American Institute of Mathematical Sciences

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

Chance-constrained multiperiod mean absolute deviation uncertain portfolio selection

Peng Zhang ,

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.

Keywords: Chance-constrained programming, uncertainty measure, multiperiod portfolio optimization, mean absolute deviation model, the discrete iteration method.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
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
References:
[1]

K. P. Anagnostopoulos and G. Mamanis, The mean-variance cardinality constrained portfolio optimization problem: an experimental evaluation of five multiobjective evolutionary algorithms, Expert Systems with Applications, 38 (2011), 14208-14217. doi: 10.1016/j.eswa.2011.04.233. Google Scholar

[2]

D. Bertsimas and R. Shioda, Algorithms for cardinality-constrained quadratic optimization, Computational Optimization and Applications, 43 (2009), 1-12. doi: 10.1007/s10589-007-9126-9. Google Scholar

[3]

F. Cesarone, A. Scozzari and F. Tardella, A new method for mean-variance portfolio optimization with cardinality constraints, Annals of Operations Research, 205 (2013), 213-234. doi: 10.1007/s10479-012-1165-7. Google Scholar

[4]

Z. Chen, G. Li and Y. Zhao, Time-consistent investment policies in Markovian markets: A case of mean-variance analysis, Journal of Economic Dynamics and Control, 40 (2014), 293-316. doi: 10.1016/j.jedc.2014.01.011. Google Scholar

[5]

Z. Chen, J. Liu, G. Li and Z. Yan, Composite time-consistent multi-period risk measure and its application in optimal portfolio selection, Journal of Economic Dynamics and Control, 24 (2016), 515-540. doi: 10.1007/s11750-015-0407-7. Google Scholar

[6]

X. Y. Cui, D. Li, S. Y. Wang and S. S. Zhu, Better than dynamic mean-variance: Time inconsistency and free cash flow stream, Mathematical Finance, 22 (2012), 346-378. doi: 10.1111/j.1467-9965.2010.00461.x. Google Scholar

[7]

X. Y. Cui, X. Li and D. Li, Unified framework of mean-field formulations for optimal multi-period mean-variance portfolio selection, IEEE Transactions on Automatic Control, 59 (2014), 1833-1844. doi: 10.1109/TAC.2014.2311875. Google Scholar

[8]

X. Y. Cui, D. Li and X. Li, Mean variance policy for discrete time cone-constrained markets: time consistency in efficiency and the minimum-variance signed supermartingale measure, Mathematical Finance, 27 (2017), 471-504. doi: 10.1111/mafi.12093. Google Scholar

[9]

X. T. Cui, X. J. Zheng, S. S. Zhu and X. L. Sun, Convex relaxations and MIQCQP reformulations for a class of cardinality-constrained portfolio selection problems, Journal of Global Optimization, 56 (2013), 1409-1423. doi: 10.1007/s10898-012-9842-2. Google Scholar

[10]

G. F. Deng, W. T. Lin and C. C. Lo, Markowitz-based portfolio selection with cardinality constraints using improved particle swarm optimization, Expert Systems with Applications, 39 (2012), 4558-4566. Google Scholar

[11]

J. J. Gao, D. Li, X. Y. Cui and S. Y. Wang, Time cardinality constrained mean-variance dynamic portfolio selection and market timing: A stochastic control approach, Automatica, 54 (2015), 91-99. doi: 10.1016/j.automatica.2015.01.040. Google Scholar

[12]

N. Gülpinar 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

[13]

P. Gupta, M. Inuiguchi, M. Kumar Mehlawat and G. Mittal, Multiobjective credibilistic portfolio selection model with fuzzy chance-constraints, Information Sciences, 229 (2013), 1-17. doi: 10.1016/j.ins.2012.12.011. Google Scholar

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[15]

X. Huang, Fuzzy chance-constrained portfolio selection, Applied Mathematics and Computation, 177 (2006), 500-507. doi: 10.1016/j.amc.2005.11.027. Google Scholar

[16]

X. Huang and L. Qiao, A risk index model for multi-period uncertain portfolio selection, Information Sciences, 217 (2012), 108-116. doi: 10.1016/j.ins.2012.06.017. Google Scholar

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M. Köksalan and C. T. Şakar, An interactive approach to stochastic programming-based portfolio optimization, Annals of Operations Research, 245 (2016), 47-66. doi: 10.1007/s10479-014-1719-y. Google Scholar

[18]

H. Konno and H. Yamazaki, Mean absolute portfolio optimization model and its application to Tokyo stock market, Management Science, 37 (1991), 519-531. doi: 10.1287/mnsc.37.5.519. Google Scholar

[19]

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

[20]

D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Mathematical Finance, 10 (2000), 387-406. doi: 10.1111/1467-9965.00100. Google Scholar

