# American Institute of Mathematical Sciences

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

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

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

Received  August 2017 Revised  December 2018 Published  April 2018

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

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

Citation: Peng Zhang. Chance-constrained multiperiod mean absolute deviation uncertain portfolio selection. Journal of Industrial & Management Optimization, 2019, 15 (2) : 537-564. doi: 10.3934/jimo.2018056
##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [15] X. Huang, Fuzzy chance-constrained portfolio selection, Applied Mathematics and Computation, 177 (2006), 500-507. doi: 10.1016/j.amc.2005.11.027. [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. [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. [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. [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. [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. [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. [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. [23] B. Liu, Theory and Practice of Uncertain Programming, Physics-verlag, Heidelberg, 2002. [24] B. Liu, Uncertainty Theory, 2nd edition, Springer-Verlag, Berlin, 2007. [25] B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10. [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. [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. [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. [29] H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. [30] H. M. Markowitz, Portfolio Selection: Efficient Diversification of Investments, New York, Wiley, 1959. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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), 341-349. doi: 10.1016/j.ejor.2012.04.023. [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. [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. [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. [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. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [15] X. Huang, Fuzzy chance-constrained portfolio selection, Applied Mathematics and Computation, 177 (2006), 500-507. doi: 10.1016/j.amc.2005.11.027. [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. [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. [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. [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. [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. [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. [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. [23] B. Liu, Theory