# American Institute of Mathematical Sciences

April  2019, 15(2): 537-564. doi: 10.3934/jimo.2018056

## Chance-constrained multiperiod mean absolute deviation uncertain portfolio selection

 School of Economics and Management, South China Normal University, Guangzhou 510006, China

Received  August 2017 Revised  December 2018 Published  April 2018

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

In this paper, we propose a new multiperiod mean absolute deviation uncertain chance-constrained portfolio selection model with transaction costs, borrowing constraints, threshold constraints and cardinality constraints. In proposed model, the return rate of asset is quantified by uncertain expected value and the risk is characterized by uncertain absolute deviation. The chance constraints are that the uncertain expected return of the portfolio selection is bigger than the preset return of investors under the given confidence level. Cardinality constraints limit the number of assets in the optimal portfolio and threshold constraints limit the amount of capital to be invested in each asset and prevent very small investments in any asset. Based on uncertain theories, the model is converted to a dynamic optimization problem. Because of the transaction costs and cardinality constraints, the multiperiod portfolio selection is a mix integer dynamic optimization problem with path dependence, which is "NP hard" problem. The proposed model is approximated to a mix integer dynamic programming model. A novel discrete iteration method is designed to obtain the optimal portfolio strategy, and is proved linearly convergent. Finally, an example is given to illustrate the behavior of the proposed model and the designed algorithm using real data from the Shanghai Stock Exchange.

Citation: Peng Zhang. Chance-constrained multiperiod mean absolute deviation uncertain portfolio selection. Journal of Industrial & Management Optimization, 2019, 15 (2) : 537-564. doi: 10.3934/jimo.2018056
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##### References:
The multiperiod weighted digraph
The optimal solution when $K = 3, AD_t = 0.02, r_{0t} = 0.1, \delta = 95\%$
 The optimal investment proportions 1 Asset3 Asset 13 Asset 17 $x_{f1}$ 0.2 0.171581 0.2 0.428419 2 Asset15 Asset 17 Asset 29 $x_{f2}$ 0.2 0.2 0.2 0.4 3 Asset3 Asset 15 Asset 24 $x_{f3}$ 0.149406 0.2 0.2 0.450594 4 Asset13 Asset 20 Asset 25 $x_{f4}$ 0.169388 0.2 0.2 0.430612 5 Asset13 Asset 17 Asset 20 $x_{f5}$ 0.170902 0.2 0.2 0.429098
