doi: 10.3934/jimo.2018082

Robust and sparse portfolio model for index tracking

1. 

Dept. of Applied Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, China

2. 

Dept. of Applied Mathematics, School of Science, Beijing Jiaotong University, The Primary School Attached to Beijing Jiaotong University, Beijing 100044, China

* Corresponding author: Chao Zhang

Received  December 2017 Revised  January 2018 Published  June 2018

Fund Project: The first author is supported by NSFC grant 11571033, and the Fundamental Research Funds for the Central Universities of China under Grant 2016JBZ012; The third author is supported by NSFC grant 11431002

In the context of index tracking, the tracking error measures the difference between the return an investor receives and that of the benchmark he was attempting to imitate. In this paper, we use the weighted $\ell_{2}$ and $\ell_{p}$ $(0 < p < 1)$ norm penalties as well as the shortsale constraints ($\ell_2-\ell_p$ model for short) to the tracking portfolio model in order to get a robust and sparse portfolio for index tracking. The $\ell_{2}$ norm penalty imposes smoothness to alleviate the effect of the existence of highly correlated variables and hence has better out-of-sample performance and the $\ell_{p}$ norm penalty achieves sparsity to account for transaction costs. We enroll in the model explicitly the non-negativity constraints, that is, the shortsale constraints appeared in practice. The $\ell_p$ norm penalty is non-Lipschitz, nonconvex which leads to computational difficulty. We adopt the smoothing projected gradient (SPG) method to solve the robust and sparse portfolio model. We show that any accumulation point of the SPG method is a special limiting stationary point. We find our proposed $\ell_2-\ell_p$ model outperforms the $\ell_2 +\ell_0$ model proposed by Takeda et al. [26] for real stock data set S&P500 in terms of in-sample and out-of-sample errors.

Citation: Chao Zhang, Jingjing Wang, Naihua Xiu. Robust and sparse portfolio model for index tracking. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018082
References:
[1]

S. D. Babacan, L. Mancera, R. Molina and A. Katsaggelos, Non-convex priors in Bayesian compressed sensing, The 17th European Signal Processing Conference, Glasgow (UK), (2009), 110-114. doi: 10.1109/TIP.2009.2032894.

[2]

W. BianX. J. Chen and Y. Y. Ye, Complexity analysis of interior point algorithms for non-Lipschitz and nonconvex minimization, Mathematical Programming, 149 (2015), 301-327. doi: 10.1007/s10107-014-0753-5.

[3]

P. Bonami and M. 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. BrodieI. DaubechiesC. D. MolD. Giannone and I. Loris, Sparse and stable Markowitz portfolios, Proceedings of the National Academy of Science USA, 106 (2009), 12267-12272. doi: 10.1073/pnas.0904287106.

[5]

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

[6]

R. Chartrand and W. T. Yin, Iteratively reweighted algorithms for compressive sensing, IEEE International Conference on Acoustics, Speech, and Signal Processing, (2008), 3869-3872.

[7]

C. Chen, X. Li, C. Tolman, S. Wang and Y. Y. Ye, Sparse portfolio selection via quasi-norm regularization, preprint, 2014, arXiv: 1312. 6350.

[8]

X. J. ChenD. D. GeZ. Wang and Y. Y. Ye, Complexity of unconstrained $\ell_{2}-\ell_{p}$ minimization, Mathematical Programming, 143 (2014), 371-383. doi: 10.1007/s10107-012-0613-0.

[9]

X. J. ChenL. GuoZ. S. Lu and J. J. Ye, An augmented Lagrangian method for non-Lipschitz nonconvex programming, SIAM Journal on Numerical Analysis, 55 (2017), 168-193. doi: 10.1137/15M1052834.

[10]

X. J. ChenM. K. Ng and C. Zhang, Non-Lipschitz $\ell_{p}$-regularization and box constrained model for image restoration, IEEE Transactions on Image Processing, 21 (2012), 4709-4721. doi: 10.1109/TIP.2012.2214051.