and Practice of Uncertain Programming, Physics-verlag, Heidelberg, 2002. [24] B. Liu, Uncertainty Theory, 2nd edition, Springer-Verlag, Berlin, 2007. [25] B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10. [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. [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. [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. [29] H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. [30] H. M. Markowitz, Portfolio Selection: Efficient Diversification of Investments, New York, Wiley, 1959. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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), 341-349. doi: 10.1016/j.ejor.2012.04.023. [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. [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. [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. [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. [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.
The multiperiod weighted digraph
The optimal solution when $K = 3, AD_t = 0.02, r_{0t} = 0.1, \delta = 95\%$
 The optimal investment proportions 1 Asset3 Asset 13 Asset 17 $x_{f1}$ 0.2 0.171581 0.2 0.428419 2 Asset15 Asset 17 Asset 29 $x_{f2}$ 0.2 0.2 0.2 0.4 3 Asset3 Asset 15 Asset 24 $x_{f3}$ 0.149406 0.2 0.2 0.450594 4 Asset13 Asset 20 Asset 25 $x_{f4}$ 0.169388 0.2 0.2 0.430612 5 Asset13 Asset 17 Asset 20 $x_{f5}$ 0.170902 0.2 0.2 0.429098
 The optimal investment proportions 1 Asset3 Asset 13 Asset 17 $x_{f1}$ 0.2 0.171581 0.2 0.428419 2 Asset15 Asset 17 Asset 29 $x_{f2}$ 0.2 0.2 0.2 0.4 3 Asset3 Asset 15 Asset 24 $x_{f3}$ 0.149406 0.2 0.2 0.450594 4 Asset13 Asset 20 Asset 25 $x_{f4}$ 0.169388 0.2 0.2 0.430612 5 Asset13 Asset 17 Asset 20 $x_{f5}$ 0.170902 0.2 0.2 0.429098
The optimal solution when $K = 6, AD_t = 0.02, r_{0t} = 0.1, \delta = 95\%$
 The optimal investment proportions 1 Asset3 Asset 13 Asset 17 Asset 22 Asset 25 $x_{f1}$ 0.2 0.004973 0.2 0.2 0.2 0.195027 2 Asset15 Asset 17 Asset 24 Asset 30 $x_{f2}$ 0.2 0.2 0.2 0.07038627 0.329614 3 Asset3 Asset 13 Asset 15 Asset 24 $x_{f3}$ 0.2 0.060781 0.2 0.2 0.339219 3 Asset8 Asset 15 Asset 20 Asset 25 $x_{f4}$ 0.055307 0.2 0.2 0.2 0.344693 4 Asset15 Asset 17 Asset 20 Asset 25 Asset 30 $x_{f5}$ 0.04597 0.2 0.2 0.2 0.2 0.15403
 The optimal investment proportions 1 Asset3 Asset 13 Asset 17 Asset 22 Asset 25 $x_{f1}$ 0.2 0.004973 0.2 0.2 0.2 0.195027 2 Asset15 Asset 17 Asset 24 Asset 30 $x_{f2}$ 0.2 0.2 0.2 0.07038627 0.329614 3 Asset3 Asset 13 Asset 15 Asset 24 $x_{f3}$ 0.2 0.060781 0.2 0.2 0.339219 3 Asset8 Asset 15 Asset 20 Asset 25 $x_{f4}$ 0.055307 0.2 0.2 0.2 0.344693 4 Asset15 Asset 17 Asset 20 Asset 25 Asset 30 $x_{f5}$ 0.04597 0.2 0.2 0.2 0.2 0.15403