[21]

D. Li, X. Sun and J. Wang, Optimal lot solution to cardinality constrained mean-variance formulation for portfolio selection, Mathematical Finance, 16 (2006), 83-101. doi: 10.1111/j.1467-9965.2006.00262.x. Google Scholar

[22]

X. Li, Z. Qin and L. Yang, A chance-constrained portfolio selection model with risk constraints, Applied Mathematics and Computation, 217 (2010), 949-951. doi: 10.1016/j.amc.2010.06.035. Google Scholar

[23]

B. Liu, Theory and Practice of Uncertain Programming, Physics-verlag, Heidelberg, 2002.Google Scholar

[24]

B. Liu, Uncertainty Theory, 2nd edition, Springer-Verlag, Berlin, 2007. Google Scholar

[25]

B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10. Google Scholar

[26]

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

[27]

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

[28]

R. Mansini, W. Ogryczak and M. G. Speranza, Conditional value at risk and related linear programming models for portfolio optimization, Annals of Operations Research, 152 (2007), 227-256. doi: 10.1007/s10479-006-0142-4. Google Scholar

[29]

H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. Google Scholar

[30]

H. M. Markowitz, Portfolio Selection: Efficient Diversification of Investments, New York, Wiley, 1959. Google Scholar

[31]

W. Murray and H. Shek, A local relaxation method for the cardinality constrained portfolio optimization problem, Computational Optimization and Applications, 53 (2012), 681-709. doi: 10.1007/s10589-012-9471-1. Google Scholar

[32]

F. Omidi, B. Abbasi and A. Nazemi, An efficient dynamic model for solving a portfolio selection with uncertain chance constraint models, Journal of Computational and Applied Mathematics, 319 (2017), 43-55. doi: 10.1016/j.cam.2016.12.020. Google Scholar

[33]

Z. Qin, M. Wen and C. Gu, Mean-absolute deviation portfolio selection model with fuzzy returns, Iranian Journal of Fuzzy Systems, 8 (2011), 61-75. Google Scholar

[34]

Z. Qin and S. Kar, Single-period inventory problem under uncertain environment, Journal of Applied Mathematics and Computing, 219 (2013), 9630-9638. doi: 10.1016/j.amc.2013.02.015. Google Scholar

[35]

Z. Qin, Random fuzzy mean-absolute deviation models for portfolio optimization problem with hybrid uncertainty, Applied Soft Computing, 56 (2017), 597-603. doi: 10.1016/j.asoc.2016.06.017. Google Scholar

[36]

R. Ruiz-Torrubiano and A. Suarez, Hybrid approaches and dimensionality reduction for portfolio selection with cardinality constrains, IEEE Computational Intelligence Magazine, 5 (2010), 92-107. Google Scholar

[37]

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

[38]

H. A. Le Thi and M. Moeini, Long-short portfolio optimization under cardinality constraints by difference of convex functions algorithm, Journal of Optimization Theory and Applications, 161 (2014), 199-224. doi: 10.1007/s10957-012-0197-0. Google Scholar

[39]

J. H. van Binsbergen and M. Brandt, Solving dynamic portfolio choice problems by recursing on optimized portfolio weights or on the value function?, Computational Economics, 29 (2007), 355-367. doi: 10.1007/s10614-006-9073-z. Google Scholar

[40]

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

[41]

M. Woodside-Oriakhi, C. Lucas and J. E. Beasley, Heuristic algorithms for the cardinality constrained efficient frontier, European Journal of Operational Research, 213 (2011), 538-550. doi: 10.1016/j.ejor.2011.03.030. Google Scholar

[42]

H. Wu and Y. Zeng, Equilibrium investment strategy for defined-contribution pension schemes with generalized mean-variance criterion and mortality risk, Insurance: Mathematics and Economics, 64 (2015), 396-408. doi: 10.1016/j.insmatheco.2015.07.007. 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), 341-349. doi: 10.1016/j.ejor.2012.04.023. Google Scholar

[46]

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

[47]

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

[48]

P. Zhang, Multiperiod mean absolute deviation uncertain portfolio selection, Industrial Engineering & Management Systems, 15 (2016), 63-76. doi: 10.7232/iems.2016.15.1.063. Google Scholar

[49]

Z. Zhou, H. Xiao, J. Yin, X. Zeng and L. Lin, Pre-commitment vs. time-consistent strategies for the generalized multi-period portfolio optimization with stochastic cash flows, Insurance: Mathematics and Economics, 68 (2016), 187-202. doi: 10.1016/j.insmatheco.2016.03.002. Google Scholar

[50]

