Time-consistent multiperiod mean semivariance portfolio selection with the real constraints

Peng Zhang; Yongquan Zeng; Guotai Chi

doi:10.3934/jimo.2020039

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July 2021, 17(4): 1663-1680. doi: 10.3934/jimo.2020039

Time-consistent multiperiod mean semivariance portfolio selection with the real constraints

Peng Zhang ^1,, , Yongquan Zeng ^2,3, and Guotai Chi ^2,3,

1.	School of Economics and Management, South China Normal University, Guangzhou 510006, China
2.	College of Humanities and Social sciences, Zhongkai University of Agriculture and Engineering, Guangzhou 510225, China
3.	Faculty of Management and Economics, Dalian University of Technology, Dalian 116024, China

^* Corresponding author: Tel.:+86 18971194382

Received July 2019 Revised September 2019 Published July 2021 Early access February 2020

Fund Project: This research was supported by the Key Projects of National Natural Science Foundation (nos. 71731003)

In this paper, a new multiperiod mean semivariance portfolio selection with the transaction costs, borrowing constraints, threshold constraints and cardinality constraints is proposed. In the model, the return and risk of assets are characterized by mean value and semivariance, respectively. Because the semivariance operator is not separable, the optimal solution of the model is not time-consistent. The time-consistent strategy for this model can be obtained by using game approach. The time-consistent strategy, which is a mix integer dynamic optimization problem with path dependence, is approximately turned into a dynamic programming problem by approximate dynamic programming method. A novel discrete approximate iteration method is designed to obtain the optimal time-consistent strategy, and is proved linearly convergent. Finally, the comparison analysis of trade-off parameters is given to illustrate the idea of our model and the effectiveness of the designed algorithm.

Keywords: Multiperiod mean semivariance portfolio selection, transaction costs, cardinality constraints, time-consistency, a discrete approximate iteration method.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation: Peng Zhang, Yongquan Zeng, Guotai Chi. Time-consistent multiperiod mean semivariance portfolio selection with the real constraints. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1663-1680. doi: 10.3934/jimo.2020039