[11]

X. J. ChenF. M. Xu and Y. Y. Ye, Lower bound theory of nonzero entries in solutions of $\ell_{2}-\ell_{p}$ minimization, SIAM Journal on Scientific Computing, 32 (2010), 2832-2852. doi: 10.1137/090761471.

[12]

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

[13]

V. DeMiguel, L. Garlappi and R. Uppal, Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy? Review of Financial Studies, 22 (2009b), 1915-1953.

[14]

D. W. Diamond and R. E. Verrecchia, Constraints on short-selling and asset price adjustment to private information, Journal of Financial Economics, 18 (2006), 277-311.

[15]

B. FastrichS. Paterlini and P. Winker, Cardinality versus $q$-norm constraints for index tracking, Quantitative Finance, 14 (2014), 2019-2032. doi: 10.1080/14697688.2012.691986.

[16]

J. Gothoh and A. Takeda, On the role of norm constraints in portfolio selection, Computational Management Science, 8 (2011), 323-353. doi: 10.1007/s10287-011-0130-2.

[17]

A. E. Hoerl, Application of ridge analysis to regression problems, Chemical Engineering Progress, 58 (1962), 54-59.

[18]

A. E. Hoerl and R. W. Kennard, Ridge regression: Biased estimation for nonorthogonal problems, Technometrics, 12 (1970), 55-67.

[19]

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

[20]

R. Jarrow, Heterogeneous expectations, restrictions on short sales, and equilibrium asset prices, Journal of Finance, 35 (1980), 1105-1113.

[21]

M. KloftU. BrefeldS. SonnenburgP. LaskovK.-R. M$\ddot{\mbox{u}}$ller and A. Zien, Efficient and accurate $L_p$-norm multiple kernel learning, Advances in Neural Information Processing Systems, 22 (2009), 997-1005.

[22]

C. Michelot, A finite algorithm for finding the projection of a point onto the canonical simplex of ${R}^{n}$, Journal of Optimization Theory and Applications, 50 (1986), 195-200. doi: 10.1007/BF00938486.

[23] S. J. Qin, A statistical perspective of neural networks for process modelling and control, Proceedings of the 1993 International Symposium on Intelligent Control, Chicago, Illinois, USA, 1993.
[24]

A. RakotomamonjyR. FlamaryG. Gasso and S. Canu, $\ell_{p}-\ell_{q}$ penalty for sparse linear and sparse multiple kernel multitask learning, IEEE Transactions on Neural Networks, 22 (2011), 1307-1320.

[25]

R. Ruiz-Torrubiano and A. Su$\acute{\rm a}$rez, A hybrid optimization approach to index tracking, Annals of Operations Research, 166 (2009), 57-71. doi: 10.1007/s10479-008-0404-4.

[26]

A. TakedaM. NiranjanJ. Gotoh and Y. Kawahara, Simultaneous pursuit of out-of-sample performance and sparsity in index tracking portfolios, Computational Management Science, 10 (2012), 21-49. doi: 10.1007/s10287-012-0158-y.

[27]

R. Tibshirani, Optimal reinsertion: Regression shrinkage and selection via the lasso, Royal Statistical Society, 58 (1996), 267-288.

[28] D. H. TrinhM. LuongJ. M. RocchisaniC. D. Pham and F. Dibos, Medical image denoising using kernel ridge regression, Processing of 2011 IEEE International Conference on Image Processing, Brussels, 2011.
[29]

M. Woodside-OriakhiC. Lucas and J. Beasley, Heuristic algorithms for the cardinality constrained efficient frontier, European Journal of Operational Research, 213 (2011), 538-550. doi: 10.1016/j.ejor.2011.03.030.

[30]

F. M. Xu, Z. Xu and H. Xue, Sparse index tracking: An $\ell_{1/2}$ regularization model and solution, 2011. Available from: gr.xjtu.edu.cn/LiferayFCKeditor/UserFiles/File/NewAOR.pdf.

[31]

F. M. XuZ. Lu 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.

[32]

M. W. XuS. Y. Wu and J. J. Ye, Solving semi-infinite programs by smoothing projected gradient method, Computational Optimization and Applications, 59 (2014), 591-616. doi: 10.1007/s10589-014-9654-z.

