# 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 and Management Optimization, 2017, 13 (3) : 1169-1187. doi: 10.3934/jimo.2016067
##### References:
 [1] G. Alefeld and G. Mayer, Interval analysis: theory and applications, Journal of Computational and Applied Mathematics, 121 (2000), 421-464.  doi: 10.1016/S0377-0427(00)00342-3. [2] R. D. Arnott and W. H. Wagner, The measurement and control of trading costs, Financial Analysts Journal, 46 (1990), 73-80.  doi: 10.2469/faj.v46.n6.73. [3] D. Bertsimas and D. Pachamanova, Robust multiperiod portfolio management in the presence of transaction costs, Computers and Operations Research, 35 (2008), 3-17.  doi: 10.1016/j.cor.2006.02.011. [4] R. Bhattacharyya, S. Kar and D. Majumder, Majumder, Fuzzymean-variance-skewness portfolio selection models by interval analysis, Computers & Mathematics with Applications, 61 (2011), 126-137.  doi: 10.1016/j.camwa.2010.10.039. [5] G. C. Calafiore, Multi-period portfolio optimization with linear control policies, Automatica, 44 (2008), 2463-2473.  doi: 10.1016/j.automatica.2008.02.007. [6] C. Carlsson and R. Fullér, On possibilistic mean value and variance of fuzzy numbers, Fuzzy Sets and Systems, 122 (2001), 315-326.  doi: 10.1016/S0165-0114(00)00043-9. [7] C. Carlsson, R. Fulleér and P. Majlender, A possibilistic approach to selecting portfolios with highest utility score, Fuzzy Sets and Systems, 131 (2002), 13-21.  doi: 10.1016/S0165-0114(01)00251-2. [8] U. Çlikyurt and S. Öekici, Multiperiod portfolio optimization models in stochastic markets using the mean-variance approach, European Journal of Operational Research, 179 (2007), 186-202. [9] X. Deng and R. J. Li, A portfolio selection model with borrowing constraint based on possibility theory, Applied Soft Computing, 12 (2012), 754-758.  doi: 10.1016/j.asoc.2011.10.017. [10] D. Dubois and H. Prade, Possibility Theory, Plenum Perss, New York, 1988. doi: 10.1007/978-1-4684-5287-7. [11] Y. Fang, K. K. Lai and S. Y. Wang, Portfolio rebalancing model with transaction costs based on fuzzy decision theory, European Journal of Operational Research, 175 (2006), 879-893. [12] S. C. Fang and S. Puthenpura, Linear Optimization and Extensions: Theory and Algorithms, Prentice-Hall Inc, 1993. [13] C. D. Feinstein and M. N. Thapa, Notes: A reformation of a mean-absolute deviation portfolio optimization, Management Science, 39 (1993), 1552-1558. [14] S. Giove and S. Funari, Nardelli, An interval portfolio selection problems based on regret function, European Journal of Operational Research, 170 (2006), 253-264. [15] N. Güpinar and B. Rustem, Worst-case robust decisions for multi-period mean-variance portfolio optimization, European Journal of Operational Research, 183 (2007), 981-1000.  doi: 10.1016/j.ejor.2006.02.046. [16] N. Güpinar, B. Rustem and R. Settergren, Multistage stochastic mean-variance portfolio analysis with transaction cost, Innovations, in Financial and Economic Networks, 3 (2003), 46-63. [17] B. Heidergott, G. J. Olsder and J. V. Woude, Max Plus at Work Modeling and Analysis of Press, synchronized systems: a course on max-plus algebra and its applications, Princeton University, 2006. [18] B. Hu and S. Wang, A novel approach in uncertain programming Part 1: New arithmetic and order relation for interval numbers, Journal of Industrial and Management Optimization, 2 (2006), 351-371.  doi: 10.3934/jimo.2006.2.351. [19] X. Huang, Risk curve and fuzzy portfolio selection, Computers and Mathematics with Applications, 55 (2008), 1102-1112.  