 The optimal investment proportions 1 Asset3 Asset 13 Asset 17 $x_{f1}$ 0.2 0.171581 0.2 0.428419 2 Asset15 Asset 17 Asset 29 $x_{f2}$ 0.2 0.2 0.2 0.4 3 Asset3 Asset 15 Asset 24 $x_{f3}$ 0.149406 0.2 0.2 0.450594 4 Asset13 Asset 20 Asset 25 $x_{f4}$ 0.169388 0.2 0.2 0.430612 5 Asset13 Asset 17 Asset 20 $x_{f5}$ 0.170902 0.2 0.2 0.429098
The optimal solution when $K = 6, AD_t = 0.02, r_{0t} = 0.1, \delta = 95\%$
 The optimal investment proportions 1 Asset3 Asset 13 Asset 17 Asset 22 Asset 25 $x_{f1}$ 0.2 0.004973 0.2 0.2 0.2 0.195027 2 Asset15 Asset 17 Asset 24 Asset 30 $x_{f2}$ 0.2 0.2 0.2 0.07038627 0.329614 3 Asset3 Asset 13 Asset 15 Asset 24 $x_{f3}$ 0.2 0.060781 0.2 0.2 0.339219 3 Asset8 Asset 15 Asset 20 Asset 25 $x_{f4}$ 0.055307 0.2 0.2 0.2 0.344693 4 Asset15 Asset 17 Asset 20 Asset 25 Asset 30 $x_{f5}$ 0.04597 0.2 0.2 0.2 0.2 0.15403
 The optimal investment proportions 1 Asset3 Asset 13 Asset 17 Asset 22 Asset 25 $x_{f1}$ 0.2 0.004973 0.2 0.2 0.2 0.195027 2 Asset15 Asset 17 Asset 24 Asset 30 $x_{f2}$ 0.2 0.2 0.2 0.07038627 0.329614 3 Asset3 Asset 13 Asset 15 Asset 24 $x_{f3}$ 0.2 0.060781 0.2 0.2 0.339219 3 Asset8 Asset 15 Asset 20 Asset 25 $x_{f4}$ 0.055307 0.2 0.2 0.2 0.344693 4 Asset15 Asset 17 Asset 20 Asset 25 Asset 30 $x_{f5}$ 0.04597 0.2 0.2 0.2 0.2 0.15403
The optimal solution when $K = 6, AD_t = 0.03, r_{0t} = 0.1, \delta = 95\%$
 The optimal investment proportions 1 Asset3 Asset 13 Asset 17 Asset 22 Asset 25 $x_{f1}$ 0.2 0.182593 0.2 0.2 0.2 0.017407 2 Asset5 Asset15 Asset 17 Asset 24 Asset 29 $x_{f2}$ 0.2 0.2 0.2 0.2 0.121612 0.078388 3 Asset3 Asset 13 Asset 15 Asset 17 Asset 24 $x_{f3}$ 0.2 0.2 0.2 0.04580153 0.2 0.154198 4 Asset6 Asset 8 Asset 15 Asset 20 Asset 25 $x_{f4}$ 0.196735 0.2 0.2 0.2 0.2 0.003265 5 Asset15 Asset 17 Asset 20 Asset 22 Asset 25 Asset 30 $x_{f5}$ 0.2 0.2 0.2 0.1686411 0.2 0.2 -0.16864
 The optimal investment proportions 1 Asset3 Asset 13 Asset 17 Asset 22 Asset 25 $x_{f1}$ 0.2 0.182593 0.2 0.2 0.2 0.017407 2 Asset5 Asset15 Asset 17 Asset 24 Asset 29 $x_{f2}$ 0.2 0.2 0.2 0.2 0.121612 0.078388 3 Asset3 Asset 13 Asset 15 Asset 17 Asset 24 $x_{f3}$ 0.2 0.2 0.2 0.04580153 0.2 0.154198 4 Asset6 Asset 8 Asset 15 Asset 20 Asset 25 $x_{f4}$ 0.196735 0.2 0.2 0.2 0.2 0.003265 5 Asset15 Asset 17 Asset 20 Asset 22 Asset 25 Asset 30 $x_{f5}$ 0.2 0.2 0.2 0.1686411 0.2 0.2 -0.16864
