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A closer look at the minimum-variance portfolio optimization model. (English) Zbl 1435.91169

Summary: Recently, by imposing the regularization term to objective function or additional norm constraint to portfolio weights, a number of alternative portfolio strategies have been proposed to improve the empirical performance of the minimum-variance portfolio. In this paper, we firstly examine the relation between the weight norm-constrained method and the objective function regularization method in minimum-variance problems by analyzing the Karush-Kuhn-Tucker conditions of their Lagrangian functions. We give the range of parameters for the two models and the corresponding relationship of parameters. Given the range and manner of parameter selection, it will help researchers and practitioners better understand and apply the relevant portfolio models. We apply these models to construct optimal portfolios and test the proposed propositions by employing real market data.

MSC:

91G10 Portfolio theory
90C20 Quadratic programming
62P05 Applications of statistics to actuarial sciences and financial mathematics

Software:

CVX
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Full Text: DOI

References:

[1] Markowitz, H., Portfolio selection, The Journal of Finance, 7, 1, 77-91 (1952) · doi:10.2307/2975974
[2] Black, F.; Litterman, R., Global portfolio optimization, Financial Analysts Journal, 48, 5, 28-43 (1992) · doi:10.2469/faj.v48.n5.28
[3] Ledoit, O.; Wolf, M., Improved estimation of the covariance matrix of stock returns with an application to portfolio selection, Journal of Empirical Finance, 10, 5, 603-621 (2003) · doi:10.1016/s0927-5398(03)00007-0
[4] Ledoit, O.; Wolf, M., A well-conditioned estimator for large-dimensional covariance matrices, Journal of Multivariate Analysis, 88, 2, 365-411 (2004) · Zbl 1032.62050 · doi:10.1016/s0047-259x(03)00096-4
[5] Jagannathan, R.; Ma, T., Risk reduction in large portfolios: why imposing the wrong constraints helps, The Journal of Finance, 58, 4, 1651-1683 (2003) · doi:10.1111/1540-6261.00580
[6] DeMiguel, V.; Garlappi, L.; Nogales, F. J.; Uppal, R., A generalized approach to portfolio optimization: improving performance by constraining portfolio norms, Management Science, 55, 5, 798-812 (2009) · Zbl 1232.91617 · doi:10.1287/mnsc.1080.0986
[7] Brodie, J.; Daubechies, I.; De Mol, C.; Giannone, D.; Loris, I., Sparse and stable Markowitz portfolios, Proceedings of the National Academy of Sciences, 106, 30, 12267-12272 (2009) · Zbl 1203.91271 · doi:10.1073/pnas.0904287106
[8] Behr, P.; Guettler, A.; Miebs, F., On portfolio optimization: imposing the right constraints, Journal of Banking & Finance, 37, 4, 1232-1242 (2013) · doi:10.1016/j.jbankfin.2012.11.020
[9] Chen, C. H.; Ye, Y. Y., Sparse Portfolio Selection via Quasi-Norm Regularization (2015), https://arxiv.org/abs/1312.6350
[10] Dai, Z.; Wen, F., Some improved sparse and stable portfolio optimization problems, Finance Research Letters, 27, 46-52 (2018) · doi:10.1016/j.frl.2018.02.026
[11] Dai, Z.; Wen, F., New efficient mean-variance portfolio selection models, Risk Management
[12] Fan, J.; Zhang, J.; Yu, K., Vast portfolio selection with gross-exposure constraints, Journal of the American Statistical Association, 107, 498, 592-606 (2012) · Zbl 1261.62091 · doi:10.1080/01621459.2012.682825
[13] Fastrich, B.; Paterlini, S.; Winker, P., Constructing optimal sparse portfolios using regularization methods, Computational Management Science, 12, 3, 417-434 (2015) · Zbl 1355.91077 · doi:10.1007/s10287-014-0227-5