The optimal solution when $K = 6, AD_t = 0.03, r_{0t} = 0.1, \delta = 95\%$
 The optimal investment proportions 1 Asset3 Asset 13 Asset 17 Asset 22 Asset 25 $x_{f1}$ 0.2 0.182593 0.2 0.2 0.2 0.017407 2 Asset5 Asset15 Asset 17 Asset 24 Asset 29 $x_{f2}$ 0.2 0.2 0.2 0.2 0.121612 0.078388 3 Asset3 Asset 13 Asset 15 Asset 17 Asset 24 $x_{f3}$ 0.2 0.2 0.2 0.04580153 0.2 0.154198 4 Asset6 Asset 8 Asset 15 Asset 20 Asset 25 $x_{f4}$ 0.196735 0.2 0.2 0.2 0.2 0.003265 5 Asset15 Asset 17 Asset 20 Asset 22 Asset 25 Asset 30 $x_{f5}$ 0.2 0.2 0.2 0.1686411 0.2 0.2 -0.16864
 The optimal investment proportions 1 Asset3 Asset 13 Asset 17 Asset 22 Asset 25 $x_{f1}$ 0.2 0.182593 0.2 0.2 0.2 0.017407 2 Asset5 Asset15 Asset 17 Asset 24 Asset 29 $x_{f2}$ 0.2 0.2 0.2 0.2 0.121612 0.078388 3 Asset3 Asset 13 Asset 15 Asset 17 Asset 24 $x_{f3}$ 0.2 0.2 0.2 0.04580153 0.2 0.154198 4 Asset6 Asset 8 Asset 15 Asset 20 Asset 25 $x_{f4}$ 0.196735 0.2 0.2 0.2 0.2 0.003265 5 Asset15 Asset 17 Asset 20 Asset 22 Asset 25 Asset 30 $x_{f5}$ 0.2 0.2 0.2 0.1686411 0.2 0.2 -0.16864
the optimal terminal wealth and risk of the portfolio when $AD_t = 0.07, r_{0t} = 0.15, \delta = 95\%, K = 2, \ldots, 9$
 $K$ 2 3 4 5 6 7 8 9 $\delta=95\%, W_6$ 1.50572 1.70608 1.92375 2.15888 2.40841 2.65356 2.65933 2.65933
 $K$ 2 3 4 5 6 7 8 9 $\delta=95\%, W_6$ 1.50572 1.70608 1.92375 2.15888 2.40841 2.65356 2.65933 2.65933
The optimal solution when $AD_t = 0.07, K = 3, r_{0t} = 0.18, \delta = 95\%$
 The optimal investment proportions 1 Asset12 Asset 13 Asset 28 $x_{f1}$ 0.2 2 0.2 0.4 2 Asset1 Asset 12 Asset 13 $x_{f2}$ 0.2 0.2 0.2 0.4 3 Asset12 Asset 13 Asset 17 $x_{f3}$ 0.2 0.2 0.2 0.4 4 Asset12 Asset 13 Asset 18 $x_{f4}$ 0.2 0.2 0.2 0.4 5 Asset12 Asset 13 Asset 18 $x_{f5}$ 0.2 0.2 0.2 0.4
 The optimal investment proportions 1 Asset12 Asset 13 Asset 28 $x_{f1}$ 0.2 2 0.2 0.4 2 Asset1 Asset 12 Asset 13 $x_{f2}$ 0.2 0.2 0.2 0.4 3 Asset12 Asset 13 Asset 17 $x_{f3}$ 0.2 0.2 0.2 0.4 4 Asset12 Asset 13 Asset 18 $x_{f4}$ 0.2 0.2 0.2 0.4 5 Asset12 Asset 13 Asset 18 $x_{f5}$ 0.2 0.2 0.2 0.4
The optimal solution when $AD_t = 0.07, K = 3, r_{0t} = 0.18, \delta = 99\%$
 The optimal investment proportions 1 Asset13 Asset 16 Asset 28 $x_{f1}$ 0.2 2 0.2 0.4 2 Asset12 Asset 13 Asset 16 $x_{f2}$ 0.2 0.2 0.2 0.4 3 Asset12 Asset 13 Asset 17 $x_{f3}$ 0.2 0.2 0.2 0.4 4 Asset12 Asset 13 Asset 18 $x_{f4}$ 0.2 0.2 0.2 0.4 5 Asset12 Asset 13 Asset 18 $x_{f5}$ 0.2 0.2 0.2 0.4