the optimal terminal wealth and risk of the portfolio when $AD_t = 0.07, r_{0t} = 0.15, \delta = 95\%, K = 2, \ldots, 9$
 $K$ 2 3 4 5 6 7 8 9 $\delta=95\%, W_6$ 1.50572 1.70608 1.92375 2.15888 2.40841 2.65356 2.65933 2.65933
 $K$ 2 3 4 5 6 7 8 9 $\delta=95\%, W_6$ 1.50572 1.70608 1.92375 2.15888 2.40841 2.65356 2.65933 2.65933
The optimal solution when $AD_t = 0.07, K = 3, r_{0t} = 0.18, \delta = 95\%$
 The optimal investment proportions 1 Asset12 Asset 13 Asset 28 $x_{f1}$ 0.2 2 0.2 0.4 2 Asset1 Asset 12 Asset 13 $x_{f2}$ 0.2 0.2 0.2 0.4 3 Asset12 Asset 13 Asset 17 $x_{f3}$ 0.2 0.2 0.2 0.4 4 Asset12 Asset 13 Asset 18 $x_{f4}$ 0.2 0.2 0.2 0.4 5 Asset12 Asset 13 Asset 18 $x_{f5}$ 0.2 0.2 0.2 0.4
 The optimal investment proportions 1 Asset12 Asset 13 Asset 28 $x_{f1}$ 0.2 2 0.2 0.4 2 Asset1 Asset 12 Asset 13 $x_{f2}$ 0.2 0.2 0.2 0.4 3 Asset12 Asset 13 Asset 17 $x_{f3}$ 0.2 0.2 0.2 0.4 4 Asset12 Asset 13 Asset 18 $x_{f4}$ 0.2 0.2 0.2 0.4 5 Asset12 Asset 13 Asset 18 $x_{f5}$ 0.2 0.2 0.2 0.4
The optimal solution when $AD_t = 0.07, K = 3, r_{0t} = 0.18, \delta = 99\%$
 The optimal investment proportions 1 Asset13 Asset 16 Asset 28 $x_{f1}$ 0.2 2 0.2 0.4 2 Asset12 Asset 13 Asset 16 $x_{f2}$ 0.2 0.2 0.2 0.4 3 Asset12 Asset 13 Asset 17 $x_{f3}$ 0.2 0.2 0.2 0.4 4 Asset12 Asset 13 Asset 18 $x_{f4}$ 0.2 0.2 0.2 0.4 5 Asset12 Asset 13 Asset 18 $x_{f5}$ 0.2 0.2 0.2 0.4
 The optimal investment proportions 1 Asset13 Asset 16 Asset 28 $x_{f1}$ 0.2 2 0.2 0.4 2 Asset12 Asset 13 Asset 16 $x_{f2}$ 0.2 0.2 0.2 0.4 3 Asset12 Asset 13 Asset 17 $x_{f3}$ 0.2 0.2 0.2 0.4 4 Asset12 Asset 13 Asset 18 $x_{f4}$ 0.2 0.2 0.2 0.4 5 Asset12 Asset 13 Asset 18 $x_{f5}$ 0.2 0.2 0.2 0.4
The uncertain return rates on assets of five periods investment
 Asset 1 Asset 2 Asset 3 1 0.143 0.1049 0.1156 0.075 0.0657 0.1664 0.1083 0.0832 0.06 2 0.1449 0.0881 0.1136 0.0813 0.0708 0.16 0.1085 0.0681 0.0603 3 0.1458 0.08 0.1127 0.0857 0.0666 0.1556 0.1139 0.0725 0.0548 4 0.1516 0.062 0.107 0.093 0.0579 0.1483 0.1152 0.056 0.054 5 0.1532 0.0609 0.1054 0.1053 0.0662 0.1359 0.1172 0.057 0.0516
 Asset 1 Asset 2 Asset 3 1 0.143 0.1049 0.1156 0.075 0.0657 0.1664 0.1083 0.0832 0.06 2 0.1449 0.0881 0.1136 0.0813 0.0708 0.16 0.1085 0.0681 0.0603 3 0.1458 0.08 0.1127 0.0857 0.0666 0.1556 0.1139 0.0725 0.0548 4 0.1516 0.062 0.107 0.093 0.0579 0.1483 0.1152 0.056 0.054 5 0.1532 0.0609 0.1054 0.1053 0.0662 0.1359 0.1172 0.057 0.0516
The uncertain return rates on assets of five periods investment
