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July  2021, 17(4): 1663-1680. doi: 10.3934/jimo.2020039

## Time-consistent multiperiod mean semivariance portfolio selection with the real constraints

 1 School of Economics and Management, South China Normal University, Guangzhou 510006, China 2 College of Humanities and Social sciences, Zhongkai University of Agriculture and Engineering, Guangzhou 510225, China 3 Faculty of Management and Economics, Dalian University of Technology, Dalian 116024, China

* Corresponding author: Tel.:+86 18971194382

Received  July 2019 Revised  September 2019 Published  February 2020

Fund Project: This research was supported by the Key Projects of National Natural Science Foundation (nos. 71731003)

In this paper, a new multiperiod mean semivariance portfolio selection with the transaction costs, borrowing constraints, threshold constraints and cardinality constraints is proposed. In the model, the return and risk of assets are characterized by mean value and semivariance, respectively. Because the semivariance operator is not separable, the optimal solution of the model is not time-consistent. The time-consistent strategy for this model can be obtained by using game approach. The time-consistent strategy, which is a mix integer dynamic optimization problem with path dependence, is approximately turned into a dynamic programming problem by approximate dynamic programming method. A novel discrete approximate iteration method is designed to obtain the optimal time-consistent strategy, and is proved linearly convergent. Finally, the comparison analysis of trade-off parameters is given to illustrate the idea of our model and the effectiveness of the designed algorithm.

