of cardinality-constrained portfolio selection problems, Journal of Global Optimization, 56 (2013), 1409-1423.  doi: 10.1007/s10898-012-9842-2.

[9]

Z. F. DaiD. H. Li and F. H. Wen, Worse-case conditional value-at-risk for asymmetrically distributed asset scenarios returns, Journal of Computational Analysis and Application, 20 (2016), 237-251. 

[10]

Z. F. DaiX. H. Chen and F. H. Wen, A modified Perry's conjugate gradient method-based derivative-free method for solving large-scale nonlinear monotone equations, Applied Mathematics and Computation, 270 (2015), 378-386.  doi: 10.1016/j.amc.2015.08.014.

[11]

V. DeMiguelL. Garlappi and R. Uppal, Optimal versus naive diversification: How ineffecient is the 1/n portfolio strategy?, Review of Financial Studies, 22 (2009), 1915-1953.  doi: 10.1093/acprof:oso/9780199744282.003.0034.

[12]

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

[13]

J. FanJ. Zhang and K. Yu, Vast portfolio selection with gross-exposure constraints, Journal of the American Statistical Association, 107 (2012), 592-606.  doi: 10.1080/01621459.2012.682825.

[14]

B. FastrichS. Paterlini and P. Winker, Constructing optimal sparse portfolios using regularization methods, Computational Management Science, 12 (2015), 417-434.  doi: 10.1007/s10287-014-0227-5.

[15]

A. Frangioni and C. Gentile, Perspective cuts for a class of convex 0-1 mixed integer programs, Mathematical Programming, 106 (2006), 225-236.  doi: 10.1007/s10107-005-0594-3.

[16]

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

[17]

R. Green and B. Hollifield, When will mean-variance efficient portfolios be well diversified?, Journal of Finance, 47 (1992), 1785-1809.  doi: 10.1111/j.1540-6261.1992.tb04683.x.

[18]

M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1. 21., http://cvxr.com/cvx. 2010.

[19]

W. James and C. Stein, Estimation with quadratic loss, Proc. 4th Berkeley Sympos. Probab. Statist., University of California Press, Berkeley, 1 (1961), 361--379.

[20]

R. Jagannathan and T. Ma, Risk reduction in large portfolios: Why imposing the wrong constraints helps, Journal of Finance, 58 (2003), 1651-1684. 

[21]

O. Ledoit and M. Wolf, Improved estimation of the covariance matrix of stock returns with an application to portfolio selection, Journal of Empirical Finance, 10 (2003), 603-621.  doi: 10.1016/S0927-5398(03)00007-0.

[22]

O. Ledoit and M. Wolf, A well-conditioned estimator for large-dimensional covariance matrices, Journal of Multivariate Analysis, 88 (2004), 365-411.  doi: 10.1016/S0047-259X(03)00096-4.

[23]

D. LiX. L. Sun and J. Wang, Optimal lot solution to cardinality constrained mean-variance formulation for portfolio selection, Mathematical Finance, 16 (2006), 83-101.  doi: 10.1111/j.1467-9965.2006.00262.x.

[24]

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

[25]

D. Maringer and H. Kellerer, Optimization of cardinality constrained portfolios with a hybrid local search algorithm, OR Spectrum, 25 (2003), 481-495.  doi: 10.1007/s00291-003-0139-1.

[26]

D. X. ShawS. Liu and L. Kopman, Lagrangian relaxation procedure for cardinality-constrained portfolio optimization, Optimization Methods and Software, 23 (2008), 411-420.  doi: 10.1080/10556780701722542.

[27]

F. H. WenZ. He and Z. Dai, Characteristics of Investors' Risk Preference for Stock Markets, Economic Computation and Economic Cybernetics Studies and Research, 48 (2014), 235-254. 

[28]

F. H. WenX. Gong and S. Cai, Forecasting the volatility of crude oil futures using HAR-type models with structural breaks, Energy Economics, 59 (2016), 400-413.  doi: 10.1016/j.eneco.2016.07.014.

[29]

F. H. WenJ. Xiao and C. Huang, Interaction between oil and US dollar exchange rate: Nonlinear causality, time-varying influence and structural breaks in volatility, Applied Economics, 50 (2018), 319-334.  doi: 10.1080/00036846.2017.1321838.

[30]

J. XieS. He and S. Zhang, Randomized portfolio selection, with constraints, Pacific Journal of Optimization, 4 (2008), 87-112. 