S. S. Zhu, D. Li and S. Y. Wang, Risk control over bankruptcy in dynamic portfolio selection: A generalized mean-variance formulation, IEEE Transactions on Automatic Control, 49 (2004), 447-457. doi: 10.1109/TAC.2004.824474. Google Scholar

show all references

References:
[1]

K. P. Anagnostopoulos and G. Mamanis, The mean-variance cardinality constrained portfolio optimization problem: an experimental evaluation of five multiobjective evolutionary algorithms, Expert Systems with Applications, 38 (2011), 14208-14217. doi: 10.1016/j.eswa.2011.04.233. Google Scholar

[2]

D. Bertsimas and R. Shioda, Algorithms for cardinality-constrained quadratic optimization, Computational Optimization and Applications, 43 (2009), 1-12. doi: 10.1007/s10589-007-9126-9. Google Scholar

[3]

F. Cesarone, A. Scozzari and F. Tardella, A new method for mean-variance portfolio optimization with cardinality constraints, Annals of Operations Research, 205 (2013), 213-234. doi: 10.1007/s10479-012-1165-7. Google Scholar

[4]

Z. Chen, G. Li and Y. Zhao, Time-consistent investment policies in Markovian markets: A case of mean-variance analysis, Journal of Economic Dynamics and Control, 40 (2014), 293-316. doi: 10.1016/j.jedc.2014.01.011. Google Scholar

[5]

Z. Chen, J. Liu, G. Li and Z. Yan, Composite time-consistent multi-period risk measure and its application in optimal portfolio selection, Journal of Economic Dynamics and Control, 24 (2016), 515-540. doi: 10.1007/s11750-015-0407-7. Google Scholar

[6]

X. Y. Cui, D. Li, S. Y. Wang and S. S. Zhu, Better than dynamic mean-variance: Time inconsistency and free cash flow stream, Mathematical Finance, 22 (2012), 346-378. doi: 10.1111/j.1467-9965.2010.00461.x. Google Scholar

[7]

X. Y. Cui, X. Li and D. Li, Unified framework of mean-field formulations for optimal multi-period mean-variance portfolio selection, IEEE Transactions on Automatic Control, 59 (2014), 1833-1844. doi: 10.1109/TAC.2014.2311875. Google Scholar

[8]

X. Y. Cui, D. Li and X. Li, Mean variance policy for discrete time cone-constrained markets: time consistency in efficiency and the minimum-variance signed supermartingale measure, Mathematical Finance, 27 (2017), 471-504. doi: 10.1111/mafi.12093. Google Scholar

[9]

X. T. Cui, X. J. Zheng, S. S. Zhu and X. L. Sun, Convex relaxations and MIQCQP reformulations for a class of cardinality-constrained portfolio selection problems, Journal of Global Optimization, 56 (2013), 1409-1423. doi: 10.1007/s10898-012-9842-2. Google Scholar

[10]

G. F. Deng, W. T. Lin and C. C. Lo, Markowitz-based portfolio selection with cardinality constraints using improved particle swarm optimization, Expert Systems with Applications, 39 (2012), 4558-4566. Google Scholar

[11]

J. J. Gao, D. Li, X. Y. Cui and S. Y. Wang, Time cardinality constrained mean-variance dynamic portfolio selection and market timing: A stochastic control approach, Automatica, 54 (2015), 91-99. doi: 10.1016/j.automatica.2015.01.040. Google Scholar

[12]

N. Gülpinar 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

[13]

P. Gupta, M. Inuiguchi, M. Kumar Mehlawat and G. Mittal, Multiobjective credibilistic portfolio selection model with fuzzy chance-constraints, Information Sciences, 229 (2013), 1-17. doi: 10.1016/j.ins.2012.12.011. Google Scholar

[14]

B. Heidergott, G. J. Olsder and J. V. Woude, Max Plus at Work-Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications, Princeton University Press, 2006. Google Scholar

[15]

X. Huang, Fuzzy chance-constrained portfolio selection, Applied Mathematics and Computation, 177 (2006), 500-507. doi: 10.1016/j.amc.2005.11.027. Google Scholar

[16]

X. Huang and L. Qiao, A risk index model for multi-period uncertain portfolio selection, Information Sciences, 217 (2012), 108-116. doi: 10.1016/j.ins.2012.06.017. Google Scholar

[17]

M. Köksalan and C. T. Şakar, An interactive approach to stochastic programming-based portfolio optimization, Annals of Operations Research, 245 (2016), 47-66. doi: 10.1007/s10479-014-1719-y. Google Scholar

[18]

H. Konno and H. Yamazaki, Mean absolute portfolio optimization model and its application to Tokyo stock market, Management Science, 37 (1991), 519-531. doi: 10.1287/mnsc.37.5.519. Google Scholar

[19]

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

[20]