Citation: Peng Zhang, Yongquan Zeng, Guotai Chi. Time-consistent multiperiod mean semivariance portfolio selection with the real constraints. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1663-1680. doi: 10.3934/jimo.2020039
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##### References:
The multiperiod weighted digraph
The optimal solutions when $K = 8$, $u_{ft}^b$ = -500000 dollars, $\eta _t$ = 0.000001
 The optimal investment proportions $X_t$ 1 Asset3 Asset 6 Asset 7 Asset 8 Asset 9 Asset 10 1044290 200000.0 200000.0 200000.0 200000.0 200000.0 200000.0 Asset11 Asset28 other risk asset 100000.0 200000.0 0 2 Asset6 Asset 7 Asset 8 Asset 9 Asset 10 Asset 11 1091060 200000.0 200000.0 200000.0 200000.0 200000.0 200000.0 Asset 12 Asset 28 other risk asset 144290.0 200000.0 0 3 Asset3 Asset 6 Asset 7 Asset 8 Asset 9 Asset 10 1140219 191060.0 200000.0 200000.0 200000.0 200000.0 200000.0 Asset11 Asset28 other risk asset 200000.0 200000.0 0 4 Asset3 Asset 6 Asset 7 Asset 8 Asset 9 Asset 10 1188760 200000.0 200000.0 200000.0 200000.0 200000.0 200000.0 Asset11 Asset28 other risk asset 200000.0 200000.0 0 5 Asset3 Asset 6 Asset 7 Asset 8 Asset 9 Asset 10 1235266 200000.0 200000.0 200000.0 200000.0 200000.0 200000.0 Asset11 Asset28 other risk asset 200000.0 200000.0 0
 The optimal investment proportions $X_t$ 1 Asset3 Asset 6 Asset 7 Asset 8 Asset 9 Asset 10 1044290 200000.0 200000.0 200000.0 200000.0 200000.0 200000.0 Asset11 Asset28 other risk asset 100000.0 200000.0 0 2 Asset6 Asset 7 Asset 8 Asset 9 Asset 10 Asset 11 1091060 200000.0 200000.0 200000.0 200000.0 200000.0 200000.0 Asset 12 Asset 28 other risk asset 144290.0 200000.0 0 3 Asset3 Asset 6 Asset 7 Asset 8 Asset 9 Asset 10 1140219 191060.0 200000.0 200000.0 200000.0 200000.0 200000.0 Asset11 Asset28 other risk asset 200000.0 200000.0 0 4 Asset3 Asset 6 Asset 7 Asset 8 Asset 9 Asset 10 1188760 200000.0 200000.0 200000.0 200000.0 200000.0 200000.0 Asset11 Asset28 other risk asset 200000.0 200000.0 0 5 Asset3 Asset 6 Asset 7 Asset 8 Asset 9 Asset 10 1235266 200000.0 200000.0 200000.0 200000.0 200000.0 200000.0 Asset11 Asset28 other risk asset 200000.0 200000.0 0
The terminal wealth when $K$ = 6, and $K = 8$, $u_{ft}^b$ = -500000 dollars, $\eta_{t} = 0, 0.000001, \dots, 0.08$
 $\eta_{t}$ 0 1e-06 2e-06 3e-06 4e-06 5e-06 6e-06 7e-06 8e-06 $X_5$ 1.20334e+06 1.19988e+06 1.19265e+06 1.18715e+06 1.18114e+06 1.17113e+06 1.15525e+06 1.1462e+06 1.129e+06 $X_5^{'}$ 1.23717e+06 1.23368e+06 1.22494e+06 1.21576e+06 1.20173e+06 1.18519e+06 1.16904e+06 1.15098e+06 1.13789e+06 $\eta_{t}$ 9e-06 1e-05 2e-05 3e-05 4e-05 5e-05 6e-05 7e-05 8e-05 $X_5$ 1.11937e+06 1.11052e+06 1.06571e+06 1.04884e+06 1.0404e+06 1.03534e+06 1.03196e+06 1.02955e+06 1.02774e+06 $X_5^{'}$ 1.12717e+06 1.12252e+06 1.06751e+06 1.05004e+06 1.0413e+06 1.03606e+06 1.03256e+06 1.03007e+06 1.02819e+06 $\eta_{t}$ 9e-05 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 $X_5$ 1.02634e+06 1.02521e+06 1.02015e+06 1.01847e+06 1.01762e+06 1.01714e+06 1.01677e+06 1.01653e+06 1.01635e+06 $X_5^{'}$ 1.02703e+06 1.02558e+06 1.02033e+06 1.01858e+06 1.01771e+06 1.01719e+06 1.01684e+06 1.01659e+06 1.0164e+06 $\eta_{t}$ 0.0009 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 $X_5$ 1.01621e+06 1.0161e+06 1.01559e+06 1.01542e+06 1.01534e+06 1.01529e+06 1.01526e+06 1.01523e+06 1.01522e+06 $X_5^{'}$ 1.01626e+06 1.01614e+06 1.01561e+06 1.01544e+06 1.01535e+06 1.0153e+06 1.01527e+06 1.01524e+06 1.01522e+06 $\eta_{t}$ 0.009 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 $X_5$ 1.0152e+06 1.01519e+06 1.01514e+06 1.01512e+06 1.01512e+06 1.01511e+06 1.01511e+06 1.01511e+06 1.01511e+06 $X_5^{'}$ 1.01521e+06 1.0152e+06 1.01514e+06 1.01513e+06 1.01512e+06 1.01511e+06 1.01511e+06 1.01511e+06 1.01511e+06
