American Institute of Mathematical Sciences

July  2017, 13(3): 1169-1187. doi: 10.3934/jimo.2016067

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

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

* Corresponding author: Peng Zhang

Received  February 2015 Published  October 2016

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

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.

Citation: Peng Zhang. Multiperiod mean semi-absolute deviation interval portfolio selection with entropy constraints. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1169-1187. doi: 10.3934/jimo.2016067
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References:
The multiperiod weighted digraph
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
 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
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 $W_6$ 2.176 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
 $H_t$ 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 $W_6$ 2.176 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
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
 $\theta$ 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
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
 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
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
 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
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
 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
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
 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
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
 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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