 The optimal investment proportions 1 Asset13 Asset 16 Asset 28 $x_{f1}$ 0.2 2 0.2 0.4 2 Asset12 Asset 13 Asset 16 $x_{f2}$ 0.2 0.2 0.2 0.4 3 Asset12 Asset 13 Asset 17 $x_{f3}$ 0.2 0.2 0.2 0.4 4 Asset12 Asset 13 Asset 18 $x_{f4}$ 0.2 0.2 0.2 0.4 5 Asset12 Asset 13 Asset 18 $x_{f5}$ 0.2 0.2 0.2 0.4
The uncertain return rates on assets of five periods investment
 Asset 1 Asset 2 Asset 3 1 0.143 0.1049 0.1156 0.075 0.0657 0.1664 0.1083 0.0832 0.06 2 0.1449 0.0881 0.1136 0.0813 0.0708 0.16 0.1085 0.0681 0.0603 3 0.1458 0.08 0.1127 0.0857 0.0666 0.1556 0.1139 0.0725 0.0548 4 0.1516 0.062 0.107 0.093 0.0579 0.1483 0.1152 0.056 0.054 5 0.1532 0.0609 0.1054 0.1053 0.0662 0.1359 0.1172 0.057 0.0516
 Asset 1 Asset 2 Asset 3 1 0.143 0.1049 0.1156 0.075 0.0657 0.1664 0.1083 0.0832 0.06 2 0.1449 0.0881 0.1136 0.0813 0.0708 0.16 0.1085 0.0681 0.0603 3 0.1458 0.08 0.1127 0.0857 0.0666 0.1556 0.1139 0.0725 0.0548 4 0.1516 0.062 0.107 0.093 0.0579 0.1483 0.1152 0.056 0.054 5 0.1532 0.0609 0.1054 0.1053 0.0662 0.1359 0.1172 0.057 0.0516
The uncertain return rates on assets of five periods investment
 Asset 4 Asset 5 Asset 6 1 0.1172 0.0731 0.0813 0.0801 0.0791 0.0616 0.1064 0.1635 0.0616 2 0.1203 0.0743 0.0782 0.0847 0.0772 0.0571 0.1073 0.1634 0.0608 3 0.1255 0.0749 0.073 0.09 0.0503 0.0517 0.1083 0.1093 0.0598 4 0.1274 0.0733 0.071 0.0906 0.0507 0.0512 0.1091 0.0763 0.059 5 0.1289 0.0633 0.07 0.0926 0.0495 0.0492 0.1129 0.0727 0.0551
 Asset 4 Asset 5 Asset 6 1 0.1172 0.0731 0.0813 0.0801 0.0791 0.0616 0.1064 0.1635 0.0616 2 0.1203 0.0743 0.0782 0.0847 0.0772 0.0571 0.1073 0.1634 0.0608 3 0.1255 0.0749 0.073 0.09 0.0503 0.0517 0.1083 0.1093 0.0598 4 0.1274 0.0733 0.071 0.0906 0.0507 0.0512 0.1091 0.0763 0.059 5 0.1289 0.0633 0.07 0.0926 0.0495 0.0492 0.1129 0.0727 0.0551
The uncertain return rates on assets of five periods investment
 Asset 7 Asset 8 Asset 9 1 0.0798 0.0562 0.1694 0.1238 0.0815 0.1023 0.0639 0.1522 0.0951 2 0.0907 0.0643 0.175 0.1259 0.076 0.1003 0.079 0.0673 0.0866 3 0.0992 0.0555 0.15 0.1277 0.0765 0.0985 0.0818 0.0606 0.0839 4 0.1029 0.0551 0.1462 0.1383 0.0538 0.0878 0.0861 0.0645 0.08 5 0.1069 0.0534 0.1423 0.1457 0.0612 0.0805 0.0884 0.065 0.0773
 Asset 7 Asset 8 Asset 9 1 0.0798 0.0562 0.1694 0.1238 0.0815 0.1023 0.0639 0.1522 0.0951 2 0.0907 0.0643 0.175 0.1259 0.076 0.1003 0.079 0.0673 0.0866 3 0.0992 0.0555 0.15 0.1277 0.0765 0.0985 0.0818 0.0606 0.0839 4 0.1029 0.0551 0.1462 0.1383 0.0538 0.0878 0.0861 0.0645 0.08 5 0.1069 0.0534 0.1423 0.1457 0.0612 0.0805 0.0884 0.065 0.0773
The uncertain return rates on assets of five periods investment
 Asset 10 Asset 11 Asset 12 1 0.0377 0.0325 0.0414 0.0575 0.0403 0.1282 0.1243 0.117 0.184 2 0.041 0.0292 0.0379 0.0592 0.0403 0.1264 0.1303 0.1007 0.1781 3 0.0469 0.0318 0.0321 0.0669 0.0374 0.1188 0.138 0.0827 0.1704 4 0.048 0.0314 0.0309 0.0724 0.04 0.1133 0.1491 0.0843 0.1593 5 0.0492 0.0318 0.0298 0.0741 0.0453 0.1116 0.154 0.0752 0.1544
