American Institute of Mathematical Sciences

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

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

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.

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: 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
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

D. Dubois and H. Prade, Possibility Theory, Plenum Perss, New York, 1988. doi: 10.1007/978-1-4684-5287-7.  Google Scholar

[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.   Google Scholar

[12]

S. C. Fang and S. Puthenpura, Linear Optimization and Extensions: Theory and Algorithms, Prentice-Hall Inc, 1993. Google Scholar

[13]

C. D. Feinstein and M. N. Thapa, Notes: A reformation of a mean-absolute deviation portfolio optimization, Management Science, 39 (1993), 1552-1558.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

H. Ishibuchi and H. Tanaka, Multiobjective programming in optimization of the interval objective function, European Journal of Operational Research, 48 (1990), 219-225.   Google Scholar

[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.  Google Scholar

[22]

J. N. Kapur, Maximum Entropy Models in Science and Engineering, John Wiley & Sons, Inc. , New York, 1989.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x.  Google Scholar

[33]

R. Moore, Interval Analysis: Prentice Hall, New York: Englewood Cliffs, 1966.  Google Scholar

[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.  Google Scholar

[35]

M. G. Speranza, Linear programming models for portfolio optimization, The Journal of Finance, 14 (1993), 107-123.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

D. Dubois and H. Prade, Possibility Theory, Plenum Perss, New York, 1988. doi: 10.1007/978-1-4684-5287-7.  Google Scholar

[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.   Google Scholar

[12]

S. C. Fang and S. Puthenpura, Linear Optimization and Extensions: Theory and Algorithms, Prentice-Hall Inc, 1993. Google Scholar

[13]

C. D. Feinstein and M. N. Thapa, Notes: A reformation of a mean-absolute deviation portfolio optimization, Management Science, 39 (1993), 1552-1558.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

H. Ishibuchi and H. Tanaka, Multiobjective programming in optimization of the interval objective function, European Journal of Operational Research, 48 (1990), 219-225.   Google Scholar

[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.  Google Scholar

[22]

J. N. Kapur, Maximum Entropy Models in Science and Engineering, John Wiley & Sons, Inc. , New York, 1989.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x.  Google Scholar

[33]

R. Moore, Interval Analysis: Prentice Hall, New York: Englewood Cliffs, 1966.  Google Scholar

[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.  Google Scholar

[35]

M. G. Speranza, Linear programming models for portfolio optimization, The Journal of Finance, 14 (1993), 107-123.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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
1Asset 1Asset 13Asset 15Asset 17Asset 28otherwise 0
0.30.30.30.30.3
2Asset 1Asset 13Asset 15Asset 17Asset 28otherwise 0
0.30.30.30.30.3
3Asset 1Asset 13Asset 15Asset 17Asset 28otherwise 0
0.30.30.30.30.3
4Asset 1Asset 12Asset 13Asset 15Asset 17otherwise 0
0.30.30.30.30.3
5Asset 1Asset 12Asset 13Asset 15Asset 17otherwise 0
0.30.30.30.30.3
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The optimal investment proportions
1Asset 1Asset 13Asset 15Asset 17Asset 28otherwise 0
0.30.30.30.30.3
2Asset 1Asset 13Asset 15Asset 17Asset 28otherwise 0
0.30.30.30.30.3
3Asset 1Asset 13Asset 15Asset 17Asset 28otherwise 0
0.30.30.30.30.3
4Asset 1Asset 12Asset 13Asset 15Asset 17otherwise 0
0.30.30.30.30.3
5Asset 1Asset 12Asset 13Asset 15Asset 17otherwise 0
0.30.30.30.30.3
Table 2.  the optimal terminal wealth when $\theta=0.5, H_t =0,0.2,...,4.4$
$H_t$00.20.40.60.811.21.41.61.8
$W_6$1.0851.93662.17922.17922.17922.17922.17922.17922.17922.1792
$H_t$22.22.42.62.833.23.43.63.8
$W_6$2.17602.17132.16452.15572.14462.13092.11482.09582.07282.0438
$H_t$44.24.4
$W_6$2.00221.94471.6974
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$H_t$00.20.40.60.811.21.41.61.8
$W_6$1.0851.93662.17922.17922.17922.17922.17922.17922.17922.1792
$H_t$22.22.42.62.833.23.43.63.8
$W_6$2.17602.17132.16452.15572.14462.13092.11482.09582.07282.0438
$H_t$44.24.4
$W_6$2.00221.94471.6974
Table 3.  the optimal terminal wealth when $H_t=0.5,\theta=0,0.1,...,1$
$\theta$00.10.20.30.40.50.60.70.80.9
$W_6$2.18322.18322.18292.17922.17922.17922.17922.16602.15152.0674
$\theta$1
$W_6$1.1368
Table Options