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 Appl., 38 (2011), 14208-14217. doi: 10.1016/j.eswa.2011.04.233. Google Scholar
[2]	S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Rev. Financial Studies, 23 (2010), 2970-3016. doi: 10.1093/rfs/hhq028. Google Scholar
[3]	A. Bensoussan, K. C. Wong, S. C. P. Yam and S. P. Yung, Time-consistent portfolio selection under short-selling prohibition: From discrete to continuous setting, SIAM J. Financial Math., 5 (2014), 153-190. doi: 10.1137/130914139. Google Scholar
[4]	D. Bertsimas and R. Shioda, Algorithms for cardinality-constrained quadratic optimization, Comput. Optim. Appl., 43 (2009), 1-22. doi: 10.1007/s10589-007-9126-9. Google Scholar
[5]	D. Bienstock, Computational study of a family of mixed-integer quadratic programming problems, Math. Programming, 74 (1996), 121-140. doi: 10.1007/BF02592208. Google Scholar
[6]	T. Björk, M. H. A. Davis and C. Landén, Optimal investment under partial information, Math. Methods Oper. Res., 71 (2010), 371-399. doi: 10.1007/s00186-010-0301-x. Google Scholar
[7]	T. Björk, A. Murgoci and X. Y. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Math. Finance, 24 (2014), 1-24. doi: 10.1111/j.1467-9965.2011.00515.x. Google Scholar
[8]	F. Cesarone, A. Scozzari and F. Tardella, A new method for mean-variance portfolio optimization with cardinality constraints, Ann. Oper. Res., 205 (2013), 213-234. doi: 10.1007/s10479-012-1165-7. Google Scholar
[9]	Z. Chen, G. Li and Y. Zhao, Time-consistent investment policies in Markovian markets: A case of mean-variance analysis, J. Econom. Dynam. Control, 40 (2014), 293-316. doi: 10.1016/j.jedc.2014.01.011. Google Scholar
[10]	F. Cong and C. W. Oosterlee, Multi-period mean-variance portfolio optimization based on Monte-Carlo simulation, J. Econom. Dynam. Control, 64 (2016), 23-38. doi: 10.1016/j.jedc.2016.01.001. Google Scholar
[11]	X. 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, Math. Finance, 27 (2017), 471-504. doi: 10.1111/mafi.12093. Google Scholar
[12]	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, J. Global Optim., 56 (2013), 1409-1423. doi: 10.1007/s10898-012-9842-2. Google Scholar
[13]	X. Y. Cui, D. Li, S. Y. Wang and S. S. Zhu, Better than dynamic mean-variance: Time inconsistency and free cash flow stream, Math. Finance, 22 (2012), 346-378. doi: 10.1111/j.1467-9965.2010.00461.x. Google Scholar
[14]	X. Y. Cui, X. Li and D. Li, Unified framework of mean-field formulations for optimal multi-period mean-variance portfolio selection, IEEE Trans. Automat. Control, 59 (2014), 1833-1844. doi: 10.1109/TAC.2014.2311875. Google Scholar
[15]	C. Czichowsky, Time-consistent mean-variance portfolio selection in discrete and continuous time, Finance Stoch., 17 (2013), 227-271. doi: 10.1007/s00780-012-0189-9. Google Scholar
[16]	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 Appl., 39 (2012), 4558-4566. doi: 10.1016/j.eswa.2011.09.129. Google Scholar
[17]	A. Fernández and S. Gómez, Portfolio selection using neural networks, Comput. Oper. Res., 34 (2005), 1177-1191. doi: 10.1016/j.cor.2005.06.017. Google Scholar
[18]	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 J., 54 (2015), 91-99. doi: 10.1016/j.automatica.2015.01.040. Google Scholar
[19]	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 Series in Applied Mathematics, 48, Princeton University Press, Princeton, NJ, 2006. doi: 10.1515/9781400865239. Google Scholar
[20]	H. A. Le Thi and M. Moeini, Long-short portfolio optimization under cardinality constraints by difference of convex functions algorithm, J. Optim. Theory Appl., 161 (2014), 199-224. doi: 10.1007/s10957-012-0197-0. Google Scholar
[21]	H. A. Le Thi, M. Moeini and T. P. Dinh, Portfolio selection under downside risk measures and cardinality constraints based on DC programming and DCA, Comput. Manag. Sci., 6 (2009), 459-475. doi: 10.1007/s10287-009-0098-3. Google Scholar
[22]	D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Math. Finance, 10 (2000), 387-406. doi: 10.1111/1467-9965.00100. Google Scholar
[23]	D. Li, X. Sun and J. Wang, Optimal lot solution to cardinality constrained mean-variance formulation for portfolio selection, Math. Finance, 16 (2006), 83-101. doi: 10.1111/j.1467-9965.2006.00262.x. Google Scholar
[24]	A. Lioui, Time consistent vs. time inconsistent dynamic asset allocation: Some utility cost calculations for mean variance preferences, J. Econom. Dynam. Control, 37 (2013), 1066-1096. doi: 10.1016/j.jedc.2013.01.007. Google Scholar
[25]	J. Liu and Z. Chen, Time consistent multi-period robust risk measures and portfolio selection models with regime-switching, European J. Oper. Res., 268 (2018), 373-385. doi: 10.1016/j.ejor.2018.01.009. Google Scholar
[26]	F. M. Longin, From value at risk to stress testing: The extreme value approach, J. Banking Finance, 24 (2000), 1097-1130. doi: 10.1016/S0378-4266(99)00077-1. Google Scholar
[27]	H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, Monograph, 16, John Wiley & Sons, Inc., New York, 1959. Google Scholar
