Article Contents
Article Contents

# Robust multi-period and multi-objective portfolio selection

• * Corresponding author: lydiajianglin@gmail.com
• In this paper, a multi-period multi-objective portfolio selection problem with uncertainty is studied. Under the assumption that the uncertainty set is ellipsoidal, the robust counterpart of the proposed problem can be transformed into a standard multi-objective optimization problem. A weighted-sum approach is then introduced to obtain Pareto front of the problem. Numerical examples will be presented to illustrate the proposed method and validate the effectiveness and efficiency of the model developed.

Mathematics Subject Classification: Primary: 90C26, 90C29, 65K05; Secondary: 91G10.

 Citation:

• Figure 1.  The variation of $g(\kappa)$ in terms of $\kappa$ when $\lambda = 0.1$

Figure 2.  The variation of $g(\kappa)$ in terms of $\kappa$ when $\lambda=0.5$

Figure 3.  The variation of $g(\kappa)$ in terms of $\kappa$ when $\lambda = 0.9$

Figure 4.  The variation of $f_{\lambda}(\Delta w_t, \kappa^*)$ in terms of $\delta$ when $\lambda = 0.1$

Figure 5.  The variation of $f_{\lambda}(\Delta w_t, \kappa^*)$ in terms of $\delta$ when $\lambda = 0.5$

Figure 6.  The variation of $f_{\lambda}(\Delta w_t, \kappa^*)$ in terms of $\delta$ when $\lambda = 0.9$

Figure 7.  The variation of $f_{\lambda}(\Delta w_t, \kappa^*)$ in terms of $\varsigma$ when $\lambda = 0.5$

Figure 8.  The pareton front with $\delta = 1$

Figure 9.  The pareton front with $\delta = 5$

Figure 10.  The pareton front with $\delta = 10$

Table 1.  Solution with five period; $\varsigma = 0.001, \delta = 0.01$

 $k_1$ $k_2$ $k_3$ $k_4$ $k_5$ $\Delta w_1$ 1.6062540e+002 1.6005381e+002 1.5941714e+002 1.5883923e+002 1.5824019e+002 $\Delta w_2$ -4.3718793e+001 -4.3458314e+001 -4.3193335e+001 -4.2953594e+001 -4.2647292e+001 $\Delta w_3$ -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 $\Delta w_4$ -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 $\Delta w_5$ 1.9724480e+003 1.9722713e+003 1.9720766e+003 1.9718941e+003 1.9717006e+003 $\Delta w_6$ -1.3675740e+002 -1.3651295e+002 -1.3620477e+002 -1.3596164e+002 -1.3572250e+002 $\Delta w_7$ -9.7855928e+002 -9.7836228e+002 -9.7827192e+002 -9.7813905e+002 -9.7802742e+002 $\Delta w_8$ -4.6542445e+002 -4.6521487e+002 -4.6500508e+002 -4.6479322e+002 -4.6454631e+002 $\Delta w_9$ 1.5230230e+003 1.5227606e+003 1.5225158e+003 1.5222595e+003 1.5219708e+003 $\Delta w_{10}$ -3.8955914e+001 -3.8854705e+001 -3.8649778e+001 -3.8458643e+001 -3.8279222e+001

Table 2.  Solution with five period; $\varsigma = 0.01, \delta = 0.01$

 $k_1$ $k_2$ $k_3$ $k_4$ $k_5$ $\Delta w_1$ 2.1589291e-002 -3.4351365e-002 5.7805964e-002 2.3686226e-003 1.0158904e-003 $\Delta w_2$ -2.8283011e-001 -2.7323758e-004 1.6033220e-003 -3.5364786e-003 -2.6401259e-004 $\Delta w_3$ -3.3036266e+002 -3.5623656e+002 -2.4031564e+002 -3.7257428e+002 -3.4108233e+002 $\Delta w_4$ -3.0865547e+002 -2.8842569e+002 -2.9824681e+002 -2.6018254e+002 -2.7533841e+002 $\Delta w_5$ 7.0391477e+002 6.9146259e+002 6.8271055e+002 6.6623463e+002 6.5626173e+002 $\Delta w_6$ -2.7364462e-002 -3.5663105e-002 -4.9939414e-002 1.0528198e-003 -3.5473078e-001 $\Delta w_7$ -1.3859574e+002 -1.1951280e+002 -2.1609022e+002 -1.0360357e+002 -1.0937146e+002 $\Delta w_8$ -1.1308908e-001 -2.4663699e-003 -2.2421419e-003 -7.7830259e-003 -1.7166429e-007 $\Delta w_9$ 2.1891752e-003 1.6847602e-007 5.8640523e-002 3.0980424e-003 7.3055641e-001 $\Delta w_{10}$ -2.8203111e-007 -3.2609527e-004 -7.4650111e-008 -1.1673710e-006 -3.2439030e-003