 Asset 10 Asset 11 Asset 12 1 0.0377 0.0325 0.0414 0.0575 0.0403 0.1282 0.1243 0.117 0.184 2 0.041 0.0292 0.0379 0.0592 0.0403 0.1264 0.1303 0.1007 0.1781 3 0.0469 0.0318 0.0321 0.0669 0.0374 0.1188 0.138 0.0827 0.1704 4 0.048 0.0314 0.0309 0.0724 0.04 0.1133 0.1491 0.0843 0.1593 5 0.0492 0.0318 0.0298 0.0741 0.0453 0.1116 0.154 0.0752 0.1544
The uncertain return rates on assets of five periods investment
 Asset 13 Asset 14 Asset 15 1 0.2049 0.1244 0.1331 0.0254 0.1 0.0743 0.0893 0.2079 0.1463 2 0.2102 0.1182 0.1277 0.0667 0.0604 0.0443 0.1518 0.1015 0.0859 3 0.2194 0.1236 0.1186 0.07 0.0563 0.0411 0.1538 0.1033 0.084 4 0.2225 0.1248 0.1154 0.0716 0.05 0.0395 0.1565 0.0534 0.0812 5 0.2238 0.1029 0.1142 0.0731 0.0399 0.0379 0.16 0.0553 0.0778
 Asset 13 Asset 14 Asset 15 1 0.2049 0.1244 0.1331 0.0254 0.1 0.0743 0.0893 0.2079 0.1463 2 0.2102 0.1182 0.1277 0.0667 0.0604 0.0443 0.1518 0.1015 0.0859 3 0.2194 0.1236 0.1186 0.07 0.0563 0.0411 0.1538 0.1033 0.084 4 0.2225 0.1248 0.1154 0.0716 0.05 0.0395 0.1565 0.0534 0.0812 5 0.2238 0.1029 0.1142 0.0731 0.0399 0.0379 0.16 0.0553 0.0778
The uncertain return rates on assets of five periods investment
 Asset 16 Asset 17 Asset 18 1 0.0615 0.0622 0.4819 0.1665 0.1188 0.0375 0.0675 0.1157 0.0586 2 0.0625 0.2795 0.2306 0.155 0.0984 0.0772 0.1183 0.1137 0.4232 3 0.0656 0.0514 0.2276 0.1553 0.0893 0.0769 0.1349 0.1165 0.4066 4 0.0747 0.046 0.2185 0.1575 0.0844 0.0747 0.1467 0.1225 0.3947 5 0.0835 0.0506 0.2096 0.1579 0.0535 0.0744 0.1664 0.1122 0.375
 Asset 16 Asset 17 Asset 18 1 0.0615 0.0622 0.4819 0.1665 0.1188 0.0375 0.0675 0.1157 0.0586 2 0.0625 0.2795 0.2306 0.155 0.0984 0.0772 0.1183 0.1137 0.4232 3 0.0656 0.0514 0.2276 0.1553 0.0893 0.0769 0.1349 0.1165 0.4066 4 0.0747 0.046 0.2185 0.1575 0.0844 0.0747 0.1467 0.1225 0.3947 5 0.0835 0.0506 0.2096 0.1579 0.0535 0.0744 0.1664 0.1122 0.375
The uncertain return rates on assets of five periods investment
 Asset 19 Asset 20 Asset 21 1 0.25 0.0736 0.0896 0.0825 0.1559 0.0853 0.0536 0.0817 0.0403 2 0.0916 0.0716 0.0634 0.1217 0.0633 0.0619 0.0704 0.0544 0.122 3 0.0928 0.0708 0.0622 0.1218 0.062 0.0618 0.0838 0.0666 0.1087 4 0.094 0.05 0.061 0.1243 0.0516 0.0593 0.088 0.0681 0.1045 5 0.0952 0.0472 0.06 0.1269 0.0267 0.0568 0.0917 0.0703 0.1007
 Asset 19 Asset 20 Asset 21 1 0.25 0.0736 0.0896 0.0825 0.1559 0.0853 0.0536 0.0817 0.0403 2 0.0916 0.0716 0.0634 0.1217 0.0633 0.0619 0.0704 0.0544 0.122 3 0.0928 0.0708 0.0622 0.1218 0.062 0.0618 0.0838 0.0666 0.1087 4 0.094 0.05 0.061 0.1243 0.0516 0.0593 0.088 0.0681 0.1045 5 0.0952 0.0472 0.06 0.1269 0.0267 0.0568 0.0917 0.0703 0.1007