D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Mathematical Finance, 10 (2000), 387-406. doi: 10.1111/1467-9965.00100. Google Scholar

[21]

D. Li, X. Sun and J. Wang, Optimal lot solution to cardinality constrained mean-variance formulation for portfolio selection, Mathematical Finance, 16 (2006), 83-101. doi: 10.1111/j.1467-9965.2006.00262.x. Google Scholar

[22]

X. Li, Z. Qin and L. Yang, A chance-constrained portfolio selection model with risk constraints, Applied Mathematics and Computation, 217 (2010), 949-951. doi: 10.1016/j.amc.2010.06.035. Google Scholar

[23]

B. Liu, Theory and Practice of Uncertain Programming, Physics-verlag, Heidelberg, 2002.Google Scholar

[24]

B. Liu, Uncertainty Theory, 2nd edition, Springer-Verlag, Berlin, 2007. Google Scholar

[25]

B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10. Google Scholar

[26]

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

[27]

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

[28]

R. Mansini, W. Ogryczak and M. G. Speranza, Conditional value at risk and related linear programming models for portfolio optimization, Annals of Operations Research, 152 (2007), 227-256. doi: 10.1007/s10479-006-0142-4. Google Scholar

[29]

H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. Google Scholar

[30]

H. M. Markowitz, Portfolio Selection: Efficient Diversification of Investments, New York, Wiley, 1959. Google Scholar

[31]

W. Murray and H. Shek, A local relaxation method for the cardinality constrained portfolio optimization problem, Computational Optimization and Applications, 53 (2012), 681-709. doi: 10.1007/s10589-012-9471-1. Google Scholar

[32]

F. Omidi, B. Abbasi and A. Nazemi, An efficient dynamic model for solving a portfolio selection with uncertain chance constraint models, Journal of Computational and Applied Mathematics, 319 (2017), 43-55. doi: 10.1016/j.cam.2016.12.020. Google Scholar

[33]

Z. Qin, M. Wen and C. Gu, Mean-absolute deviation portfolio selection model with fuzzy returns, Iranian Journal of Fuzzy Systems, 8 (2011), 61-75. Google Scholar

[34]

Z. Qin and S. Kar, Single-period inventory problem under uncertain environment, Journal of Applied Mathematics and Computing, 219 (2013), 9630-9638. doi: 10.1016/j.amc.2013.02.015. Google Scholar

[35]

Z. Qin, Random fuzzy mean-absolute deviation models for portfolio optimization problem with hybrid uncertainty, Applied Soft Computing, 56 (2017), 597-603. doi: 10.1016/j.asoc.2016.06.017. Google Scholar

[36]

R. Ruiz-Torrubiano and A. Suarez, Hybrid approaches and dimensionality reduction for portfolio selection with cardinality constrains, IEEE Computational Intelligence Magazine, 5 (2010), 92-107. Google Scholar

[37]

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

[38]

H. A. Le Thi and M. Moeini, Long-short portfolio optimization under cardinality constraints by difference of convex functions algorithm, Journal of Optimization Theory and Applications, 161 (2014), 199-224. doi: 10.1007/s10957-012-0197-0. Google Scholar

[39]

J. H. van Binsbergen and M. Brandt, Solving dynamic portfolio choice problems by recursing on optimized portfolio weights or on the value function?, Computational Economics, 29 (2007), 355-367. doi: 10.1007/s10614-006-9073-z. Google Scholar

[40]

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

[41]

M. Woodside-Oriakhi, C. Lucas and J. E. Beasley, Heuristic algorithms for the cardinality constrained efficient frontier, European Journal of Operational Research, 213 (2011), 538-550. doi: 10.1016/j.ejor.2011.03.030. Google Scholar

[42]

H. Wu and Y. Zeng, Equilibrium investment strategy for defined-contribution pension schemes with generalized mean-variance criterion and mortality risk, Insurance: Mathematics and Economics, 64 (2015), 396-408. doi: 10.1016/j.insmatheco.2015.07.007. 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), 341-349. doi: 10.1016/j.ejor.2012.04.023. Google Scholar

[46]

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

[47]

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

[48]

P. Zhang, Multiperiod mean absolute deviation uncertain portfolio selection, Industrial Engineering & Management Systems, 15 (2016), 63-76. doi: 10.7232/iems.2016.15.1.063. Google Scholar

[49]

Z. Zhou, H. Xiao, J. Yin, X. Zeng and L. Lin, Pre-commitment vs. time-consistent strategies for the generalized multi-period portfolio optimization with stochastic cash flows, Insurance: Mathematics and Economics, 68 (2016), 187-202. doi: 10.1016/j.insmatheco.2016.03.002. Google Scholar

[50]