doi: 10.1016/j.camwa.2007.06.019. [20] H. Ishibuchi and H. Tanaka, Multiobjective programming in optimization of the interval objective function, European Journal of Operational Research, 48 (1990), 219-225. [21] P. Jana, T. K. Roy and S. K. Mazumder, Multi-objective possibilistic model for portfolio selection with transaction cost, Journal of Computational and Applied Mathematics, 228 (2009), 188-196.  doi: 10.1016/j.cam.2008.09.008. [22] J. N. Kapur, Maximum Entropy Models in Science and Engineering, John Wiley & Sons, Inc. , New York, 1989. [23] H. Konno and H. Yamazaki, Mean-absolute Deviation Portfolio Optimization model and its applications to Tokyo stock market, Management Science, 37 (1991), 519-531. [24] K. K. Lai, S. Y. Wang, J. P. Xu, S. S. Zhu and Y. Fang, A class of linear interval programming problems and its application to portfolio selection, IEEE Transactions on Fuzzy Systems, 10 (2002), 698-704.  doi: 10.1109/TFUZZ.2002.805902. [25] T. León, V. Liem and E. Vercher, Viability of infeasible portfolio selection problems: A fuzzy approach, European Journal of Operational Research, 139 (2002), 178-189. [26] C. J. Li and Z. F. Li, Multi-period portfolio optimization for asset-liability management with bankrupt control, Applied Mathematics and Computation, 218 (2012), 11196-11208.  doi: 10.1016/j.amc.2012.05.010. [27] X. Li, Z. Qin and S. Kar, Mean-variance-skewness model for portfolio selection with fuzzy returns, European Journal of operational Research, 202 (2010), 239-247.  doi: 10.1016/j.ejor.2009.05.003. [28] J. Li and J. P. Xu, A class of possibilistic portfolio selection model with interval coefficients and its application, Fuzzy Optimization Decision Making, 6 (2007), 123-137.  doi: 10.1007/s10700-007-9005-y. [29] S. T. Liu, The mean-absolute deviation portfolio selection problem with interval valued returns, Journal of Computational and Applied Mathematics, 235 (2011), 4149-4157.  doi: 10.1016/j.cam.2011.03.008. [30] Y. J. Liu, W. G. Zhang and W. J. Xu, Fuzzy multi-period portfolio selection optimization models using multiple criteria, Automatica, 48 (2012), 3042-3053.  doi: 10.1016/j.automatica.2012.08.036. [31] Y. J. Liu, W. G. Zhang and P. Zhang, A multi-period portfolio selection optimization model by using interval analysis, Economic Modelling, 33 (2013), 113-119.  doi: 10.1016/j.econmod.2013.03.006. [32] H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x. [33] R. Moore, Interval Analysis: Prentice Hall, New York: Englewood Cliffs, 1966. [34] S. J. Sadjadi, S. M. Seyedhosseini and Kh. Hassanlou, Fuzzy multi period portfolio selection with different rates for borrowing and Lending, Applied Soft Computing, 11 (2011), 3821-3826.  doi: 10.1016/j.asoc.2011.02.015. [35] M. G. Speranza, Linear programming models for portfolio optimization, The Journal of Finance, 14 (1993), 107-123. [36] E. Vercher, J. Bermudez and J. Segura, Fuzzy portfolio optimization under downside risk measures, Fuzzy Sets and Systems, 158 (2007), 769-782.  doi: 10.1016/j.fss.2006.10.026. [37] M. Wu, D. Kong, J. Xu and N. Huang, On interval portfolio selection problem, Fuzzy Optimization and Decision Making, 12 (2013), 289-304.  doi: 10.1007/s10700-013-9155-z. [38] H. L. Wu and Z. F. Li, Multi-period mean-variance portfolio selection with regime switching and a stochastic cash flow, Insurance: Mathematics and Economics, 50 (2012), 371-384.  doi: 10.1016/j.insmatheco.2012.01.003. [39] W. Yan and S. Li, A class of multi-period semi-variance portfolio selection with a four-factor futures price model, Journal of Applied Mathematics and Computing, 29 (2009), 19-34.  doi: 10.1007/s12190-008-0086-8. [40] W. Yan, R. Miao and S. R. Li, Multi-period semi-variance portfolio selection: Model and numerical solution, Applied Mathematics and Computation, 194 (2007), 128-134.  