[33]

Y. M. Yen and T. J. Yen, Solving norm constrained portfolio optimization via coordinate-wise descent algorithms, Computational Statistics and Data Analysis, 76 (2014), 737-759. doi: 10.1016/j.csda.2013.07.010.

[34]

C. Zhang and X. J. Chen, Smoothing projected gradient method and its application to stochastic linear complementarity problems, SIAM Journal on Optimization, 20 (2009), 627-649. doi: 10.1137/070702187.

[35]

Z. ZhaoF. M. Xu and X. Li, Adaptive projected gradient thresholding methods for constrained $l_0$ problems, Science China Mathematics, 58 (2015), 2205-2224. doi: 10.1007/s11425-015-5038-9.

show all references

References:
[1]

S. D. Babacan, L. Mancera, R. Molina and A. Katsaggelos, Non-convex priors in Bayesian compressed sensing, The 17th European Signal Processing Conference, Glasgow (UK), (2009), 110-114. doi: 10.1109/TIP.2009.2032894.

[2]

W. BianX. J. Chen and Y. Y. Ye, Complexity analysis of interior point algorithms for non-Lipschitz and nonconvex minimization, Mathematical Programming, 149 (2015), 301-327. doi: 10.1007/s10107-014-0753-5.

[3]

P. Bonami and M. 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. BrodieI. DaubechiesC. D. MolD. Giannone and I. Loris, Sparse and stable Markowitz portfolios, Proceedings of the National Academy of Science USA, 106 (2009), 12267-12272. doi: 10.1073/pnas.0904287106.

[5]

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

[6]

R. Chartrand and W. T. Yin, Iteratively reweighted algorithms for compressive sensing, IEEE International Conference on Acoustics, Speech, and Signal Processing, (2008), 3869-3872.

[7]

C. Chen, X. Li, C. Tolman, S. Wang and Y. Y. Ye, Sparse portfolio selection via quasi-norm regularization, preprint, 2014, arXiv: 1312. 6350.

[8]

X. J. ChenD. D. GeZ. Wang and Y. Y. Ye, Complexity of unconstrained $\ell_{2}-\ell_{p}$ minimization, Mathematical Programming, 143 (2014), 371-383. doi: 10.1007/s10107-012-0613-0.

[9]

X. J. ChenL. GuoZ. S. Lu and J. J. Ye, An augmented Lagrangian method for non-Lipschitz nonconvex programming, SIAM Journal on Numerical Analysis, 55 (2017), 168-193. doi: 10.1137/15M1052834.

[10]

X. J. ChenM. K. Ng and C. Zhang, Non-Lipschitz $\ell_{p}$-regularization and box constrained model for image restoration, IEEE Transactions on Image Processing, 21 (2012), 4709-4721. doi: 10.1109/TIP.2012.2214051.

[11]

X. J. ChenF. M. Xu and Y. Y. Ye, Lower bound theory of nonzero entries in solutions of $\ell_{2}-\ell_{p}$ minimization, SIAM Journal on Scientific Computing, 32 (2010), 2832-2852. doi: 10.1137/090761471.

[12]

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

[13]

V. DeMiguel, L. Garlappi and R. Uppal, Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy? Review of Financial Studies, 22 (2009b), 1915-1953.

[14]

D. W. Diamond and R. E. Verrecchia, Constraints on short-selling and asset price adjustment to private information, Journal of Financial Economics, 18 (2006), 277-311.

[15]

B. FastrichS. Paterlini and P. Winker, Cardinality versus $q$-norm constraints for index tracking, Quantitative Finance, 14 (2014), 2019-2032. doi: 10.1080/14697688.2012.691986.

[16]

J. Gothoh and A. Takeda, On the role of norm constraints in portfolio selection, Computational Management Science, 8 (2011), 323-353. doi: 10.1007/s10287-011-0130-2.

[17]

A. E. Hoerl, Application of ridge analysis to regression problems, Chemical Engineering Progress, 58 (1962), 54-59.

[18]

A. E. Hoerl and R. W. Kennard, Ridge regression: Biased estimation for nonorthogonal problems, Technometrics, 12 (1970), 55-67.