S. S. Zhu, D. Li and S. Y. Wang, Risk control over bankruptcy in dynamic portfolio selection: A generalized mean-variance formulation, IEEE Transactions on Automatic Control, 49 (2004), 447-457. doi: 10.1109/TAC.2004.824474. Google Scholar

Figure 1.  The multiperiod weighted digraph
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Table 1.  The optimal solution when $ K = 3, AD_t = 0.02, r_{0t} = 0.1, \delta = 95\%$
The optimal investment proportions
1Asset3Asset 13Asset 17 $x_{f1}$
0.20.1715810.20.428419
2Asset15Asset 17Asset 29 $x_{f2}$
0.20.20.20.4
3Asset3Asset 15Asset 24 $x_{f3}$
0.1494060.20.20.450594
4Asset13Asset 20Asset 25 $x_{f4}$
0.1693880.20.20.430612
5Asset13Asset 17Asset 20 $x_{f5}$
0.1709020.20.20.429098
Table Options

Download as excel

The optimal investment proportions
1Asset3Asset 13Asset 17 $x_{f1}$
0.20.1715810.20.428419
2Asset15Asset 17Asset 29 $x_{f2}$
0.20.20.20.4
3Asset3Asset 15Asset 24 $x_{f3}$
0.1494060.20.20.450594
4Asset13Asset 20Asset 25 $x_{f4}$
0.1693880.20.20.430612
5Asset13Asset 17Asset 20 $x_{f5}$
0.1709020.20.20.429098
Table 2.  The optimal solution when $K = 6, AD_t = 0.02, r_{0t} = 0.1, \delta = 95\%$
The optimal investment proportions
1Asset3Asset 13Asset 17Asset 22Asset 25 $x_{f1}$
0.20.0049730.20.20.20.195027
2Asset15Asset 17Asset 24Asset 30 $x_{f2}$
0.20.20.20.070386270.329614
3Asset3Asset 13Asset 15Asset 24 $x_{f3}$
0.20.0607810.20.20.339219
3Asset8Asset 15Asset 20Asset 25 $x_{f4}$
0.0553070.20.20.20.344693
4Asset15Asset 17Asset 20Asset 25Asset 30 $x_{f5}$
0.045970.20.20.20.20.15403
Table Options

