Article Contents
Article Contents

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

• * Corresponding author: Peng Zhang
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.

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

 Citation:

• Figure 1.  The multiperiod weighted digraph

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

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 $W_6$ 2.176 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

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

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

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

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

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

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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