[19]

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

[20]

R. Jarrow, Heterogeneous expectations, restrictions on short sales, and equilibrium asset prices, Journal of Finance, 35 (1980), 1105-1113.

[21]

M. KloftU. BrefeldS. SonnenburgP. LaskovK.-R. M$\ddot{\mbox{u}}$ller and A. Zien, Efficient and accurate $L_p$-norm multiple kernel learning, Advances in Neural Information Processing Systems, 22 (2009), 997-1005.

[22]

C. Michelot, A finite algorithm for finding the projection of a point onto the canonical simplex of ${R}^{n}$, Journal of Optimization Theory and Applications, 50 (1986), 195-200. doi: 10.1007/BF00938486.

[23] S. J. Qin, A statistical perspective of neural networks for process modelling and control, Proceedings of the 1993 International Symposium on Intelligent Control, Chicago, Illinois, USA, 1993.
[24]

A. RakotomamonjyR. FlamaryG. Gasso and S. Canu, $\ell_{p}-\ell_{q}$ penalty for sparse linear and sparse multiple kernel multitask learning, IEEE Transactions on Neural Networks, 22 (2011), 1307-1320.

[25]

R. Ruiz-Torrubiano and A. Su$\acute{\rm a}$rez, A hybrid optimization approach to index tracking, Annals of Operations Research, 166 (2009), 57-71. doi: 10.1007/s10479-008-0404-4.

[26]

A. TakedaM. NiranjanJ. Gotoh and Y. Kawahara, Simultaneous pursuit of out-of-sample performance and sparsity in index tracking portfolios, Computational Management Science, 10 (2012), 21-49. doi: 10.1007/s10287-012-0158-y.

[27]

R. Tibshirani, Optimal reinsertion: Regression shrinkage and selection via the lasso, Royal Statistical Society, 58 (1996), 267-288.

[28] D. H. TrinhM. LuongJ. M. RocchisaniC. D. Pham and F. Dibos, Medical image denoising using kernel ridge regression, Processing of 2011 IEEE International Conference on Image Processing, Brussels, 2011.
[29]

M. Woodside-OriakhiC. Lucas and J. Beasley, Heuristic algorithms for the cardinality constrained efficient frontier, European Journal of Operational Research, 213 (2011), 538-550. doi: 10.1016/j.ejor.2011.03.030.

[30]

F. M. Xu, Z. Xu and H. Xue, Sparse index tracking: An $\ell_{1/2}$ regularization model and solution, 2011. Available from: gr.xjtu.edu.cn/LiferayFCKeditor/UserFiles/File/NewAOR.pdf.

[31]

F. M. XuZ. Lu 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.

[32]

M. W. XuS. Y. Wu and J. J. Ye, Solving semi-infinite programs by smoothing projected gradient method, Computational Optimization and Applications, 59 (2014), 591-616. doi: 10.1007/s10589-014-9654-z.

[33]

Y. M. Yen and T. J. Yen, Solving norm constrained portfolio optimization via coordinate-wise descent algorithms, Computational Statistics and Data Analysis, 76 (2014), 737-759. doi: 10.1016/j.csda.2013.07.010.

[34]

C. Zhang and X. J. Chen, Smoothing projected gradient method and its application to stochastic linear complementarity problems, SIAM Journal on Optimization, 20 (2009), 627-649. doi: 10.1137/070702187.

[35]

Z. ZhaoF. M. Xu and X. Li, Adaptive projected gradient thresholding methods for constrained $l_0$ problems, Science China Mathematics, 58 (2015), 2205-2224. doi: 10.1007/s11425-015-5038-9.