Download as excel

The optimal investment proportions
1Asset3Asset 13Asset 17Asset 22Asset 25 $x_{f1}$
0.20.0049730.20.20.20.195027
2Asset15Asset 17Asset 24Asset 30 $x_{f2}$
0.20.20.20.070386270.329614
3Asset3Asset 13Asset 15Asset 24 $x_{f3}$
0.20.0607810.20.20.339219
3Asset8Asset 15Asset 20Asset 25 $x_{f4}$
0.0553070.20.20.20.344693
4Asset15Asset 17Asset 20Asset 25Asset 30 $x_{f5}$
0.045970.20.20.20.20.15403
Table 3.  The optimal solution when $K = 6, AD_t = 0.03, r_{0t} = 0.1, \delta = 95\%$
The optimal investment proportions
1Asset3Asset 13Asset 17Asset 22Asset 25 $x_{f1}$
0.20.1825930.20.20.20.017407
2Asset5Asset15Asset 17Asset 24Asset 29 $x_{f2}$
0.20.20.20.20.1216120.078388
3Asset3Asset 13Asset 15Asset 17Asset 24 $x_{f3}$
0.20.20.20.045801530.20.154198
4Asset6Asset 8Asset 15Asset 20Asset 25 $x_{f4}$
0.1967350.20.20.20.20.003265
5Asset15Asset 17Asset 20Asset 22Asset 25Asset 30 $x_{f5}$
0.20.20.20.16864110.20.2-0.16864
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The optimal investment proportions
1Asset3Asset 13Asset 17Asset 22Asset 25 $x_{f1}$
0.20.1825930.20.20.20.017407
2Asset5Asset15Asset 17Asset 24Asset 29 $x_{f2}$
0.20.20.20.20.1216120.078388
3Asset3Asset 13Asset 15Asset 17Asset 24 $x_{f3}$
0.20.20.20.045801530.20.154198
4Asset6Asset 8Asset 15Asset 20Asset 25 $x_{f4}$
0.1967350.20.20.20.20.003265
5Asset15Asset 17Asset 20Asset 22Asset 25Asset 30 $x_{f5}$
0.20.20.20.16864110.20.2-0.16864
Table 4.  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$23456789
$\delta=95\%, W_6$1.5057251.706081.9237462.1588842.4084062.6535632.6593342.659334
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$K$23456789
$\delta=95\%, W_6$1.5057251.706081.9237462.1588842.4084062.6535632.6593342.659334
Table 5.  The optimal solution when $AD_t = 0.07, K = 3, r_{0t} = 0.18, \delta = 95\%$
The optimal investment proportions
1Asset12Asset 13Asset 28 $x_{f1}$
0.220.20.4
2Asset1Asset 12Asset 13 $x_{f2}$
0.20.20.20.4
3Asset12Asset 13Asset 17 $x_{f3}$
0.20.20.20.4
4Asset12Asset 13Asset 18 $x_{f4}$
0.20.20.20.4
5Asset12Asset 13Asset 18 $x_{f5}$
0.20.20.20.4
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The optimal investment proportions
1Asset12Asset 13Asset 28 $x_{f1}$
0.220.20.4
2Asset1Asset 12Asset 13 $x_{f2}$
0.20.20.20.4
3Asset12Asset 13Asset 17 $x_{f3}$
0.20.20.20.4
4Asset12Asset 13Asset 18 $x_{f4}$
0.20.20.20.4
5Asset12Asset 13Asset 18 $x_{f5}$
0.20.20.20.4
Table 6.  The optimal solution when $AD_t = 0.07, K = 3, r_{0t} = 0.18, \delta = 99\%$
The optimal investment proportions
1Asset13Asset 16Asset 28 $x_{f1}$
0.220.20.4
2Asset12Asset 13Asset 16 $x_{f2}$
0.20.20.20.4
3Asset12Asset 13Asset 17 $x_{f3}$
0.20.20.20.4
4Asset12Asset 13Asset 18$x_{f4}$
0.20.20.20.4
5Asset12Asset 13Asset 18 $x_{f5}$
0.20.20.20.4
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The optimal investment proportions
1Asset13Asset 16Asset 28 $x_{f1}$
0.220.20.4
2Asset12Asset 13Asset 16 $x_{f2}$
0.20.20.20.4
3Asset12Asset 13Asset 17 $x_{f3}$
0.20.20.20.4
4Asset12Asset 13Asset 18$x_{f4}$
0.20.20.20.4
5Asset12Asset 13Asset 18 $x_{f5}$
0.20.20.20.4
Table 7.  The uncertain return rates on assets of five periods investment
Asset 1Asset 2Asset 3
10.1430.10490.11560.0750.06570.16640.10830.08320.06
20.14490.08810.11360.08130.07080.160.10850.06810.0603
30.14580.080.11270.08570.06660.15560.11390.07250.0548
40.15160.0620.1070.0930.05790.14830.11520.0560.054
50.15320.06090.10540.10530.06620.13590.11720.0570.0516
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Asset 1Asset 2Asset 3
10.1430.10490.11560.0750.06570.16640.10830.08320.06
20.14490.08810.11360.08130.07080.160.10850.06810.0603
30.14580.080.11270.08570.06660.15560.11390.07250.0548
40.15160.0620.1070.0930.05790.14830.11520.0560.054
50.15320.06090.10540.10530.06620.13590.11720.0570.0516
Table 8.  The uncertain return rates on assets of five periods investment
Asset 4Asset 5Asset 6
10.11720.07310.08130.08010.07910.06160.10640.16350.0616
20.12030.07430.07820.08470.07720.05710.10730.16340.0608
30.12550.07490.0730.090.05030.05170.10830.10930.0598
40.12740.07330.0710.09060.05070.05120.10910.07630.059
50.12890.06330.070.09260.04950.04920.11290.07270.0551
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Asset 4Asset 5Asset 6
10.11720.07310.08130.08010.07910.06160.10640.16350.0616
20.12030.07430.07820.08470.07720.05710.10730.16340.0608
30.12550.07490.0730.090.05030.05170.10830.10930.0598