[14] Gotoh, J.-y.; Takeda, A., On the role of norm constraints in portfolio selection, Computational Management Science, 8, 4, 323-353 (2011) · Zbl 1225.91060 · doi:10.1007/s10287-011-0130-2
[15] Xing, X.; Hu, J.; Yang, Y., Robust minimum variance portfolio with L-infinity constraints, Journal of Banking & Finance, 46, 107-117 (2014) · doi:10.1016/j.jbankfin.2014.05.004
[16] Xu, F. M.; Wang, G.; Gao, Y. L., Nonconvex \(L_{1 / 2}\) regularization for sparse portfolio selection, Pacific Journal Optimization, 10, 1, 163-176 (2014) · Zbl 1288.90066
[17] Andrecut, M., Portfolio optimization in R (2013), https://arxiv.org/pdf/1307.0450.pdf · Zbl 1398.91499
[18] DeMiguel, V.; Garlappi, L.; Uppal, R., Optimal versus naive diversification: how inefficient is the 1/N portfolio strategy?, Review of Financial Studies, 22, 5, 1915-1953 (2009) · doi:10.1093/rfs/hhm075
[19] Dai, Z.; Zhu, H.; Wen, F., Two nonparametric approaches to mean absolute deviation portfolio selection model, Journal of Industrial & Management Optimization (2019) · Zbl 1476.90307 · doi:10.3934/jimo.2019054
[20] Ling, A.; Sun, J.; Yang, X., Robust tracking error portfolio selection with worst-case downside risk measures, Journal of Economic Dynamics and Control, 39, 178-207 (2015) · Zbl 1402.91713 · doi:10.1016/j.jedc.2013.11.011
[21] Natarajan, K.; Pachamanova, D.; Sim, M., Incorporating asymmetric distributional information in robust value-at-risk optimization, Management Science, 54, 3, 573-585 (2008) · Zbl 1142.91593 · doi:10.1287/mnsc.1070.0769
[22] Yan, L.; Xu, F.; Liu, J.; Teo, K. L.; Lai, M., Stability strategies of demand-driven supply networks with transportation delay, Applied Mathematical Modelling, 76, 109-121 (2019) · Zbl 1481.90077 · doi:10.1016/j.apm.2019.06.015
[23] Yu, X. J.; Xie, S. Y.; Xu, W. J., Optimal portfolio strategy under rolling economic maximum drawdown constraints, Mathematical Problems in Engineering, 2014 (2014) · Zbl 1407.91241 · doi:10.1155/2014/787943
[24] Zhang, L.; Li, Z., Multi-period mean-variance portfolio selection with uncertain time horizon when returns are serially correlated, Mathematical Problems in Engineering, 2012 (2012) · Zbl 1264.91121 · doi:10.1155/2012/216891
[25] Zhu, S.; Fukushima, M., Worst-case conditional value-at-risk with application to robust portfolio management, Operations Research, 57, 5, 1155-1168 (2009) · Zbl 1233.91254 · doi:10.1287/opre.1080.0684
[26] Chopra, V. K.; Ziemba, W. T., The effect of errors in means, variances, and covariances on optimal portfolio choice, The Journal of Portfolio Management, 19, 2, 6-11 (1993) · doi:10.3905/jpm.1993.409440
[27] Ghaoui, L. E.; Oks, M.; Oustry, F., Worst-case value-at-risk and robust portfolio optimization: a conic programming approach, Operations Research, 51, 4, 543-556 (2003) · Zbl 1165.91397 · doi:10.1287/opre.51.4.543.16101
[28] Green, R. C.; Hollifield, B., When will mean-variance efficient portfolios be well diversified?, The Journal of Finance, 47, 5, 1785-1809 (1992) · doi:10.2307/2328996
[29] Kwok, Y. K., Mean-variance portfolio theory (2007), https://www.math.ust.hk/maykwok/courses/ma362/Topic2.pdf
[30] Dai, Z.; Wen, F., A generalized approach to sparse and stable portfolio optimization problem, Journal of Industrial & Management Optimization, 14, 4, 1651-1666 (2018) · doi:10.3934/jimo.2018025
[31] Grant, M.; Boyd, S., CVX: matlab software for disciplined convex programming (2010), http://cvxr.com/cvx
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