doi: 10.1016/j.amc.2007.04.036. [41] A. Yoshimoto, The mean-variance approach to portfolio optimization subject to transaction costs, Journal of the Operational Research Society of Japan, 39 (1996), 99-117. [42] M. Yu, S. Takahashi, H. Inoue and S. Y. Wang, Dynamic portfolio optimization with risk control for absolute deviation model, European Journal of Operational Research, 201 (2010), 349-364.  doi: 10.1016/j.ejor.2009.03.009. [43] M. Yu and S. Y. Wang, Dynamic optimal portfolio with maximum absolute deviation model, Journal of Global Optimization, 53 (2012), 363-380.  doi: 10.1007/s10898-012-9887-2. [44] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X. [45] W. G. Zhang, Y. J. Liu and W. J. Xu, A possibilistic mean-semivariance-entropy model for multi-period portfolio selection with transaction costs, European Journal of Operational Research, 222 (2012), 41-349.  doi: 10.1016/j.ejor.2012.04.023. [46] W. G. Zhang, Y. L. Wang, Z. P. Nie and Z. K. Chen, Possibilistic mean-variance models and efficient frontiers for portfolio selection problem, Information Sciences, 177 (2007), 2787-2801.  doi: 10.1016/j.ins.2007.01.030. [47] W. G. Zhang, X. L. Zhang and W. L. Xiao, Portfolio selection under possibilistic mean-variance utility and a SMO algorithm, European Journal of Operational Research, 197 (2009), 693-700.  doi: 10.1016/j.ejor.2008.07.011. [48] W. G. Zhang, Y. J. Liu and W. J. Xu, A new fuzzy programming approach for multi-period portfolio Optimization with return demand and risk control, Fuzzy Sets and Systems, 246 (2014), 107-126.  doi: 10.1016/j.fss.2013.09.002. [49] P. Zhang and W. G. Zhang, Multiperiod mean absolute deviation fuzzy portfolio selection model with risk control and cardinality constraints, Fuzzy Sets and Systems, 255 (2014), 74-91.  doi: 10.1016/j.fss.2014.07.018.

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##### References:
 [1] G. Alefeld and G. Mayer, Interval analysis: theory and applications, Journal of Computational and Applied Mathematics, 121 (2000), 421-464.  doi: 10.1016/S0377-0427(00)00342-3. [2] R. D. Arnott and W. H. Wagner, The measurement and control of trading costs, Financial Analysts Journal, 46 (1990), 73-80.  doi: 10.2469/faj.v46.n6.73. [3] D. Bertsimas and D. Pachamanova, Robust multiperiod portfolio management in the presence of transaction costs, Computers and Operations Research, 35 (2008), 3-17.  doi: 10.1016/j.cor.2006.02.011. [4] R. Bhattacharyya, S. Kar and D. Majumder, Majumder, Fuzzymean-variance-skewness portfolio selection models by interval analysis, Computers & Mathematics with Applications, 61 (2011), 126-137.  doi: 10.1016/j.camwa.2010.10.039. [5] G. C. Calafiore, Multi-period portfolio optimization with linear control policies, Automatica, 44 (2008), 2463-2473.  doi: 10.1016/j.automatica.2008.02.007. [6] C. Carlsson and R. Fullér, On possibilistic mean value and variance of fuzzy numbers, Fuzzy Sets and Systems, 122 (2001), 315-326.  doi: 10.1016/S0165-0114(00)00043-9. [7] C. Carlsson, R. Fulleér and P. Majlender, A possibilistic approach to selecting portfolios with highest utility score, Fuzzy Sets and Systems, 131 (2002), 13-21.  doi: 10.1016/S0165-0114(01)00251-2. [8] U. Çlikyurt and S. Öekici, Multiperiod portfolio optimization models in stochastic markets using the mean-variance approach, European Journal of Operational Research, 179 (2007), 186-202. [9] X. Deng and R. J. Li, A portfolio selection model with borrowing constraint based on possibility theory, Applied Soft Computing, 12 (2012), 754-758.  