Multiperiod mean semi-absolute deviation interval portfolio selection with entropy constraints

Peng Zhang

doi:10.3934/jimo.2016067

Article Contents

2017, Volume 13, Issue 3: 1169-1187. Doi: 10.3934/jimo.2016067

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Multiperiod mean semi-absolute deviation interval portfolio selection with entropy constraints

Peng Zhang^,

School of Economics, Wuhan University of Technology, Wuhan 430070, China

* Corresponding author: Peng Zhang
* Corresponding author: Peng Zhang

Received: January 31, 2015

Published: September 2017

This research was supported by the National Natural Science Foundation of China (nos. 71271161).

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Abstract

In this paper, we discuss the uncertain portfolio selection problem where the asset returns are represented by interval data. Since the parameters are interval values, the gain of returns is interval value as well. A new multiperiod mean semi-absolute deviation interval portfolio selection model with the transaction costs, borrowing constraints, threshold constraints and diversification degree of portfolio has been proposed, where the return and risk are characterized by the interval mean and interval semi-absolute deviation of return, respectively. The diversification degree of portfolio is measured by the presented possibilistic entropy. Threshold constraints limit the amount of capital to be invested in each stock and prevent very small investments in any stock. Based on interval theories, the model is converted to a dynamic optimization problem. Because of the transaction costs, the model is a dynamic optimization problem with path dependence. The discrete approximate iteration method is designed to obtain the optimal portfolio strategy. Finally, the comparison analysis of differently desired number of assets and different preference coefficients are provided by numerical examples to illustrate the efficiency of the proposed approach and the designed algorithm.

Keywords:

Multiperiod portfolio selection,
mean semi-absolute deviation,
entropy constraints,
interval numbers,
the 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 solution when $\theta=0.5,H_t=0.4$

	The optimal investment proportions
1	Asset 1	Asset 13	Asset 15	Asset 17	Asset 28	otherwise 0
	0.3	0.3	0.3	0.3	0.3
2	Asset 1	Asset 13	Asset 15	Asset 17	Asset 28	otherwise 0
	0.3	0.3	0.3	0.3	0.3
3	Asset 1	Asset 13	Asset 15	Asset 17	Asset 28	otherwise 0
	0.3	0.3	0.3	0.3	0.3
4	Asset 1	Asset 12	Asset 13	Asset 15	Asset 17	otherwise 0
	0.3	0.3	0.3	0.3	0.3
5	Asset 1	Asset 12	Asset 13	Asset 15	Asset 17	otherwise 0
	0.3	0.3	0.3	0.3	0.3

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Table 2. the optimal terminal wealth when $\theta=0.5, H_t =0,0.2,...,4.4$

$H_t$	0	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6	1.8
$W_6$	1.085	1.9366	2.1792	2.1792	2.1792	2.1792	2.1792	2.1792	2.1792	2.1792
$H_t$	2	2.2	2.4	2.6	2.8	3	3.2	3.4	3.6	3.8
$W_6$	2.1760	2.1713	2.1645	2.1557	2.1446	2.1309	2.1148	2.0958	2.0728	2.0438
$H_t$	4	4.2	4.4
$W_6$	2.0022	1.9447	1.6974

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Table 3. the optimal terminal wealth when $H_t=0.5,\theta=0,0.1,...,1$

$\theta$	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
$W_6$	2.1832	2.1832	2.1829	2.1792	2.1792	2.1792	2.1792	2.1660	2.1515	2.0674
$\theta$	1
$W_6$	1.1368

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Table 4. The fuzzy return rates on assets of five periods investment

	Asset 1	Asset 2	Asset 3	Asset 4	Asset 5	Asset 6
1	0.1300 0.1559	0.0556 0.0943	0.0921 0.1244	0.1044 0.1299	0.0611 0.0991	0.0899 0.1229
2	0.1339 0.1559	0.0603 0.1022	0.0925 0.1244	0.1106 0.1299	0.0702 0.0991	0.0916 0.1229
3	0.1357	0.1559	0.0645 0.1069	0.1034 0.1244	0.1210 0.1299	0.0809 0.0991
4	0.1449 0.1582	0.0742 0.1117	0.1059 0.1244	0.1249 0.1299	0.0820 0.0991	0.0952 0.1229
5	0.1480 0.1583	0.0943 0.1163	0.1099 0.1244	0.1250 0.1327	0.0860 0.0991	0.1029 0.1229

