Robust multi-period and multi-objective portfolio selection

Lin Jiang; Song Wang

doi:10.3934/jimo.2019130

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2021, Volume 17, Issue 2: 695-709. Doi: 10.3934/jimo.2019130

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Robust multi-period and multi-objective portfolio selection

Lin Jiang^, and
Song Wang

Department of Mathematics and Statistics, Curtin University, GPO Box U1987, Perth, WA 6845, Australia

^* Corresponding author: lydiajianglin@gmail.com
^* Corresponding author: lydiajianglin@gmail.com

Received: December 31, 2018

Revised: May 31, 2019

Early access: October 2019

Published: March 2021

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Abstract

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.

Keywords:

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

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

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

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

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

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

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

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

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

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

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

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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

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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

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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

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