A generalized approach to sparse and stable portfolio optimization problem

Zhifeng Dai; Fenghua Wen

doi:10.3934/jimo.2018025

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

A generalized approach to sparse and stable portfolio optimization problem

Zhifeng Dai ^1, and Fenghua Wen ^2,3,,

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 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.

Keywords: Portfolio optimization, minimum-variance model, sparse and stable portfolios.

Mathematics Subject Classification: Primary: 90C15, 90C90; Secondary: 91G10.

Citation: Zhifeng Dai, Fenghua Wen. A generalized approach to sparse and stable portfolio optimization problem. Journal of Industrial & 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. Google Scholar
[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. Google Scholar
[3]	J. Brodie, I. Daubechies, C. De Mol, D. 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. Google Scholar
[4]	P. Behr, A. Guettler and F. Miebs, On portfolio optimization: Imposing the right constraints, Journal of Banking and Finance, 37 (2013), 1232-1242. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	C. H. Chen and Y. Y. Ye, Sparse portfolio selection via quasi-norm regularization, preprint, arXiv: 1312.6350, 2015.Google Scholar
[8]	X. T. Cui, X. J. Zheng, S. 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. Google Scholar
[9]	Z. F. Dai, D. 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. Google Scholar
[10]	Z. F. Dai, X. 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. Google Scholar
[11]	V. DeMiguel, L. 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. Google Scholar
[12]	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
[13]	J. Fan, J. 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. Google Scholar
[14]	B. Fastrich, S. 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. Google Scholar
[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. Google Scholar
[16]	J. J. Gao and D. Li, Optimal cardinality constrained portfolio selection, Operational Research, 61 (2013), 745-761. doi: 10.1287/opre.2013.1170. Google Scholar
[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. Google Scholar
[18]	M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1. 21., http://cvxr.com/cvx. 2010.Google Scholar
[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. Google Scholar
[20]	R. Jagannathan and T. Ma, Risk reduction in large portfolios: Why imposing the wrong constraints helps, Journal of Finance, 58 (2003), 1651-1684. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	D. Li, X. 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. Google Scholar
[24]	H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. Google Scholar
[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. Google Scholar
[26]	D. X. Shaw, S. Liu and L. Kopman, Lagrangian relaxation procedure for cardinality-constrained portfolio optimization, Optimization Methods and Software, 23 (2008), 411-420. doi: 10.1080/10556780701722542. Google Scholar
[27]	F. H. Wen, Z. He and Z. Dai, Characteristics of Investors' Risk Preference for Stock Markets, Economic Computation and Economic Cybernetics Studies and Research, 48 (2014), 235-254. Google Scholar
[28]	F. H. Wen, X. 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. Google Scholar
[29]	F. H. Wen, J. 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. Google Scholar
[30]	J. Xie, S. He and S. Zhang, Randomized portfolio selection, with constraints, Pacific Journal of Optimization, 4 (2008), 87-112. Google Scholar
[31]	X. Xing, J. 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. Google Scholar
[32]	F. M. Xu, G. Wang and Y. L. Gao, Nonconvex $L_{1/2}$ regularization for sparse portfolio selection, Pacific Journal of Optimization, 10 (2014), 163-176. Google Scholar
[33]	X. J. Zheng, X. 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. Google Scholar

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. Google Scholar
[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. Google Scholar
[3]	J. Brodie, I. Daubechies, C. De Mol, D. 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. Google Scholar
[4]	P. Behr, A. Guettler and F. Miebs, On portfolio optimization: Imposing the right constraints, Journal of Banking and Finance, 37 (2013), 1232-1242. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	C. H. Chen and Y. Y. Ye, Sparse portfolio selection via quasi-norm regularization, preprint, arXiv: 1312.6350, 2015.Google Scholar
[8]	X. T. Cui, X. J. Zheng, S. 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. Google Scholar
[9]	Z. F. Dai, D. 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. Google Scholar
[10]	Z. F. Dai, X. 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. Google Scholar
[11]	V. DeMiguel, L. 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. Google Scholar
[12]	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
[13]	J. Fan, J. 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. Google Scholar
[14]	B. Fastrich, S. 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. Google Scholar
[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. Google Scholar
[16]	J. J. Gao and D. Li, Optimal cardinality constrained portfolio selection, Operational Research, 61 (2013), 745-761. doi: 10.1287/opre.2013.1170. Google Scholar
[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. Google Scholar
[18]	M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1. 21., http://cvxr.com/cvx. 2010.Google Scholar
[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. Google Scholar
[20]	R. Jagannathan and T. Ma, Risk reduction in large portfolios: Why imposing the wrong constraints helps, Journal of Finance, 58 (2003), 1651-1684. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	D. Li, X. 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. Google Scholar
[24]	H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. Google Scholar
[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. Google Scholar
[26]	D. X. Shaw, S. Liu and L. Kopman, Lagrangian relaxation procedure for cardinality-constrained portfolio optimization, Optimization Methods and Software, 23 (2008), 411-420. doi: 10.1080/10556780701722542. Google Scholar
[27]	F. H. Wen, Z. He and Z. Dai, Characteristics of Investors' Risk Preference for Stock Markets, Economic Computation and Economic Cybernetics Studies and Research, 48 (2014), 235-254. Google Scholar
[28]	F. H. Wen, X. 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. Google Scholar
[29]	F. H. Wen, J. 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. Google Scholar
[30]	J. Xie, S. He and S. Zhang, Randomized portfolio selection, with constraints, Pacific Journal of Optimization, 4 (2008), 87-112. Google Scholar
[31]	X. Xing, J. 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. Google Scholar
[32]	F. M. Xu, G. Wang and Y. L. Gao, Nonconvex $L_{1/2}$ regularization for sparse portfolio selection, Pacific Journal of Optimization, 10 (2014), 163-176. Google Scholar
[33]	X. J. Zheng, X. 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. Google Scholar