The uncertain return rates on assets of five periods investment
 Asset 22 Asset 23 Asset 24 1 0.1083 0.1197 0.0684 0.0413 0.0149 0.0749 0.143 0.0458 0.0141 2 0.117 0.0618 0.0739 0.0454 0.1137 0.204 0.0947 0.0637 0.0393 3 0.12 0.0623 0.0708 0.0531 0.1209 0.1963 0.0984 0.0628 0.0355 4 0.1235 0.0656 0.0674 0.0574 0.093 0.192 0.1005 0.0548 0.0334 5 0.1254 0.0488 0.0654 0.0718 0.0716 0.1777 0.1021 0.0493 0.0319
 Asset 22 Asset 23 Asset 24 1 0.1083 0.1197 0.0684 0.0413 0.0149 0.0749 0.143 0.0458 0.0141 2 0.117 0.0618 0.0739 0.0454 0.1137 0.204 0.0947 0.0637 0.0393 3 0.12 0.0623 0.0708 0.0531 0.1209 0.1963 0.0984 0.0628 0.0355 4 0.1235 0.0656 0.0674 0.0574 0.093 0.192 0.1005 0.0548 0.0334 5 0.1254 0.0488 0.0654 0.0718 0.0716 0.1777 0.1021 0.0493 0.0319
The uncertain return rates on assets of five periods investment
 Asset 25 Asset 26 Asset 27 1 0.0589 0.1432 0.1024 0.0783 0.1712 0.1096 0.069 0.0047 0.1634 2 0.1021 0.0652 0.0591 0.1276 0.0906 0.1263 0.0438 0.1623 0.1828 3 0.1037 0.0567 0.0574 0.1329 0.0949 0.121 0.0506 0.1526 0.176 4 0.1044 0.0314 0.0567 0.1432 0.0812 0.1105 0.0562 0.1232 0.1694 5 0.109 0.0243 0.0521 0.1445 0.0777 0.1093 0.0619 0.0511 0.1647
 Asset 25 Asset 26 Asset 27 1 0.0589 0.1432 0.1024 0.0783 0.1712 0.1096 0.069 0.0047 0.1634 2 0.1021 0.0652 0.0591 0.1276 0.0906 0.1263 0.0438 0.1623 0.1828 3 0.1037 0.0567 0.0574 0.1329 0.0949 0.121 0.0506 0.1526 0.176 4 0.1044 0.0314 0.0567 0.1432 0.0812 0.1105 0.0562 0.1232 0.1694 5 0.109 0.0243 0.0521 0.1445 0.0777 0.1093 0.0619 0.0511 0.1647
The uncertain return rates on assets of five periods investment
 Asset 28 Asset 29 Asset 30 1 0.1551 0.1128 0.0498 0.0994 0.1233 0.0677 0.0674 0.1355 0.0854 2 0.1382 0.0789 0.0969 0.1123 0.0641 0.0498 0.1037 0.0636 0.0438 3 0.1395 0.0679 0.0956 0.1134 0.0648 0.0488 0.1048 0.0645 0.0426 4 0.1426 0.0379 0.0924 0.1157 0.0416 0.0464 0.106 0.0574 0.0414 5 0.147 0.0364 0.088 0.1175 0.0312 0.0446 0.1061 0.035 0.0413
 Asset 28 Asset 29 Asset 30 1 0.1551 0.1128 0.0498 0.0994 0.1233 0.0677 0.0674 0.1355 0.0854 2 0.1382 0.0789 0.0969 0.1123 0.0641 0.0498 0.1037 0.0636 0.0438 3 0.1395 0.0679 0.0956 0.1134 0.0648 0.0488 0.1048 0.0645 0.0426 4 0.1426 0.0379 0.0924 0.1157 0.0416 0.0464 0.106 0.0574 0.0414 5 0.147 0.0364 0.088 0.1175 0.0312 0.0446 0.1061 0.035 0.0413
The uncertain absolute deviation of assets of five periods investment
 Asset 1 Asset 2 Asset 3 Asset 4 Asset 5 Asset 6 Asset 7 Asset 8 1 0.0551 0.0599 0.0232 0.0386 0.0179 0.028 0.0588 0.0461 2 0.0506 0.0593 0.0268 0.0381 0.0183 0.0282 0.062 0.0443 3 0.0504 0.0571 0.0261 0.0331 0.0255 0.0263 0.0532 0.0439 4 0.0428 0.0533 0.0333 0.034 0.0255 0.0245 0.0521 0.0358 5 0.0422 0.0516 0.033 0.0395 0.0259 0.0258 0.0507 0.0356