Table 1.  Comparison of our $\ell_2-\ell_{3/4}$ and the $\ell_2 + \ell_0$ model.
sparsity T1 T2 TEI TEO Method
49 3e-4 3e-3 6.1954e-7 1.7899e-8 ours
0 - 1.5546e-6 7.4781e-5 Takeda's
0.01 - 1.9824e-6 2.8132e-5
1 - 4.9711e-6 1.0041e-5
10 - 6.1191e-6 2.6618e-6
47 6e-4 3e-3 6.2410e-7 2.7493e-8 ours
0 - 1.6880e-6 4.6893e-5 Takeda's
0.01 - 2.1284e-6 2.9751e-5
1 - 5.1254e-6 1.5820e-5
10 - 6.2630e-6 8.8319e-6
46 5e-4 5e-3 2.1752e-6 3.6834e-7 ours
0 - 1.6203e-6 3.6957e-5 Takeda's
0.01 - 2.2539e-6 2.8785e-5
1 - 5.3248e-6 1.4463e-5
10 - 6.2100e-6 6.7308e-6
45 6e-4 5e-3 2.2408e-6 8.8737e-7 ours
0 - 1.9140e-6 3.6539e-5
0.01 - 2.2558e-6 2.8113e-5 Takeda's
1 - 5.2872e-6 1.2206e-5
10 - 6.1558e-6 6.6423e-6
39 8e-4 9e-3 2.2261e-6 6.9414e-7 ours
0 - 2.5681e-6 3.4804e-5 Takeda's
0.01 - 3.2549e-6 1.5205e-5
1 - 6.5316e-6 1.7354e-5
10 - 7.5129e-6 9.8518e-6
30 1e-4 6e-3 1.6756e-7 3.3065e-7 ours
0 - 5.2831e-6 5.6541e-5 Takeda's
0.01 - 4.9822e-6 1.2508e-5
1 - 9.8457e-6 3.3260e-5
10 - 1.0897e-5 5.2070e-6
9 9e-4 1e-2 1.5735e-6 1.1466e-7 ours
0 - 3.5971e-5 1.1370e-5 Takeda's
0.01 - 3.5029e-5 1.0898e-4
1 - 4.7174e-5 2.0478e-5
10 - 4.7787e-5 9.5118e-5
sparsity T1 T2 TEI TEO Method
49 3e-4 3e-3 6.1954e-7 1.7899e-8 ours
0 - 1.5546e-6 7.4781e-5 Takeda's
0.01 - 1.9824e-6 2.8132e-5
1 - 4.9711e-6 1.0041e-5
10 - 6.1191e-6 2.6618e-6
47 6e-4 3e-3 6.2410e-7 2.7493e-8 ours
0 - 1.6880e-6 4.6893e-5 Takeda's
0.01 - 2.1284e-6 2.9751e-5
1 - 5.1254e-6 1.5820e-5
10 - 6.2630e-6 8.8319e-6
46 5e-4 5e-3 2.1752e-6 3.6834e-7 ours
0 - 1.6203e-6 3.6957e-5 Takeda's
0.01 - 2.2539e-6 2.8785e-5
1 - 5.3248e-6 1.4463e-5
10 - 6.2100e-6 6.7308e-6
45 6e-4 5e-3 2.2408e-6 8.8737e-7 ours
0 - 1.9140e-6 3.6539e-5
0.01 - 2.2558e-6 2.8113e-5 Takeda's
1 - 5.2872e-6 1.2206e-5
10 - 6.1558e-6 6.6423e-6
39 8e-4 9e-3 2.2261e-6 6.9414e-7 ours
0 - 2.5681e-6 3.4804e-5 Takeda's
0.01 - 3.2549e-6 1.5205e-5
1 - 6.5316e-6 1.7354e-5
10 - 7.5129e-6 9.8518e-6
30 1e-4 6e-3 1.6756e-7 3.3065e-7 ours
0 - 5.2831e-6 5.6541e-5 Takeda's
0.01 - 4.9822e-6 1.2508e-5
1 - 9.8457e-6 3.3260e-5
10 - 1.0897e-5 5.2070e-6
9 9e-4 1e-2 1.5735e-6 1.1466e-7 ours
0 - 3.5971e-5 1.1370e-5 Takeda's
0.01 - 3.5029e-5 1.0898e-4
1 - 4.7174e-5 2.0478e-5
10 - 4.7787e-5 9.5118e-5
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