doi: 10.1016/j.asoc.2011.10.017. [10] D. Dubois and H. Prade, Possibility Theory, Plenum Perss, New York, 1988. doi: 10.1007/978-1-4684-5287-7. [11] Y. Fang, K. K. Lai and S. Y. Wang, Portfolio rebalancing model with transaction costs based on fuzzy decision theory, European Journal of Operational Research, 175 (2006), 879-893. [12] S. C. Fang and S. Puthenpura, Linear Optimization and Extensions: Theory and Algorithms, Prentice-Hall Inc, 1993. [13] C. D. Feinstein and M. N. Thapa, Notes: A reformation of a mean-absolute deviation portfolio optimization, Management Science, 39 (1993), 1552-1558. [14] S. Giove and S. Funari, Nardelli, An interval portfolio selection problems based on regret function, European Journal of Operational Research, 170 (2006), 253-264. [15] N. Güpinar and B. Rustem, Worst-case robust decisions for multi-period mean-variance portfolio optimization, European Journal of Operational Research, 183 (2007), 981-1000.  doi: 10.1016/j.ejor.2006.02.046. [16] N. Güpinar, B. Rustem and R. Settergren, Multistage stochastic mean-variance portfolio analysis with transaction cost, Innovations, in Financial and Economic Networks, 3 (2003), 46-63. [17] B. Heidergott, G. J. Olsder and J. V. Woude, Max Plus at Work Modeling and Analysis of Press, synchronized systems: a course on max-plus algebra and its applications, Princeton University, 2006. [18] B. Hu and S. Wang, A novel approach in uncertain programming Part 1: New arithmetic and order relation for interval numbers, Journal of Industrial and Management Optimization, 2 (2006), 351-371.  doi: 10.3934/jimo.2006.2.351. [19] X. Huang, Risk curve and fuzzy portfolio selection, Computers and Mathematics with Applications, 55 (2008), 1102-1112.  doi: 10.1016/j.camwa.2007.06.019. [20] H. Ishibuchi and H. Tanaka, Multiobjective programming in optimization of the interval objective function, European Journal of Operational Research, 48 (1990), 219-225. [21] P. Jana, T. K. Roy and S. K. Mazumder, Multi-objective possibilistic model for portfolio selection with transaction cost, Journal of Computational and Applied Mathematics, 228 (2009), 188-196.  doi: 10.1016/j.cam.2008.09.008. [22] J. N. Kapur, Maximum Entropy Models in Science and Engineering, John Wiley & Sons, Inc. , New York, 1989. [23] H. Konno and H. Yamazaki, Mean-absolute Deviation Portfolio Optimization model and its applications to Tokyo stock market, Management Science, 37 (1991), 519-531. [24] K. K. Lai, S. Y. Wang, J. P. Xu, S. S. Zhu and Y. Fang, A class of linear interval programming problems and its application to portfolio selection, IEEE Transactions on Fuzzy Systems, 10 (2002), 698-704.  doi: 10.1109/TFUZZ.2002.805902. [25] T. León, V. Liem and E. Vercher, Viability of infeasible portfolio selection problems: A fuzzy approach, European Journal of Operational Research, 139 (2002), 178-189. [26] C. J. Li and Z. F. Li, Multi-period portfolio optimization for asset-liability management with bankrupt control, Applied Mathematics and Computation, 218 (2012), 11196-11208.  doi: 10.1016/j.amc.2012.05.010. [27] X. Li, Z. Qin and S. Kar, Mean-variance-skewness model for portfolio selection with fuzzy returns, European Journal of operational Research, 202 (2010), 239-247.  doi: 10.1016/j.ejor.2009.05.003. [28] J. Li and J. P. Xu, A class of possibilistic portfolio selection model with interval coefficients and its application, Fuzzy Optimization Decision Making, 6 (2007), 123-137.  doi: 10.1007/s10700-007-9005-y. [29] S. T. Liu, The mean-absolute deviation portfolio selection problem with interval valued returns, Journal of Computational and Applied Mathematics, 235 (2011), 4149-4157.  doi: 10.1016/j.cam.2011.03.008. [30] Y. J. Liu, W. G. Zhang and W. J. Xu, Fuzzy multi-period portfolio selection optimization models using multiple criteria, Automatica, 48 (2012), 3042-3053.  