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Robust multi-period and multi-objective portfolio selection

  • * Corresponding author: lydiajianglin@gmail.com

    * Corresponding author: lydiajianglin@gmail.com 
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  • 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:

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  • 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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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