 Asset 4 Asset 5 Asset 6 1 0.1172 0.0731 0.0813 0.0801 0.0791 0.0616 0.1064 0.1635 0.0616 2 0.1203 0.0743 0.0782 0.0847 0.0772 0.0571 0.1073 0.1634 0.0608 3 0.1255 0.0749 0.073 0.09 0.0503 0.0517 0.1083 0.1093 0.0598 4 0.1274 0.0733 0.071 0.0906 0.0507 0.0512 0.1091 0.0763 0.059 5 0.1289 0.0633 0.07 0.0926 0.0495 0.0492 0.1129 0.0727 0.0551
 Asset 4 Asset 5 Asset 6 1 0.1172 0.0731 0.0813 0.0801 0.0791 0.0616 0.1064 0.1635 0.0616 2 0.1203 0.0743 0.0782 0.0847 0.0772 0.0571 0.1073 0.1634 0.0608 3 0.1255 0.0749 0.073 0.09 0.0503 0.0517 0.1083 0.1093 0.0598 4 0.1274 0.0733 0.071 0.0906 0.0507 0.0512 0.1091 0.0763 0.059 5 0.1289 0.0633 0.07 0.0926 0.0495 0.0492 0.1129 0.0727 0.0551
The uncertain return rates on assets of five periods investment
 Asset 7 Asset 8 Asset 9 1 0.0798 0.0562 0.1694 0.1238 0.0815 0.1023 0.0639 0.1522 0.0951 2 0.0907 0.0643 0.175 0.1259 0.076 0.1003 0.079 0.0673 0.0866 3 0.0992 0.0555 0.15 0.1277 0.0765 0.0985 0.0818 0.0606 0.0839 4 0.1029 0.0551 0.1462 0.1383 0.0538 0.0878 0.0861 0.0645 0.08 5 0.1069 0.0534 0.1423 0.1457 0.0612 0.0805 0.0884 0.065 0.0773
 Asset 7 Asset 8 Asset 9 1 0.0798 0.0562 0.1694 0.1238 0.0815 0.1023 0.0639 0.1522 0.0951 2 0.0907 0.0643 0.175 0.1259 0.076 0.1003 0.079 0.0673 0.0866 3 0.0992 0.0555 0.15 0.1277 0.0765 0.0985 0.0818 0.0606 0.0839 4 0.1029 0.0551 0.1462 0.1383 0.0538 0.0878 0.0861 0.0645 0.08 5 0.1069 0.0534 0.1423 0.1457 0.0612 0.0805 0.0884 0.065 0.0773
The uncertain return rates on assets of five periods investment
 Asset 10 Asset 11 Asset 12 1 0.0377 0.0325 0.0414 0.0575 0.0403 0.1282 0.1243 0.117 0.184 2 0.041 0.0292 0.0379 0.0592 0.0403 0.1264 0.1303 0.1007 0.1781 3 0.0469 0.0318 0.0321 0.0669 0.0374 0.1188 0.138 0.0827 0.1704 4 0.048 0.0314 0.0309 0.0724 0.04 0.1133 0.1491 0.0843 0.1593 5 0.0492 0.0318 0.0298 0.0741 0.0453 0.1116 0.154 0.0752 0.1544
 Asset 10 Asset 11 Asset 12 1 0.0377 0.0325 0.0414 0.0575 0.0403 0.1282 0.1243 0.117 0.184 2 0.041 0.0292 0.0379 0.0592 0.0403 0.1264 0.1303 0.1007 0.1781 3 0.0469 0.0318 0.0321 0.0669 0.0374 0.1188 0.138 0.0827 0.1704 4 0.048 0.0314 0.0309 0.0724 0.04 0.1133 0.1491 0.0843 0.1593 5 0.0492 0.0318 0.0298 0.0741 0.0453 0.1116 0.154 0.0752 0.1544
The uncertain return rates on assets of five periods investment