 $\eta_{t}$ 0 1e-06 2e-06 3e-06 4e-06 5e-06 6e-06 7e-06 8e-06 $X_5$ 1.20334e+06 1.19988e+06 1.19265e+06 1.18715e+06 1.18114e+06 1.17113e+06 1.15525e+06 1.1462e+06 1.129e+06 $X_5^{'}$ 1.23717e+06 1.23368e+06 1.22494e+06 1.21576e+06 1.20173e+06 1.18519e+06 1.16904e+06 1.15098e+06 1.13789e+06 $\eta_{t}$ 9e-06 1e-05 2e-05 3e-05 4e-05 5e-05 6e-05 7e-05 8e-05 $X_5$ 1.11937e+06 1.11052e+06 1.06571e+06 1.04884e+06 1.0404e+06 1.03534e+06 1.03196e+06 1.02955e+06 1.02774e+06 $X_5^{'}$ 1.12717e+06 1.12252e+06 1.06751e+06 1.05004e+06 1.0413e+06 1.03606e+06 1.03256e+06 1.03007e+06 1.02819e+06 $\eta_{t}$ 9e-05 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 $X_5$ 1.02634e+06 1.02521e+06 1.02015e+06 1.01847e+06 1.01762e+06 1.01714e+06 1.01677e+06 1.01653e+06 1.01635e+06 $X_5^{'}$ 1.02703e+06 1.02558e+06 1.02033e+06 1.01858e+06 1.01771e+06 1.01719e+06 1.01684e+06 1.01659e+06 1.0164e+06 $\eta_{t}$ 0.0009 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 $X_5$ 1.01621e+06 1.0161e+06 1.01559e+06 1.01542e+06 1.01534e+06 1.01529e+06 1.01526e+06 1.01523e+06 1.01522e+06 $X_5^{'}$ 1.01626e+06 1.01614e+06 1.01561e+06 1.01544e+06 1.01535e+06 1.0153e+06 1.01527e+06 1.01524e+06 1.01522e+06 $\eta_{t}$ 0.009 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 $X_5$ 1.0152e+06 1.01519e+06 1.01514e+06 1.01512e+06 1.01512e+06 1.01511e+06 1.01511e+06 1.01511e+06 1.01511e+06 $X_5^{'}$ 1.01521e+06 1.0152e+06 1.01514e+06 1.01513e+06 1.01512e+06 1.01511e+06 1.01511e+06 1.01511e+06 1.01511e+06
The terminal wealth when $K = 8$, $u_{ft}^b$ = -500000 and $u_{ft}^b$ = -1000000, $\eta_{t} = 0, 0.000001, \dots, 0.000009$
 $\eta_{t}$ 0 1e-07 2e-07 3e-07 4e-07 $X_5$ 1.23717e+06 1.23717e+06 1.23703e+06 1.23683e+06 1.23582e+06 $X_5^{'}$ 1.24029e+06 240294 1.24015e+06 1.24015e+06 1.23969e+06 $\eta_{t}$ 5e-07 6e-07 7e-07 8e-07 9e-07 $X_5$ 1.23556e+06 1.23474e+06 1.23466e+06 1.23464e+06 1.23368e+06 $X_5^{'}$ 1.23969e+06 1.23713e+06 1.23713e+06 1.23713e+06 1.23713e+06
 $\eta_{t}$ 0 1e-07 2e-07 3e-07 4e-07 $X_5$ 1.23717e+06 1.23717e+06 1.23703e+06 1.23683e+06 1.23582e+06 $X_5^{'}$ 1.24029e+06 240294 1.24015e+06 1.24015e+06 1.23969e+06 $\eta_{t}$ 5e-07 6e-07 7e-07 8e-07 9e-07 $X_5$ 1.23556e+06 1.23474e+06 1.23466e+06 1.23464e+06 1.23368e+06 $X_5^{'}$ 1.23969e+06 1.23713e+06 1.23713e+06 1.23713e+06 1.23713e+06
The terminal wealth when $K = 0, 1, \dots, 13$, $u_{ft}^b$ = -500000 dollars, $\eta_{t}$ = 0.0000001
 $K$ 0 1 2 3 4 5 6 $X_5$ 1015090 1065669 1102726 1136562 1160553 1181499 1202122 $K$ 7 8 9 10 11 12 13 $X_5$ 1221479 1240294 1258686 1275590 1284427 1284833 1284833
 $K$ 0 1 2 3 4 5 6 $X_5$ 1015090 1065669 1102726 1136562 1160553 1181499 1202122 $K$ 7 8 9 10 11 12 13 $X_5$ 1221479 1240294 1258686 1275590 1284427 1284833 1284833
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