[31]

X. XingJ. J. Hub and Y. Yang, Robust minimum variance portfolio with L-infinity constraints, Journal of Banking and Finance, 46 (2014), 107-117.  doi: 10.1016/j.jbankfin.2014.05.004.

[32]

F. M. XuG. Wang and Y. L. Gao, Nonconvex $L_{1/2}$ regularization for sparse portfolio selection, Pacific Journal of Optimization, 10 (2014), 163-176. 

[33]

X. J. ZhengX. L. 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 of Computing, 26 (2014), 690-703.  doi: 10.1287/ijoc.2014.0592.

Table 1.  List of Datasets
No.DatasetNumber of assetTime PeriodSourse
1FF-10010012/1992-12/2014K. French
2FF-484812/1992-12/2014K. French
3500 CRSP50004/1992-04/2014CRSP
4100 CRSP10004/1992-04/2014CRSP
5S&P 50046112/2007-12/2014Datastream
No.DatasetNumber of assetTime PeriodSourse
1FF-10010012/1992-12/2014K. French
2FF-484812/1992-12/2014K. French
3500 CRSP50004/1992-04/2014CRSP
4100 CRSP10004/1992-04/2014CRSP
5S&P 50046112/2007-12/2014Datastream
Table 2.  Portfolio variances of all considered portfolio strategies.
Dataset500 CRSP100 CRSPS&P 500FF-100FF48
Model 10.000490.001090.000600.000980.00119
Model 20.000420.001020.000520.000910.00111
1/N0.001720.001680.001790.002080.00232
MINC0.000940.001390.001080.001420.00141
NC1V0.000810.001190.000780.001210.00129
NC2V0.000730.001210.000840.001250.00132
NCFV0.000700.001230.000870.001200.00125
CC-Minvar0.000720.001130.000820.001020.00120
Dataset500 CRSP100 CRSPS&P 500FF-100FF48
Model 10.000490.001090.000600.000980.00119
Model 20.000420.001020.000520.000910.00111
1/N0.001720.001680.001790.002080.00232
MINC0.000940.001390.001080.001420.00141
NC1V0.000810.001190.000780.001210.00129
NC2V0.000730.001210.000840.001250.00132
NCFV0.000700.001230.000870.001200.00125
CC-Minvar0.000720.001130.000820.001020.00120
Table 3.  Out-of-sample Sharpe ratio of the portfolio strategies.
Dataset500 CRSP100 CRSPS&P 500FF-100FF48
Model 10.42780.44600.43140.39980.3146
Model 20.46360.45480.45680.40240.3220
1/N0.31020.33580.35860.28080.2524
MINC0.38820.36490.37230.31420.2712
NC1V0.40010.41220.40550.32240.2922
NC2V0.41510.42120.41860.35220.2916
NCFV0.40630.41360.40280.34580.2806
CC-Minvar0.40420.43260.41010.38750.3206
Dataset500 CRSP100 CRSPS&P 500FF-100FF48
Model 10.42780.44600.43140.39980.3146
Model 20.46360.45480.45680.40240.3220
1/N0.31020.33580.35860.28080.2524
MINC0.38820.36490.37230.31420.2712
NC1V0.40010.41220.40550.32240.2922
NC2V0.41510.42120.41860.35220.2916
NCFV0.40630.41360.40280.34580.2806
CC-Minvar0.40420.43260.41010.38750.3206
Table 4.  Turnover of the portfolio strategies.
Dataset500 CRSP100 CRSPS&P 500FF-100FF48
Model 10.40280.31240.41200.30680.2580
Model 20.41450.31820.41660.30220.2528
1/N0.06250.04450.05860.05080.0324
MINC0.31250.20250.40300.22210.1822
NC1V0.66540.46700.62080.53080.2822
NC2V0.60220.42490.60300.54210.3168
NCFV0.59480.41320.58700.51340.2762
CC-Minvar0.35420.28020.39800.26320.2356
Dataset500 CRSP100 CRSPS&P 500FF-100FF48
Model 10.40280.31240.41200.30680.2580
Model 20.41450.31820.41660.30220.2528
1/N0.06250.04450.05860.05080.0324
MINC0.31250.20250.40300.22210.1822
NC1V0.66540.46700.62080.53080.2822
NC2V0.60220.42490.60300.54210.3168
NCFV0.59480.41320.58700.51340.2762
CC-Minvar0.35420.28020.39800.26320.2356
[1]