[28]	H. M. Markowitz, Portfolio selection analysis, J. Finance, 7 (1952), 77-91. Google Scholar
[29]	W. Murray and H. Shek, A local relaxation method for the cardinality constrained portfolio optimization problem, Comput. Optim. Appl., 53 (2012), 681-709. doi: 10.1007/s10589-012-9471-1. Google Scholar
[30]	M. Ç. Pınar, Robust scenario optimization based on downside-risk measure for multi-period portfolio selection, OR Spectrum, 29 (2007), 295-309. doi: 10.1007/s00291-005-0023-2. Google Scholar
[31]	B. Rudloff, A. Street and D. M. Valladō, Time consistency and risk averse dynamic decision models: Definition, interpretation and practical consequences, European J. Oper. Res., 234 (2014), 743-750. doi: 10.1016/j.ejor.2013.11.037. Google Scholar
[32]	R. Ruiz-Torrubiano and A. Suarez, Hybrid approaches and dimensionality reduction for portfolio selection with cardinality constrains, IEEE Comput. Intell. Magazine, 5 (2010), 92-107. doi: 10.1109/MCI.2010.936308. Google Scholar
[33]	D. X. Shaw, S. Liu and L. Kopman, Lagrangian relaxation procedure for cardinality-constrained portfolio optimization, Optim. Methods Softw., 23 (2008), 411-420. doi: 10.1080/10556780701722542. Google Scholar
[34]	M. Simkowitz and W. Beedles, Diversification in a three moment world, J. Financial Quantitative Anal., 13 (1978), 927-941. doi: 10.2307/2330635. Google Scholar
[35]	H. Soleimani, H. R. Golmakani and M. H. Salimi, Markowitz-based portfolio selection with minimum transaction lots, cardinality constraints and regarding sector capitalization using genetic algorithm, Expert Systems with Appl., 36 (2009), 5058-5063. doi: 10.1016/j.eswa.2008.06.007. Google Scholar
[36]	X. L. Sun, X. J. Zheng and D. Li, Recent advances in mathematical programming with semi-continuous variables and cardinality constraint, J. Oper. Res. Soc. of China, 1 (2013), 55-77. doi: 10.1007/s40305-013-0004-0. Google Scholar
[37]	E. Vercher and J. D. Bermúdez, A possibilistic mean-downside risk-skewness model for efficient portfolio selection, IEEE Transactions on Fuzzy Systems, 3 (2013), 585-595. doi: 10.1109/TFUZZ.2012.2227487. Google Scholar
[38]	J. Wang and P. A. Forsyth, Continuous time mean variance asset allocation: A time-consistent strategy, European J. Oper. Res., 209 (2011), 184-201. doi: 10.1016/j.ejor.2010.09.038. Google Scholar
[39]	J. Wei, K. C. Wong, S. C. P. Yam and S. P. Yung, Markowitz's mean-variance asset-liability management with regime switching: A time-consistent approach, Insurance Math. Econom., 53 (2013), 281-291. doi: 10.1016/j.insmatheco.2013.05.008. Google Scholar
[40]	M. Woodside-Oriakhi, C. Lucas and J. E. Beasley, Heuristic algorithms for the cardinality constrained efficient frontier, European J. Oper. Res., 213 (2011), 538-550. doi: 10.1016/j.ejor.2011.03.030. Google Scholar
[41]	H. Wu and H. Chen, Nash equilibrium strategy for a multi-period mean-variance portfolio selection problem with regime switching, Economic Modell., 46 (2015), 79-90. doi: 10.1016/j.econmod.2014.12.024. 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 Math. Econom., 64 (2015), 396-408. doi: 10.1016/j.insmatheco.2015.07.007. Google Scholar
[43]	W. Yan and S.R. Li, A class of multi-period semi-variance portfolio selection with a four-factor futures price model, J. Appl. Math. Comput., 29 (2009), 19-34. doi: 10.1007/s12190-008-0086-8. Google Scholar
[44]	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
[45]	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 Math. Econom., 68 (2016), 187-202. doi: 10.1016/j.insmatheco.2016.03.002. Google Scholar
[46]	S. S. Zhu, D. Li and S. Y. Wang, Risk control over bankruptcy in dynamic portfolio selection: A generalized mean-variance formulation, IEEE Trans. Automat. 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 Appl., 38 (2011), 14208-14217. doi: 10.1016/j.eswa.2011.04.233. Google Scholar
[2]	S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Rev. Financial Studies, 23 (2010), 2970-3016. doi: 10.1093/rfs/hhq028. Google Scholar
[3]	A. Bensoussan, K. C. Wong, S. C. P. Yam and S. P. Yung, Time-consistent portfolio selection under short-selling prohibition: From discrete to continuous setting, SIAM J. Financial Math., 5 (2014), 153-190. doi: 10.1137/130914139. Google Scholar
[4]	D. Bertsimas and R. Shioda, Algorithms for cardinality-constrained quadratic optimization, Comput. Optim. Appl., 43 (2009), 1-22. doi: 10.1007/s10589-007-9126-9. Google Scholar
[5]	D. Bienstock, Computational study of a family of mixed-integer quadratic programming problems, Math. Programming, 74 (1996), 121-140. doi: 10.1007/BF02592208. Google Scholar
[6]	T. Björk, M. H. A. Davis and C. Landén, Optimal investment under partial information, Math. Methods Oper. Res., 71 (2010), 371-399. doi: 10.1007/s00186-010-0301-x. Google Scholar
[7]	T. Björk, A. Murgoci and X. Y. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Math. Finance, 24 (2014), 1-24. doi: 10.1111/j.1467-9965.2011.00515.x. Google Scholar