doi: 10.1016/j.automatica.2012.08.036. [31] Y. J. Liu, W. G. Zhang and P. Zhang, A multi-period portfolio selection optimization model by using interval analysis, Economic Modelling, 33 (2013), 113-119.  doi: 10.1016/j.econmod.2013.03.006. [32] H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x. [33] R. Moore, Interval Analysis: Prentice Hall, New York: Englewood Cliffs, 1966. [34] S. J. Sadjadi, S. M. Seyedhosseini and Kh. Hassanlou, Fuzzy multi period portfolio selection with different rates for borrowing and Lending, Applied Soft Computing, 11 (2011), 3821-3826.  doi: 10.1016/j.asoc.2011.02.015. [35] M. G. Speranza, Linear programming models for portfolio optimization, The Journal of Finance, 14 (1993), 107-123. [36] E. Vercher, J. Bermudez and J. Segura, Fuzzy portfolio optimization under downside risk measures, Fuzzy Sets and Systems, 158 (2007), 769-782.  doi: 10.1016/j.fss.2006.10.026. [37] M. Wu, D. Kong, J. Xu and N. Huang, On interval portfolio selection problem, Fuzzy Optimization and Decision Making, 12 (2013), 289-304.  doi: 10.1007/s10700-013-9155-z. [38] H. L. Wu and Z. F. Li, Multi-period mean-variance portfolio selection with regime switching and a stochastic cash flow, Insurance: Mathematics and Economics, 50 (2012), 371-384.  doi: 10.1016/j.insmatheco.2012.01.003. [39] W. Yan and S. Li, A class of multi-period semi-variance portfolio selection with a four-factor futures price model, Journal of Applied Mathematics and Computing, 29 (2009), 19-34.  doi: 10.1007/s12190-008-0086-8. [40] W. Yan, R. Miao and S. R. Li, Multi-period semi-variance portfolio selection: Model and numerical solution, Applied Mathematics and Computation, 194 (2007), 128-134.  doi: 10.1016/j.amc.2007.04.036. [41] A. Yoshimoto, The mean-variance approach to portfolio optimization subject to transaction costs, Journal of the Operational Research Society of Japan, 39 (1996), 99-117. [42] M. Yu, S. Takahashi, H. Inoue and S. Y. Wang, Dynamic portfolio optimization with risk control for absolute deviation model, European Journal of Operational Research, 201 (2010), 349-364.  doi: 10.1016/j.ejor.2009.03.009. [43] M. Yu and S. Y. Wang, Dynamic optimal portfolio with maximum absolute deviation model, Journal of Global Optimization, 53 (2012), 363-380.  doi: 10.1007/s10898-012-9887-2. [44] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X. [45] W. G. Zhang, Y. J. Liu and W. J. Xu, A possibilistic mean-semivariance-entropy model for multi-period portfolio selection with transaction costs, European Journal of Operational Research, 222 (2012), 41-349.  doi: 10.1016/j.ejor.2012.04.023. [46] W. G. Zhang, Y. L. Wang, Z. P. Nie and Z. K. Chen, Possibilistic mean-variance models and efficient frontiers for portfolio selection problem, Information Sciences, 177 (2007), 2787-2801.  doi: 10.1016/j.ins.2007.01.030. [47] W. G. Zhang, X. L. Zhang and W. L. Xiao, Portfolio selection under possibilistic mean-variance utility and a SMO algorithm, European Journal of Operational Research, 197 (2009), 693-700.  doi: 10.1016/j.ejor.2008.07.011. [48] W. G. Zhang, Y. J. Liu and W. J. Xu, A new fuzzy programming approach for multi-period portfolio Optimization with return demand and risk control, Fuzzy Sets and Systems, 246 (2014), 107-126.  doi: 10.1016/j.fss.2013.09.002. [49] P. Zhang and W. G. Zhang, Multiperiod mean absolute deviation fuzzy portfolio selection model with risk control and cardinality constraints, Fuzzy Sets and Systems, 255 (2014), 74-91.  doi: 10.1016/j.fss.2014.07.018.
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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