40.12740.07330.0710.09060.05070.05120.10910.07630.059
50.12890.06330.070.09260.04950.04920.11290.07270.0551
Table 9.  The uncertain return rates on assets of five periods investment
Asset 7Asset 8Asset 9
10.07980.05620.16940.12380.08150.10230.06390.15220.0951
20.09070.06430.1750.12590.0760.10030.0790.06730.0866
30.09920.05550.150.12770.07650.09850.08180.06060.0839
40.10290.05510.14620.13830.05380.08780.08610.06450.08
50.10690.05340.14230.14570.06120.08050.08840.0650.0773
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Asset 7Asset 8Asset 9
10.07980.05620.16940.12380.08150.10230.06390.15220.0951
20.09070.06430.1750.12590.0760.10030.0790.06730.0866
30.09920.05550.150.12770.07650.09850.08180.06060.0839
40.10290.05510.14620.13830.05380.08780.08610.06450.08
50.10690.05340.14230.14570.06120.08050.08840.0650.0773
Table 10.  The uncertain return rates on assets of five periods investment
Asset 10Asset 11Asset 12
10.03770.03250.04140.05750.04030.12820.12430.1170.184
20.0410.02920.03790.05920.04030.12640.13030.10070.1781
30.04690.03180.03210.06690.03740.11880.1380.08270.1704
40.0480.03140.03090.07240.040.11330.14910.08430.1593
50.04920.03180.02980.07410.04530.11160.1540.07520.1544
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Asset 10Asset 11Asset 12
10.03770.03250.04140.05750.04030.12820.12430.1170.184
20.0410.02920.03790.05920.04030.12640.13030.10070.1781
30.04690.03180.03210.06690.03740.11880.1380.08270.1704
40.0480.03140.03090.07240.040.11330.14910.08430.1593
50.04920.03180.02980.07410.04530.11160.1540.07520.1544
Table 11.  The uncertain return rates on assets of five periods investment
Asset 13Asset 14Asset 15
10.20490.12440.13310.02540.10.07430.08930.20790.1463
20.21020.11820.12770.06670.06040.04430.15180.10150.0859
30.21940.12360.11860.070.05630.04110.15380.10330.084
40.22250.12480.11540.07160.050.03950.15650.05340.0812
50.22380.10290.11420.07310.03990.03790.160.05530.0778
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Asset 13Asset 14Asset 15
10.20490.12440.13310.02540.10.07430.08930.20790.1463
20.21020.11820.12770.06670.06040.04430.15180.10150.0859
30.21940.12360.11860.070.05630.04110.15380.10330.084
40.22250.12480.11540.07160.050.03950.15650.05340.0812
50.22380.10290.11420.07310.03990.03790.160.05530.0778
Table 12.  The uncertain return rates on assets of five periods investment
Asset 16Asset 17Asset 18
10.06150.06220.48190.16650.11880.03750.06750.11570.0586
20.06250.27950.23060.1550.09840.07720.11830.11370.4232
30.06560.05140.22760.15530.08930.07690.13490.11650.4066
40.07470.0460.21850.15750.08440.07470.14670.12250.3947
50.08350.05060.20960.15790.05350.07440.16640.11220.375
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Asset 16Asset 17Asset 18
10.06150.06220.48190.16650.11880.03750.06750.11570.0586
20.06250.27950.23060.1550.09840.07720.11830.11370.4232
30.06560.05140.22760.15530.08930.07690.13490.11650.4066
40.07470.0460.21850.15750.08440.07470.14670.12250.3947
50.08350.05060.20960.15790.05350.07440.16640.11220.375
Table 13.  The uncertain return rates on assets of five periods investment
Asset 19Asset 20Asset 21
10.250.07360.08960.08250.15590.08530.05360.08170.0403
20.09160.07160.06340.12170.06330.06190.07040.05440.122
30.09280.07080.06220.12180.0620.06180.08380.06660.1087
40.0940.050.0610.12430.05160.05930.0880.06810.1045
50.09520.04720.060.12690.02670.05680.09170.07030.1007
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Asset 19Asset 20Asset 21
10.250.07360.08960.08250.15590.08530.05360.08170.0403
20.09160.07160.06340.12170.06330.06190.07040.05440.122
30.09280.07080.06220.12180.0620.06180.08380.06660.1087
40.0940.050.0610.12430.05160.05930.0880.06810.1045
50.09520.04720.060.12690.02670.05680.09170.07030.1007
Table 14.  The uncertain return rates on assets of five periods investment
Asset 22Asset 23Asset 24
10.10830.11970.06840.04130.01490.07490.1430.04580.0141
20.1170.06180.07390.04540.11370.2040.09470.06370.0393
30.120.06230.07080.05310.12090.19630.09840.06280.0355
40.12350.06560.06740.05740.0930.1920.10050.05480.0334
50.12540.04880.06540.07180.07160.17770.10210.04930.0319
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Asset 22Asset 23Asset 24
10.10830.11970.06840.04130.01490.07490.1430.04580.0141
20.1170.06180.07390.04540.11370.2040.09470.06370.0393
30.120.06230.07080.05310.12090.19630.09840.06280.0355
40.12350.06560.06740.05740.0930.1920.10050.05480.0334