 Asset 13 Asset 14 Asset 15 1 0.2049 0.1244 0.1331 0.0254 0.1 0.0743 0.0893 0.2079 0.1463 2 0.2102 0.1182 0.1277 0.0667 0.0604 0.0443 0.1518 0.1015 0.0859 3 0.2194 0.1236 0.1186 0.07 0.0563 0.0411 0.1538 0.1033 0.084 4 0.2225 0.1248 0.1154 0.0716 0.05 0.0395 0.1565 0.0534 0.0812 5 0.2238 0.1029 0.1142 0.0731 0.0399 0.0379 0.16 0.0553 0.0778
 Asset 13 Asset 14 Asset 15 1 0.2049 0.1244 0.1331 0.0254 0.1 0.0743 0.0893 0.2079 0.1463 2 0.2102 0.1182 0.1277 0.0667 0.0604 0.0443 0.1518 0.1015 0.0859 3 0.2194 0.1236 0.1186 0.07 0.0563 0.0411 0.1538 0.1033 0.084 4 0.2225 0.1248 0.1154 0.0716 0.05 0.0395 0.1565 0.0534 0.0812 5 0.2238 0.1029 0.1142 0.0731 0.0399 0.0379 0.16 0.0553 0.0778
The uncertain return rates on assets of five periods investment
 Asset 16 Asset 17 Asset 18 1 0.0615 0.0622 0.4819 0.1665 0.1188 0.0375 0.0675 0.1157 0.0586 2 0.0625 0.2795 0.2306 0.155 0.0984 0.0772 0.1183 0.1137 0.4232 3 0.0656 0.0514 0.2276 0.1553 0.0893 0.0769 0.1349 0.1165 0.4066 4 0.0747 0.046 0.2185 0.1575 0.0844 0.0747 0.1467 0.1225 0.3947 5 0.0835 0.0506 0.2096 0.1579 0.0535 0.0744 0.1664 0.1122 0.375
 Asset 16 Asset 17 Asset 18 1 0.0615 0.0622 0.4819 0.1665 0.1188 0.0375 0.0675 0.1157 0.0586 2 0.0625 0.2795 0.2306 0.155 0.0984 0.0772 0.1183 0.1137 0.4232 3 0.0656 0.0514 0.2276 0.1553 0.0893 0.0769 0.1349 0.1165 0.4066 4 0.0747 0.046 0.2185 0.1575 0.0844 0.0747 0.1467 0.1225 0.3947 5 0.0835 0.0506 0.2096 0.1579 0.0535 0.0744 0.1664 0.1122 0.375
The uncertain return rates on assets of five periods investment
 Asset 19 Asset 20 Asset 21 1 0.25 0.0736 0.0896 0.0825 0.1559 0.0853 0.0536 0.0817 0.0403 2 0.0916 0.0716 0.0634 0.1217 0.0633 0.0619 0.0704 0.0544 0.122 3 0.0928 0.0708 0.0622 0.1218 0.062 0.0618 0.0838 0.0666 0.1087 4 0.094 0.05 0.061 0.1243 0.0516 0.0593 0.088 0.0681 0.1045 5 0.0952 0.0472 0.06 0.1269 0.0267 0.0568 0.0917 0.0703 0.1007
 Asset 19 Asset 20 Asset 21 1 0.25 0.0736 0.0896 0.0825 0.1559 0.0853 0.0536 0.0817 0.0403 2 0.0916 0.0716 0.0634 0.1217 0.0633 0.0619 0.0704 0.0544 0.122 3 0.0928 0.0708 0.0622 0.1218 0.062 0.0618 0.0838 0.0666 0.1087 4 0.094 0.05 0.061 0.1243 0.0516 0.0593 0.088 0.0681 0.1045 5 0.0952 0.0472 0.06 0.1269 0.0267 0.0568 0.0917 0.0703 0.1007
The uncertain return rates on assets of five periods investment
 Asset 22 Asset 23 Asset 24 1 0.1083 0.1197 0.0684 0.0413 0.0149 0.0749 0.143 0.0458 0.0141 2 0.117 0.0618 0.0739 0.0454 0.1137 0.204 0.0947 0.0637 0.0393 3 0.12 0.0623 0.0708 0.0531 0.1209 0.1963 0.0984 0.0628 0.0355 4 0.1235 0.0656 0.0674 0.0574 0.093 0.192 0.1005 0.0548 0.0334 5 0.1254 0.0488 0.0654 0.0718 0.0716 0.1777 0.1021 0.0493 0.0319