Table 1. List of Datasets

No.	Dataset	Number of asset	Time Period	Sourse
1	FF-100	100	12/1992-12/2014	K. French
2	FF-48	48	12/1992-12/2014	K. French
3	500 CRSP	500	04/1992-04/2014	CRSP
4	100 CRSP	100	04/1992-04/2014	CRSP
5	S&P 500	461	12/2007-12/2014	Datastream

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No.	Dataset	Number of asset	Time Period	Sourse
1	FF-100	100	12/1992-12/2014	K. French
2	FF-48	48	12/1992-12/2014	K. French
3	500 CRSP	500	04/1992-04/2014	CRSP
4	100 CRSP	100	04/1992-04/2014	CRSP
5	S&P 500	461	12/2007-12/2014	Datastream

Table 2. Portfolio variances of all considered portfolio strategies.

Dataset	500 CRSP	100 CRSP	S&P 500	FF-100	FF48
Model 1	0.00049	0.00109	0.00060	0.00098	0.00119
Model 2	0.00042	0.00102	0.00052	0.00091	0.00111
1/N	0.00172	0.00168	0.00179	0.00208	0.00232
MINC	0.00094	0.00139	0.00108	0.00142	0.00141
NC1V	0.00081	0.00119	0.00078	0.00121	0.00129
NC2V	0.00073	0.00121	0.00084	0.00125	0.00132
NCFV	0.00070	0.00123	0.00087	0.00120	0.00125
CC-Minvar	0.00072	0.00113	0.00082	0.00102	0.00120

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Dataset	500 CRSP	100 CRSP	S&P 500	FF-100	FF48
Model 1	0.00049	0.00109	0.00060	0.00098	0.00119
Model 2	0.00042	0.00102	0.00052	0.00091	0.00111
1/N	0.00172	0.00168	0.00179	0.00208	0.00232
MINC	0.00094	0.00139	0.00108	0.00142	0.00141
NC1V	0.00081	0.00119	0.00078	0.00121	0.00129
NC2V	0.00073	0.00121	0.00084	0.00125	0.00132
NCFV	0.00070	0.00123	0.00087	0.00120	0.00125
CC-Minvar	0.00072	0.00113	0.00082	0.00102	0.00120

Table 3. Out-of-sample Sharpe ratio of the portfolio strategies.

Dataset	500 CRSP	100 CRSP	S&P 500	FF-100	FF48
Model 1	0.4278	0.4460	0.4314	0.3998	0.3146
Model 2	0.4636	0.4548	0.4568	0.4024	0.3220
1/N	0.3102	0.3358	0.3586	0.2808	0.2524
MINC	0.3882	0.3649	0.3723	0.3142	0.2712
NC1V	0.4001	0.4122	0.4055	0.3224	0.2922
NC2V	0.4151	0.4212	0.4186	0.3522	0.2916
NCFV	0.4063	0.4136	0.4028	0.3458	0.2806
CC-Minvar	0.4042	0.4326	0.4101	0.3875	0.3206

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Dataset	500 CRSP	100 CRSP	S&P 500	FF-100	FF48
Model 1	0.4278	0.4460	0.4314	0.3998	0.3146
Model 2	0.4636	0.4548	0.4568	0.4024	0.3220
1/N	0.3102	0.3358	0.3586	0.2808	0.2524
MINC	0.3882	0.3649	0.3723	0.3142	0.2712
NC1V	0.4001	0.4122	0.4055	0.3224	0.2922
NC2V	0.4151	0.4212	0.4186	0.3522	0.2916
NCFV	0.4063	0.4136	0.4028	0.3458	0.2806
CC-Minvar	0.4042	0.4326	0.4101	0.3875	0.3206

Table 4. Turnover of the portfolio strategies.

Dataset	500 CRSP	100 CRSP	S&P 500	FF-100	FF48
Model 1	0.4028	0.3124	0.4120	0.3068	0.2580
Model 2	0.4145	0.3182	0.4166	0.3022	0.2528
1/N	0.0625	0.0445	0.0586	0.0508	0.0324
MINC	0.3125	0.2025	0.4030	0.2221	0.1822
NC1V	0.6654	0.4670	0.6208	0.5308	0.2822
NC2V	0.6022	0.4249	0.6030	0.5421	0.3168
NCFV	0.5948	0.4132	0.5870	0.5134	0.2762
CC-Minvar	0.3542	0.2802	0.3980	0.2632	0.2356

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Dataset	500 CRSP	100 CRSP	S&P 500	FF-100	FF48
Model 1	0.4028	0.3124	0.4120	0.3068	0.2580
Model 2	0.4145	0.3182	0.4166	0.3022	0.2528
1/N	0.0625	0.0445	0.0586	0.0508	0.0324
MINC	0.3125	0.2025	0.4030	0.2221	0.1822
NC1V	0.6654	0.4670	0.6208	0.5308	0.2822
NC2V	0.6022	0.4249	0.6030	0.5421	0.3168
NCFV	0.5948	0.4132	0.5870	0.5134	0.2762
CC-Minvar	0.3542	0.2802	0.3980	0.2632	0.2356

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