 Asset 1 Asset 2 Asset 3 Asset 4 Asset 5 Asset 6 Asset 7 Asset 8 1 0.0551 0.0599 0.0232 0.0386 0.0179 0.028 0.0588 0.0461 2 0.0506 0.0593 0.0268 0.0381 0.0183 0.0282 0.062 0.0443 3 0.0504 0.0571 0.0261 0.0331 0.0255 0.0263 0.0532 0.0439 4 0.0428 0.0533 0.0333 0.034 0.0255 0.0245 0.0521 0.0358 5 0.0422 0.0516 0.033 0.0395 0.0259 0.0258 0.0507 0.0356
The uncertain absolute deviation of assets of five periods investment
 Asset 9 Asset 10 Asset 11 Asset 12 Asset 13 Asset 14 Asset 15 Asset 16 1 0.0374 0.0185 0.044 0.076 0.0563 0.0228 0.0573 0.0894 2 0.0386 0.0168 0.0435 0.0708 0.0615 0.0143 0.036 0.0765 3 0.0363 0.016 0.0408 0.0647 0.0589 0.015 0.0357 0.074 4 0.0362 0.0156 0.0398 0.062 0.0588 0.0162 0.0399 0.0704 5 0.0356 0.0123 0.0405 0.0587 0.0543 0.0195 0.0335 0.0688
 Asset 9 Asset 10 Asset 11 Asset 12 Asset 13 Asset 14 Asset 15 Asset 16 1 0.0374 0.0185 0.044 0.076 0.0563 0.0228 0.0573 0.0894 2 0.0386 0.0168 0.0435 0.0708 0.0615 0.0143 0.036 0.0765 3 0.0363 0.016 0.0408 0.0647 0.0589 0.015 0.0357 0.074 4 0.0362 0.0156 0.0398 0.062 0.0588 0.0162 0.0399 0.0704 5 0.0356 0.0123 0.0405 0.0587 0.0543 0.0195 0.0335 0.0688
The uncertain absolute deviation of assets of five periods investment
 Asset 17 Asset 18 Asset 19 Asset 20 Asset21 Asset 22 Asset 23 Asset 24 1 0.0285 0.0666 0.0409 0.0356 0.0119 0.0229 0.024 0.023 2 0.0361 0.0592 0.0338 0.0342 0.0453 0.0339 0.0319 0.0197 3 0.0393 0.0608 0.0217 0.0349 0.0443 0.0333 0.0311 0.0203 4 0.0417 0.0801 0.0278 0.0278 0.0435 0.0333 0.0346 0.0229 5 0.0322 0.0735 0.0269 0.0214 0.043 0.0287 0.0373 0.0255
 Asset 17 Asset 18 Asset 19 Asset 20 Asset21 Asset 22 Asset 23 Asset 24 1 0.0285 0.0666 0.0409 0.0356 0.0119 0.0229 0.024 0.023 2 0.0361 0.0592 0.0338 0.0342 0.0453 0.0339 0.0319 0.0197 3 0.0393 0.0608 0.0217 0.0349 0.0443 0.0333 0.0311 0.0203 4 0.0417 0.0801 0.0278 0.0278 0.0435 0.0333 0.0346 0.0229 5 0.0322 0.0735 0.0269 0.0214 0.043 0.0287 0.0373 0.0255
The uncertain absolute deviation of assets of five periods investment
 Asset 25 Asset 26 Asset 27 Asset 28 Asset 29 Asset 30 1 0.024 0.0456 0.0568 0.0434 0.0279 0.0379 2 0.0254 0.0545 0.0638 0.0441 0.0273 0.0233 3 0.0285 0.0542 0.0599 0.0411 0.0272 0.0271 4 0.0224 0.0482 0.0581 0.0426 0.0288 0.0257 5 0.0196 0.047 0.0564 0.0411 0.0292 0.0191
 Asset 25 Asset 26 Asset 27 Asset 28 Asset 29 Asset 30 1 0.024 0.0456 0.0568 0.0434 0.0279 0.0379 2 0.0254 0.0545 0.0638 0.0441 0.0273 0.0233 3 0.0285 0.0542 0.0599 0.0411 0.0272 0.0271 4 0.0224 0.0482 0.0581 0.0426 0.0288 0.0257 5 0.0196 0.047 0.0564 0.0411 0.0292 0.0191
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