 Asset 22 Asset 23 Asset 24 1 0.1083 0.1197 0.0684 0.0413 0.0149 0.0749 0.143 0.0458 0.0141 2 0.117 0.0618 0.0739 0.0454 0.1137 0.204 0.0947 0.0637 0.0393 3 0.12 0.0623 0.0708 0.0531 0.1209 0.1963 0.0984 0.0628 0.0355 4 0.1235 0.0656 0.0674 0.0574 0.093 0.192 0.1005 0.0548 0.0334 5 0.1254 0.0488 0.0654 0.0718 0.0716 0.1777 0.1021 0.0493 0.0319
The uncertain return rates on assets of five periods investment
 Asset 25 Asset 26 Asset 27 1 0.0589 0.1432 0.1024 0.0783 0.1712 0.1096 0.069 0.0047 0.1634 2 0.1021 0.0652 0.0591 0.1276 0.0906 0.1263 0.0438 0.1623 0.1828 3 0.1037 0.0567 0.0574 0.1329 0.0949 0.121 0.0506 0.1526 0.176 4 0.1044 0.0314 0.0567 0.1432 0.0812 0.1105 0.0562 0.1232 0.1694 5 0.109 0.0243 0.0521 0.1445 0.0777 0.1093 0.0619 0.0511 0.1647
 Asset 25 Asset 26 Asset 27 1 0.0589 0.1432 0.1024 0.0783 0.1712 0.1096 0.069 0.0047 0.1634 2 0.1021 0.0652 0.0591 0.1276 0.0906 0.1263 0.0438 0.1623 0.1828 3 0.1037 0.0567 0.0574 0.1329 0.0949 0.121 0.0506 0.1526 0.176 4 0.1044 0.0314 0.0567 0.1432 0.0812 0.1105 0.0562 0.1232 0.1694 5 0.109 0.0243 0.0521 0.1445 0.0777 0.1093 0.0619 0.0511 0.1647
The uncertain return rates on assets of five periods investment
 Asset 28 Asset 29 Asset 30 1 0.1551 0.1128 0.0498 0.0994 0.1233 0.0677 0.0674 0.1355 0.0854 2 0.1382 0.0789 0.0969 0.1123 0.0641 0.0498 0.1037 0.0636 0.0438 3 0.1395 0.0679 0.0956 0.1134 0.0648 0.0488 0.1048 0.0645 0.0426 4 0.1426 0.0379 0.0924 0.1157 0.0416 0.0464 0.106 0.0574 0.0414 5 0.147 0.0364 0.088 0.1175 0.0312 0.0446 0.1061 0.035 0.0413
 Asset 28 Asset 29 Asset 30 1 0.1551 0.1128 0.0498 0.0994 0.1233 0.0677 0.0674 0.1355 0.0854 2 0.1382 0.0789 0.0969 0.1123 0.0641 0.0498 0.1037 0.0636 0.0438 3 0.1395 0.0679 0.0956 0.1134 0.0648 0.0488 0.1048 0.0645 0.0426 4 0.1426 0.0379 0.0924 0.1157 0.0416 0.0464 0.106 0.0574 0.0414 5 0.147 0.0364 0.088 0.1175 0.0312 0.0446 0.1061 0.035 0.0413
The uncertain absolute deviation of assets of five periods investment
 Asset 1 Asset 2 Asset 3 Asset 4 Asset 5 Asset 6 Asset 7 Asset 8 1 0.0551 0.0599 0.0232 0.0386 0.0179 0.028 0.0588 0.0461 2 0.0506 0.0593 0.0268 0.0381 0.0183 0.0282 0.062 0.0443 3 0.0504 0.0571 0.0261 0.0331 0.0255 0.0263 0.0532 0.0439 4 0.0428 0.0533 0.0333 0.034 0.0255 0.0245 0.0521 0.0358 5 0.0422 0.0516 0.033 0.0395 0.0259 0.0258 0.0507 0.0356
