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

Peng Zhang; Yongquan Zeng; Guotai Chi

doi:10.3934/jimo.2020039

Article Contents

2021, Volume 17, Issue 4: 1663-1680. Doi: 10.3934/jimo.2020039

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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
^* Corresponding author: Tel.:+86 18971194382

Received: July 2019

Revised: September 2019

Early access: February 2020

Published: July 2021

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

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Abstract

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.

Keywords:

Multiperiod mean semivariance portfolio selection,
transaction costs,
cardinality constraints,
time-consistency,
a discrete approximate iteration method.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

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Figure 1. The multiperiod weighted digraph

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Table 1. 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

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Table 2. 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	0.000001	0.000002	0.000003	0.000004	0.000005	0.000006	0.000007	0.000008
$ X_5 $	1203337	1199885	1192648	1187153	1181143	1171130	1155254	1146196	1128999
$ X_5^{'} $	1237168	1233682	1224935	1215756	1201734	1185194	1169038	1150985	1137892
$ \eta_{t} $	0.000009	0.00001	0.00002	0.00003	0.00004	0.00005	0.00006	0.00007	0.00008
$ X_5 $	1119367	1110519	1065707	1048836	1040399	1035338	1031963	1029552	1027745
$ X_5^{'} $	1127173	1122523	1067509	1050035	1041300	1036057	1032563	1030067	1028194
$ \eta_{t} $	0.00009	0.0001	0.0002	0.0003	0.0004	0.0005	0.0006	0.0007	0.0008
$ X_5 $	1026339	1025214	1020151	1018466	1017621	1017139	1016767	1016526	1016347
$ X_5^{'} $	1027030	1025575	1020331	1018585	1017711	1017187	1016837	1016588	1016400
$ \eta_{t} $	0.0009	0.001	0.002	0.003	0.004	0.005	0.006	0.007	0.008
$ X_5 $	1016208	1016095	1015593	1015425	1015343	1015291	1015257	1015234	1015216
$ X_5^{'} $	1016256	1016139	1015614	1015441	1015352	1015300	1015266	1015241	1015221
$ \eta_{t} $	0.009	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08
$ X_5 $	1015201	1015191	1015141	1015123	1015115	1015110	1015106	1015106	1015106
$ X_5^{'} $	1015207	1015195	1015142	1015126	1015117	1015111	1015106	1015106	1015106

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Table 3. 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	0.0000001	0.0000002	0.0000003	0.0000004
$ X_5 $	1237168	1237167	1237031	1236828	1235819
$ X_5^{'} $	1240294	240294	1240154	1240154	1239692
$ \eta_{t} $	0.0000005	0.0000006	0.0000007	0.0000008	0.0000009
$ X_5 $	1235560	1234737	1234664	1234642	1233682
$ X_5^{'} $	1239692	1237130	1237130	1237130	1237130

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Table 4. 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

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