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$\theta$00.10.20.30.40.50.60.70.80.9
$W_6$2.18322.18322.18292.17922.17922.17922.17922.16602.15152.0674
$\theta$1
$W_6$1.1368
Table 4.  The fuzzy return rates on assets of five periods investment
Asset 1Asset 2Asset 3Asset 4Asset 5Asset 6
10.1300 0.15590.0556 0.09430.0921 0.12440.1044 0.12990.0611 0.09910.0899 0.1229
20.1339 0.15590.0603 0.10220.0925 0.12440.1106 0.12990.0702 0.09910.0916 0.1229
30.13570.15590.0645 0.10690.1034 0.12440.1210 0.12990.0809 0.0991
40.1449 0.15820.0742 0.11170.1059 0.12440.1249 0.12990.0820 0.09910.0952 0.1229
50.1480 0.15830.0943 0.11630.1099 0.12440.1250 0.13270.0860 0.09910.1029 0.1229
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Asset 1Asset 2Asset 3Asset 4Asset 5Asset 6
10.1300 0.15590.0556 0.09430.0921 0.12440.1044 0.12990.0611 0.09910.0899 0.1229
20.1339 0.15590.0603 0.10220.0925 0.12440.1106 0.12990.0702 0.09910.0916 0.1229
30.13570.15590.0645 0.10690.1034 0.12440.1210 0.12990.0809 0.0991
40.1449 0.15820.0742 0.11170.1059 0.12440.1249 0.12990.0820 0.09910.0952 0.1229
50.1480 0.15830.0943 0.11630.1099 0.12440.1250 0.13270.0860 0.09910.1029 0.1229
Table 5.  The fuzzy return rates on assets of five periods investment
Asset 7Asset 8Asset 9Asset 10Asset 11Asset 12
10.0675 0.09200.0981 0.14950.0513 0.07650.0310 0.04430.0510 0.06390.1048 0.1438
20.0728 0.10850.1022 0.14950.0714 0.08660.0345 0.04750.0534 0.06500.1101 0.1504
30.0863 0.11200.1058 0.14950.0765 0.08700.0440 0.04970.0556 0.07810.1253 0.1506
40.0887 0.11710.1271 0.14950.0813 0.09080.0442 0.05180.0636 0.08110.1404 0.1577
50.0920 0.12170.1385 0.15280.0846 0.09210.0443 0.05400.0639 0.08420.1438 0.1641
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Asset 7Asset 8Asset 9Asset 10Asset 11Asset 12
10.0675 0.09200.0981 0.14950.0513 0.07650.0310 0.04430.0510 0.06390.1048 0.1438
20.0728 0.10850.1022 0.14950.0714 0.08660.0345 0.04750.0534 0.06500.1101 0.1504
30.0863 0.11200.1058 0.14950.0765 0.08700.0440 0.04970.0556 0.07810.1253 0.1506
40.0887 0.11710.1271 0.14950.0813 0.09080.0442 0.05180.0636 0.08110.1404 0.1577
50.0920 0.12170.1385 0.15280.0846 0.09210.0443 0.05400.0639 0.08420.1438 0.1641
Table 6.  The fuzzy return rates on assets of five periods investment
Asset 13Asset 14Asset 15Asset 16Asset 17Asset 18
10.1778 0.23190.0508 0.07460.1422 0.15500.0403 0.08330.1232 0.16210.0648 0.1183
20.1885 0.23190.0588 0.07460.1485 0.15500.0417 0.08330.1479 0.16210.0740 0.1625
30.2068 0.23190.0653 0.07460.1504 0.15710.0443 0.08680.1485 0.16210.0748 0.1949
40.2131 0.23190.0685 0.07460.1505 0.16240.0473 0.10200.1529 0.16210.0889 0.2044