Chao Zhang, Jingjing Wang, Naihua Xiu. Robust and sparse portfolio model for index tracking. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1001-1015. doi: 10.3934/jimo.2018082

[2]

Tao Pang, Azmat Hussain. An infinite time horizon portfolio optimization model with delays. Mathematical Control and Related Fields, 2016, 6 (4) : 629-651. doi: 10.3934/mcrf.2016018

[3]

Torsten Trimborn, Lorenzo Pareschi, Martin Frank. Portfolio optimization and model predictive control: A kinetic approach. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6209-6238. doi: 10.3934/dcdsb.2019136

[4]

Jiannan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model. Journal of Industrial and Management Optimization, 2021, 17 (2) : 765-777. doi: 10.3934/jimo.2019133

[5]

Chenchen Zu, Xiaoqi Yang, Carisa Kwok Wai Yu. Sparse minimax portfolio and Sharpe ratio models. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021111

[6]

Yan Zeng, Zhongfei Li, Jingjun Liu. Optimal strategies of benchmark and mean-variance portfolio selection problems for insurers. Journal of Industrial and Management Optimization, 2010, 6 (3) : 483-496. doi: 10.3934/jimo.2010.6.483

[7]

Ishak Alia, Mohamed Sofiane Alia. Open-loop equilibrium strategy for mean-variance Portfolio selection with investment constraints in a non-Markovian regime-switching jump-diffusion model. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022048

[8]

Yang Shen, Tak Kuen Siu. Consumption-portfolio optimization and filtering in a hidden Markov-modulated asset price model. Journal of Industrial and Management Optimization, 2017, 13 (1) : 23-46. doi: 10.3934/jimo.2016002

[9]

Wawan Hafid Syaifudin, Endah R. M. Putri. The application of model predictive control on stock portfolio optimization with prediction based on Geometric Brownian Motion-Kalman Filter. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021119

[10]

Xuhui Wang, Lei Hu. A new method to solve the Hamilton-Jacobi-Bellman equation for a stochastic portfolio optimization model with boundary memory. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021137

[11]

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

[12]

Yong Wang, Wanquan Liu, Guanglu Zhou. An efficient algorithm for non-convex sparse optimization. Journal of Industrial and Management Optimization, 2019, 15 (4) : 2009-2021. doi: 10.3934/jimo.2018134

[13]

Yue Qi, Zhihao Wang, Su Zhang. On analyzing and detecting multiple optima of portfolio optimization. Journal of Industrial and Management Optimization, 2018, 14 (1) : 309-323. doi: 10.3934/jimo.2017048

[14]

Yufei Sun, Grace Aw, Kok Lay Teo, Guanglu Zhou. Portfolio optimization using a new probabilistic risk measure. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1275-1283. doi: 10.3934/jimo.2015.11.1275

[15]

Xueting Cui, Xiaoling Sun, Dan Sha. An empirical study on discrete optimization models for portfolio selection. Journal of Industrial and Management Optimization, 2009, 5 (1) : 33-46. doi: 10.3934/jimo.2009.5.33

[16]

Lijun Bo. Portfolio optimization of credit swap under funding costs. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 12-. doi: 10.1186/s41546-017-0023-6

[17]

Ping Chen, Haixiang Yao. Continuous-time mean-variance portfolio selection with no-shorting constraints and regime-switching. Journal of Industrial and Management Optimization, 2020, 16 (2) : 531-551. doi: 10.3934/jimo.2018166

[18]

Shuang Li, Chuong Luong, Francisca Angkola, Yonghong Wu. Optimal asset portfolio with stochastic volatility under the mean-variance utility with state-dependent risk aversion. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1521-1533. doi: 10.3934/jimo.2016.12.1521

[19]

Zhen Wang, Sanyang Liu. Multi-period mean-variance portfolio selection with fixed and proportional transaction costs. Journal of Industrial and Management Optimization, 2013, 9 (3) : 643-657. doi: 10.3934/jimo.2013.9.643

[20]

Ning Zhang. A symmetric Gauss-Seidel based method for a class of multi-period mean-variance portfolio selection problems. Journal of Industrial and Management Optimization, 2020, 16 (2) : 991-1008. doi: 10.3934/jimo.2018189

2020 Impact Factor: 1.801

Metrics

  • PDF downloads (1616)
  • HTML views (2185)
  • Cited by (6)

Other articles
by authors

[Back to Top]