50.12540.04880.06540.07180.07160.17770.10210.04930.0319
Table 15.  The uncertain return rates on assets of five periods investment
Asset 25Asset 26Asset 27
10.05890.14320.10240.07830.17120.10960.0690.00470.1634
20.10210.06520.05910.12760.09060.12630.04380.16230.1828
30.10370.05670.05740.13290.09490.1210.05060.15260.176
40.10440.03140.05670.14320.08120.11050.05620.12320.1694
50.1090.02430.05210.14450.07770.10930.06190.05110.1647
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Asset 25Asset 26Asset 27
10.05890.14320.10240.07830.17120.10960.0690.00470.1634
20.10210.06520.05910.12760.09060.12630.04380.16230.1828
30.10370.05670.05740.13290.09490.1210.05060.15260.176
40.10440.03140.05670.14320.08120.11050.05620.12320.1694
50.1090.02430.05210.14450.07770.10930.06190.05110.1647
Table 16.  The uncertain return rates on assets of five periods investment
Asset 28Asset 29Asset 30
10.15510.11280.04980.09940.12330.06770.06740.13550.0854
20.13820.07890.09690.11230.06410.04980.10370.06360.0438
30.13950.06790.09560.11340.06480.04880.10480.06450.0426
40.14260.03790.09240.11570.04160.04640.1060.05740.0414
50.1470.03640.0880.11750.03120.04460.10610.0350.0413
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Asset 28Asset 29Asset 30
10.15510.11280.04980.09940.12330.06770.06740.13550.0854
20.13820.07890.09690.11230.06410.04980.10370.06360.0438
30.13950.06790.09560.11340.06480.04880.10480.06450.0426
40.14260.03790.09240.11570.04160.04640.1060.05740.0414
50.1470.03640.0880.11750.03120.04460.10610.0350.0413
Table 17.  The uncertain absolute deviation of assets of five periods investment
Asset 1Asset 2Asset 3Asset 4Asset 5Asset 6Asset 7Asset 8
10.05510.05990.02320.03860.01790.0280.05880.0461
20.05060.05930.02680.03810.01830.02820.0620.0443
30.05040.05710.02610.03310.02550.02630.05320.0439
40.04280.05330.03330.0340.02550.02450.05210.0358
50.04220.05160.0330.03950.02590.02580.05070.0356
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Asset 1Asset 2Asset 3Asset 4Asset 5Asset 6Asset 7Asset 8
10.05510.05990.02320.03860.01790.0280.05880.0461
20.05060.05930.02680.03810.01830.02820.0620.0443
30.05040.05710.02610.03310.02550.02630.05320.0439
40.04280.05330.03330.0340.02550.02450.05210.0358
50.04220.05160.0330.03950.02590.02580.05070.0356
Table 18.  The uncertain absolute deviation of assets of five periods investment
Asset 9Asset 10Asset 11Asset 12Asset 13Asset 14Asset 15Asset 16
10.03740.01850.0440.0760.05630.02280.05730.0894
20.03860.01680.04350.07080.06150.01430.0360.0765
30.03630.0160.04080.06470.05890.0150.03570.074
40.03620.01560.03980.0620.05880.01620.03990.0704
50.03560.01230.04050.05870.05430.01950.03350.0688
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Asset 9Asset 10Asset 11Asset 12Asset 13Asset 14Asset 15Asset 16
10.03740.01850.0440.0760.05630.02280.05730.0894
20.03860.01680.04350.07080.06150.01430.0360.0765
30.03630.0160.04080.06470.05890.0150.03570.074
40.03620.01560.03980.0620.05880.01620.03990.0704
50.03560.01230.04050.05870.05430.01950.03350.0688
Table 19.  The uncertain absolute deviation of assets of five periods investment
Asset 17Asset 18Asset 19Asset 20Asset21Asset 22Asset 23Asset 24
10.02850.06660.04090.03560.01190.02290.0240.023
20.03610.05920.03380.03420.04530.03390.03190.0197
30.03930.06080.02170.03490.04430.03330.03110.0203
40.04170.08010.02780.02780.04350.03330.03460.0229
50.03220.07350.02690.02140.0430.02870.03730.0255
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Asset 17Asset 18Asset 19Asset 20Asset21Asset 22Asset 23Asset 24
10.02850.06660.04090.03560.01190.02290.0240.023
20.03610.05920.03380.03420.04530.03390.03190.0197
30.03930.06080.02170.03490.04430.03330.03110.0203
40.04170.08010.02780.02780.04350.03330.03460.0229
50.03220.07350.02690.02140.0430.02870.03730.0255
Table 20.  The uncertain absolute deviation of assets of five periods investment
Asset 25Asset 26Asset 27Asset 28Asset 29Asset 30
10.0240.04560.05680.04340.02790.0379
20.02540.05450.06380.04410.02730.0233
30.02850.05420.05990.04110.02720.0271
40.02240.04820.05810.04260.02880.0257
50.01960.0470.05640.04110.02920.0191
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Asset 25Asset 26Asset 27Asset 28Asset 29Asset 30
10.0240.04560.05680.04340.02790.0379
20.02540.05450.06380.04410.02730.0233
30.02850.05420.05990.04110.02720.0271
40.02240.04820.05810.04260.02880.0257
50.01960.0470.05640.04110.02920.0191
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