Table 3.  Solution with five period; $\varsigma = 0.01, \delta = 0.1$

 $k_1$ $k_2$ $k_3$ $k_4$ $k_5$ $\Delta w_1$ -5.4467604e-002 -7.0948978e-007 -2.2561076e-004 -1.0737564e-002 -1.6291721e-004 $\Delta w_2$ -1.6419802e-003 -5.6357371e-007 -7.4291123e-003 4.3365254e-003 -2.9140832e-004 $\Delta w_3$ -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 $\Delta w_4$ -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 $\Delta w_5$ 1.7420318e+003 1.7397785e+003 1.7370980e+003 1.7349762e+003 1.7353447e+003 $\Delta w_6$ -9.7249593e-008 -3.9907242e-002 -1.8502745e-007 -4.0113434e-002 -3.8173914e-004 $\Delta w_7$ -7.9037071e+002 -7.9581203e+002 -7.9336516e+002 -7.8689932e+002 -7.9102920e+002 $\Delta w_8$ -2.1545195e+002 -2.0339581e+002 -2.0451219e+002 -2.0476930e+002 -1.9763968e+002 $\Delta w_9$ 1.2043583e+003 1.2000782e+003 1.2014231e+003 1.1974970e+003 1.1941434e+003 $\Delta w_{10}$ -3.4352764e-002 -2.3689203e-007 -1.2659914e-004 -1.2481443e-007 -2.0840295e-007
•  [1] A. Ben-Tal and A. Nemirovski, Robust optimization-methodology and applications, Math. Program., 92 (2002), 453-480.  doi: 10.1007/s101070100286. [2] X. Cui, X. Sun and D. Sha, An empirical study on discrete optimization models for portfolio selection, J. Ind. Manag. Optim., 5 (2009), 33-46.  doi: 10.3934/jimo.2009.5.33. [3] K. Deb, A. Pratap, S. Agarwal and T. A. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Comput., 6 (2002), 182-197.  doi: 10.1109/4235.996017. [4] D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Math. Oper. Res., 28 (2003), 1-38.  doi: 10.1287/moor.28.1.1.14260. [5] D. Huang, S. Zhu, F. J. Fabozzi and M. Fukushima, Portfolio selection under distributional uncertainty: A relative robust CVaR approach, European J. Oper. Res., 203 (2010), 185-194.  doi: 10.1016/j.ejor.2009.07.010. [6] K. Khalili-Damghani and M. Amiri, Solving binary-state multi-objective reliability redundancy allocation series-parallel problem using efficient epsilon-constraint, multi-start partial bound enumeration algorithm, and DEA, Reliability Engineering and System Safety, 103 (2012), 35-44.  doi: 10.1016/j.ress.2012.03.006. [7] C. C. Lin and Y. T. Liu, Genetic algorithms for portfolio selection problems with minimum transaction lots, European J. Oper. Res., 185 (2008), 393-404.  doi: 10.1016/j.ejor.2006.12.024. [8] H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, John Wiley & Sons, Inc., New York, 1959. [9] R. T. Marler and J. S. Arora, The weighted sum method for multi-objective optimization: New insights, Struct. Multidiscip. Optim., 41 (2010), 853-862.  doi: 10.1007/s00158-009-0460-7. [10] K. Ruan and M. Fukushima, Robust portfolio selection with a combined WCVaR and factor model, J. Ind. Manag. Optim., 8 (2012), 343-362.  doi: 10.3934/jimo.2012.8.343. [11] K. Schottle and R. Werner, Robustness properties of mean-variance portfolios, Optimization, 58 (2009), 641-663.  doi: 10.1080/02331930902819220. [12] H. Soleimani, H. R. Golmakani and M. H. Salimi, Markowitz-based portfolio selection with minimum transaction lots, cardinality constraints and regarding sector capitalization using genetic algorithm, Expert Systems with Appl., 36 (2009), 5058-5063.  doi: 10.1016/j.eswa.2008.06.007. [13] Q. Wang and H. Sun, Sparse Markowitz portfolio selection by using stochastic linear complementarity approach, J. Ind. Manag. Optim., 14 (2018), 541-559.  doi: 10.3934/jimo.2017059. [14] Z. Wang and S. Liu, Multi-period mean-variance portfolio selection with fixed and proportional transaction costs, J. Ind. Manag. Optim., 9 (2013), 643-657.  doi: 10.3934/jimo.2013.9.643. [15] C. Wu, K. L. Teo and S. Wu, Min-max optimal control of linear systems with uncertainty and terminal state constraints, Automatica J. IFAC, 49 (2013), 1809-1815.  doi: 10.1016/j.automatica.2013.02.052. [16] P. Xidonas, G. Mavrotas, C. Hassapis and C. Zopounidis, Robust multiobjective portfolio optimization: A minimax regret approach, European J. Oper. Res., 262 (2017), 299-305.  doi: 10.1016/j.ejor.2017.03.041. [17] L. Yi, Z. F. Li and D. Li, Multi-period portfolio selection for asset-liability management with uncertain investment horizon, J. Ind. Manag. Optim., 4 (2008), 535-552.  doi: 10.3934/jimo.2008.4.535. [18] N. Zhang, A symmetric Gauss-Seidel based method for a class of multi-period mean-variance portfolio selection problems, J. Ind. Manag. Optim., (2018). doi: 10.3934/jimo.2018189. [19] P. Zhang, Chance-constrained multiperiod mean absolute deviation uncertain portfolio selection, J. Ind. Manag. Optim., 15 (2019), 537-564.  doi: 10.3934/jimo.2018056. [20] C. Zhao, C. Wu, J. Chai, X. Wang, X. Yang, J. M. Lee and M. J. Kim, Decomposition-based multi-objective firefly algorithm for RFID network planning with uncertainty, Appl. Soft Comput., 55 (2017), 549-564.  doi: 10.1016/j.asoc.2017.02.009.

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