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Table 5. The fuzzy return rates on assets of five periods investment

	Asset 7	Asset 8	Asset 9	Asset 10	Asset 11	Asset 12
1	0.0675 0.0920	0.0981 0.1495	0.0513 0.0765	0.0310 0.0443	0.0510 0.0639	0.1048 0.1438
2	0.0728 0.1085	0.1022 0.1495	0.0714 0.0866	0.0345 0.0475	0.0534 0.0650	0.1101 0.1504
3	0.0863 0.1120	0.1058 0.1495	0.0765 0.0870	0.0440 0.0497	0.0556 0.0781	0.1253 0.1506
4	0.0887 0.1171	0.1271 0.1495	0.0813 0.0908	0.0442 0.0518	0.0636 0.0811	0.1404 0.1577
5	0.0920 0.1217	0.1385 0.1528	0.0846 0.0921	0.0443 0.0540	0.0639 0.0842	0.1438 0.1641

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Table 6. The fuzzy return rates on assets of five periods investment

	Asset 13	Asset 14	Asset 15	Asset 16	Asset 17	Asset 18
1	0.1778 0.2319	0.0508 0.0746	0.1422 0.1550	0.0403 0.0833	0.1232 0.1621	0.0648 0.1183
2	0.1885 0.2319	0.0588 0.0746	0.1485 0.1550	0.0417 0.0833	0.1479 0.1621	0.0740 0.1625
3	0.2068 0.2319	0.0653 0.0746	0.1504 0.1571	0.0443 0.0868	0.1485 0.1621	0.0748 0.1949
4	0.2131 0.2319	0.0685 0.0746	0.1505 0.1624	0.0473 0.1020	0.1529 0.1621	0.0889 0.2044
5	0.2156 0.2319	0.0716 0.0746	0.1519 0.1680	0.0606 0.1064	0.1531 0.1626	0.1183 0.2144

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Table 7. The fuzzy return rates on assets of five periods investment

	Asset 19	Asset 20	Asset 21	Asset 22	Asset 23	Asset 24
1	0.0760 0.1000	0.1100 0.1284	0.0519 0.0833	0.1075 0.1205	0.0123 0.0439	0.0805 0.1082
2	0.0832 0.1000	0.1150 0.1284	0.0524 0.0884	0.1134 0.1205	0.0151 0.0756	0.0811 0.1082
3	0.0856 0.1000	0.1152 0.1284	0.0752 0.0923	0.1162 0.1238	0.0221 0.0840	0.0886 0.1082
4	0.0880 0.1000	0.1200 0.1285	0.0798 0.0961	0.1197 0.1272	0.0231 0.0916	0.0928 0.1082
5	0.0903 0.1000	0.1217 0.1320	0.0833 0.1001	0.1201 0.1307	0.0439 0.0996	0.0959 0.1082

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Table 8. The fuzzy return rates on assets of five periods investment

	Asset 25	Asset 26	Asset 27	Asset 28	Asset 29	Asset 30
1	0.0921 0.1100	0.1054 0.1440	0.0282 0.0455	0.1291 0.1388	0.1026 0.1201	0.0928 0.1101
2	0.0941 0.1100	0.1111 0.1440	0.0368 0.0508	0.1303 0.1460	0.1045 0.1201	0.0972 0.1101
3	0.0974 0.1100	0.1217 0.1440	0.0390 0.0622	0.1324 0.1465	0.1066 0.1201	0.0995 0.1101
4	0.0976 0.1112	0.1377 0.1487	0.0412 0.0712	0.1345 0.1507	0.1113 0.1201	0.1019 0.1101
5	0.1036 0.1144	0.1400 0.1490	0.0455 0.0783	0.1388 0.1552	0.1133 0.1217	0.1021 0.1101

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