50.2156 0.23190.0716 0.07460.1519 0.16800.0606 0.10640.1531 0.16260.1183 0.2144
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Asset 13Asset 14Asset 15Asset 16Asset 17Asset 18
10.1778 0.23190.0508 0.07460.1422 0.15500.0403 0.08330.1232 0.16210.0648 0.1183
20.1885 0.23190.0588 0.07460.1485 0.15500.0417 0.08330.1479 0.16210.0740 0.1625
30.2068 0.23190.0653 0.07460.1504 0.15710.0443 0.08680.1485 0.16210.0748 0.1949
40.2131 0.23190.0685 0.07460.1505 0.16240.0473 0.10200.1529 0.16210.0889 0.2044
50.2156 0.23190.0716 0.07460.1519 0.16800.0606 0.10640.1531 0.16260.1183 0.2144
Table 7.  The fuzzy return rates on assets of five periods investment
Asset 19Asset 20Asset 21Asset 22Asset 23Asset 24
10.0760 0.10000.1100 0.12840.0519 0.08330.1075 0.12050.0123 0.04390.0805 0.1082
20.0832 0.10000.1150 0.12840.0524 0.08840.1134 0.12050.0151 0.07560.0811 0.1082
30.0856 0.10000.1152 0.12840.0752 0.09230.1162 0.12380.0221 0.08400.0886 0.1082
40.0880 0.10000.1200 0.12850.0798 0.09610.1197 0.12720.0231 0.09160.0928 0.1082
50.0903 0.10000.1217 0.13200.0833 0.10010.1201 0.13070.0439 0.09960.0959 0.1082
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Asset 19Asset 20Asset 21Asset 22Asset 23Asset 24
10.0760 0.10000.1100 0.12840.0519 0.08330.1075 0.12050.0123 0.04390.0805 0.1082
20.0832 0.10000.1150 0.12840.0524 0.08840.1134 0.12050.0151 0.07560.0811 0.1082
30.0856 0.10000.1152 0.12840.0752 0.09230.1162 0.12380.0221 0.08400.0886 0.1082
40.0880 0.10000.1200 0.12850.0798 0.09610.1197 0.12720.0231 0.09160.0928 0.1082
50.0903 0.10000.1217 0.13200.0833 0.10010.1201 0.13070.0439 0.09960.0959 0.1082
Table 8.  The fuzzy return rates on assets of five periods investment
Asset 25Asset 26Asset 27Asset 28Asset 29Asset 30
10.0921 0.11000.1054 0.14400.0282 0.04550.1291 0.13880.1026 0.12010.0928 0.1101
20.0941 0.11000.1111 0.14400.0368 0.05080.1303 0.14600.1045 0.12010.0972 0.1101
30.0974 0.11000.1217 0.14400.0390 0.06220.1324 0.14650.1066 0.12010.0995 0.1101
40.0976 0.11120.1377 0.14870.0412 0.07120.1345 0.15070.1113 0.12010.1019 0.1101
50.1036 0.11440.1400 0.14900.0455 0.07830.1388 0.15520.1133 0.12170.1021 0.1101
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Asset 25Asset 26Asset 27Asset 28Asset 29Asset 30
10.0921 0.11000.1054 0.14400.0282 0.04550.1291 0.13880.1026 0.12010.0928 0.1101
20.0941 0.11000.1111 0.14400.0368 0.05080.1303 0.14600.1045 0.12010.0972 0.1101
30.0974 0.11000.1217 0.14400.0390 0.06220.1324 0.14650.1066 0.12010.0995 0.1101
40.0976 0.11120.1377 0.14870.0412 0.07120.1345 0.15070.1113 0.12010.1019 0.1101
50.1036 0.11440.1400 0.14900.0455 0.07830.1388 0.15520.1133 0.12170.1021 0.1101
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