 Asset 1 Asset 2 Asset 3 Asset 4 Asset 5 Asset 6 Asset 7 Asset 8 1 0.0551 0.0599 0.0232 0.0386 0.0179 0.028 0.0588 0.0461 2 0.0506 0.0593 0.0268 0.0381 0.0183 0.0282 0.062 0.0443 3 0.0504 0.0571 0.0261 0.0331 0.0255 0.0263 0.0532 0.0439 4 0.0428 0.0533 0.0333 0.034 0.0255 0.0245 0.0521 0.0358 5 0.0422 0.0516 0.033 0.0395 0.0259 0.0258 0.0507 0.0356
The uncertain absolute deviation of assets of five periods investment
 Asset 9 Asset 10 Asset 11 Asset 12 Asset 13 Asset 14 Asset 15 Asset 16 1 0.0374 0.0185 0.044 0.076 0.0563 0.0228 0.0573 0.0894 2 0.0386 0.0168 0.0435 0.0708 0.0615 0.0143 0.036 0.0765 3 0.0363 0.016 0.0408 0.0647 0.0589 0.015 0.0357 0.074 4 0.0362 0.0156 0.0398 0.062 0.0588 0.0162 0.0399 0.0704 5 0.0356 0.0123 0.0405 0.0587 0.0543 0.0195 0.0335 0.0688
 Asset 9 Asset 10 Asset 11 Asset 12 Asset 13 Asset 14 Asset 15 Asset 16 1 0.0374 0.0185 0.044 0.076 0.0563 0.0228 0.0573 0.0894 2 0.0386 0.0168 0.0435 0.0708 0.0615 0.0143 0.036 0.0765 3 0.0363 0.016 0.0408 0.0647 0.0589 0.015 0.0357 0.074 4 0.0362 0.0156 0.0398 0.062 0.0588 0.0162 0.0399 0.0704 5 0.0356 0.0123 0.0405 0.0587 0.0543 0.0195 0.0335 0.0688
The uncertain absolute deviation of assets of five periods investment
 Asset 17 Asset 18 Asset 19 Asset 20 Asset21 Asset 22 Asset 23 Asset 24 1 0.0285 0.0666 0.0409 0.0356 0.0119 0.0229 0.024 0.023 2 0.0361 0.0592 0.0338 0.0342 0.0453 0.0339 0.0319 0.0197 3 0.0393 0.0608 0.0217 0.0349 0.0443 0.0333 0.0311 0.0203 4 0.0417 0.0801 0.0278 0.0278 0.0435 0.0333 0.0346 0.0229 5 0.0322 0.0735 0.0269 0.0214 0.043 0.0287 0.0373 0.0255
 Asset 17 Asset 18 Asset 19 Asset 20 Asset21 Asset 22 Asset 23 Asset 24 1 0.0285 0.0666 0.0409 0.0356 0.0119 0.0229 0.024 0.023 2 0.0361 0.0592 0.0338 0.0342 0.0453 0.0339 0.0319 0.0197 3 0.0393 0.0608 0.0217 0.0349 0.0443 0.0333 0.0311 0.0203 4 0.0417 0.0801 0.0278 0.0278 0.0435 0.0333 0.0346 0.0229 5 0.0322 0.0735 0.0269 0.0214 0.043 0.0287 0.0373 0.0255
The uncertain absolute deviation of assets of five periods investment
 Asset 25 Asset 26 Asset 27 Asset 28 Asset 29 Asset 30 1 0.024 0.0456 0.0568 0.0434 0.0279 0.0379 2 0.0254 0.0545 0.0638 0.0441 0.0273 0.0233 3 0.0285 0.0542 0.0599 0.0411 0.0272 0.0271 4 0.0224 0.0482 0.0581 0.0426 0.0288 0.0257 5 0.0196 0.047 0.0564 0.0411 0.0292 0.0191
 Asset 25 Asset 26 Asset 27 Asset 28 Asset 29 Asset 30 1 0.024 0.0456 0.0568 0.0434 0.0279 0.0379 2 0.0254 0.0545 0.0638 0.0441 0.0273 0.0233 3 0.0285 0.0542 0.0599 0.0411 0.0272 0.0271 4 0.0224 0.0482 0.0581 0.0426 0.0288 0.0257 5 0.0196 0.047 0.0564 0.0411 0.0292 0.0191
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