[8]	F. Cesarone, A. Scozzari and F. Tardella, A new method for mean-variance portfolio optimization with cardinality constraints, Ann. Oper. Res., 205 (2013), 213-234. doi: 10.1007/s10479-012-1165-7. Google Scholar
[9]	Z. Chen, G. Li and Y. Zhao, Time-consistent investment policies in Markovian markets: A case of mean-variance analysis, J. Econom. Dynam. Control, 40 (2014), 293-316. doi: 10.1016/j.jedc.2014.01.011. Google Scholar
[10]	F. Cong and C. W. Oosterlee, Multi-period mean-variance portfolio optimization based on Monte-Carlo simulation, J. Econom. Dynam. Control, 64 (2016), 23-38. doi: 10.1016/j.jedc.2016.01.001. Google Scholar
[11]	X. 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, Math. Finance, 27 (2017), 471-504. doi: 10.1111/mafi.12093. Google Scholar
[12]	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, J. Global Optim., 56 (2013), 1409-1423. doi: 10.1007/s10898-012-9842-2. Google Scholar
[13]	X. Y. Cui, D. Li, S. Y. Wang and S. S. Zhu, Better than dynamic mean-variance: Time inconsistency and free cash flow stream, Math. Finance, 22 (2012), 346-378. doi: 10.1111/j.1467-9965.2010.00461.x. Google Scholar
[14]	X. Y. Cui, X. Li and D. Li, Unified framework of mean-field formulations for optimal multi-period mean-variance portfolio selection, IEEE Trans. Automat. Control, 59 (2014), 1833-1844. doi: 10.1109/TAC.2014.2311875. Google Scholar
[15]	C. Czichowsky, Time-consistent mean-variance portfolio selection in discrete and continuous time, Finance Stoch., 17 (2013), 227-271. doi: 10.1007/s00780-012-0189-9. Google Scholar
[16]	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 Appl., 39 (2012), 4558-4566. doi: 10.1016/j.eswa.2011.09.129. Google Scholar
[17]	A. Fernández and S. Gómez, Portfolio selection using neural networks, Comput. Oper. Res., 34 (2005), 1177-1191. doi: 10.1016/j.cor.2005.06.017. Google Scholar
[18]	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 J., 54 (2015), 91-99. doi: 10.1016/j.automatica.2015.01.040. Google Scholar
[19]	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 Series in Applied Mathematics, 48, Princeton University Press, Princeton, NJ, 2006. doi: 10.1515/9781400865239. Google Scholar
[20]	H. A. Le Thi and M. Moeini, Long-short portfolio optimization under cardinality constraints by difference of convex functions algorithm, J. Optim. Theory Appl., 161 (2014), 199-224. doi: 10.1007/s10957-012-0197-0. Google Scholar
[21]	H. A. Le Thi, M. Moeini and T. P. Dinh, Portfolio selection under downside risk measures and cardinality constraints based on DC programming and DCA, Comput. Manag. Sci., 6 (2009), 459-475. doi: 10.1007/s10287-009-0098-3. Google Scholar
[22]	D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Math. Finance, 10 (2000), 387-406. doi: 10.1111/1467-9965.00100. Google Scholar
[23]	D. Li, X. Sun and J. Wang, Optimal lot solution to cardinality constrained mean-variance formulation for portfolio selection, Math. Finance, 16 (2006), 83-101. doi: 10.1111/j.1467-9965.2006.00262.x. Google Scholar
[24]	A. Lioui, Time consistent vs. time inconsistent dynamic asset allocation: Some utility cost calculations for mean variance preferences, J. Econom. Dynam. Control, 37 (2013), 1066-1096. doi: 10.1016/j.jedc.2013.01.007. Google Scholar
[25]	J. Liu and Z. Chen, Time consistent multi-period robust risk measures and portfolio selection models with regime-switching, European J. Oper. Res., 268 (2018), 373-385. doi: 10.1016/j.ejor.2018.01.009. Google Scholar
[26]	F. M. Longin, From value at risk to stress testing: The extreme value approach, J. Banking Finance, 24 (2000), 1097-1130. doi: 10.1016/S0378-4266(99)00077-1. Google Scholar
[27]	H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, Monograph, 16, John Wiley & Sons, Inc., New York, 1959. Google Scholar
[28]	H. M. Markowitz, Portfolio selection analysis, J. Finance, 7 (1952), 77-91. Google Scholar
[29]	W. Murray and H. Shek, A local relaxation method for the cardinality constrained portfolio optimization problem, Comput. Optim. Appl., 53 (2012), 681-709. doi: 10.1007/s10589-012-9471-1. Google Scholar
[30]	M. Ç. Pınar, Robust scenario optimization based on downside-risk measure for multi-period portfolio selection, OR Spectrum, 29 (2007), 295-309. doi: 10.1007/s00291-005-0023-2. Google Scholar
[31]	B. Rudloff, A. Street and D. M. Valladō, Time consistency and risk averse dynamic decision models: Definition, interpretation and practical consequences, European J. Oper. Res., 234 (2014), 743-750. doi: 10.1016/j.ejor.2013.11.037. Google Scholar
[32]	R. Ruiz-Torrubiano and A. Suarez, Hybrid approaches and dimensionality reduction for portfolio selection with cardinality constrains, IEEE Comput. Intell. Magazine, 5 (2010), 92-107. doi: 10.1109/MCI.2010.936308. Google Scholar
[33]	D. X. Shaw, S. Liu and L. Kopman, Lagrangian relaxation procedure for cardinality-constrained portfolio optimization, Optim. Methods Softw., 23 (2008), 411-420. doi: 10.1080/10556780701722542. Google Scholar
[34]	M. Simkowitz and W. Beedles, Diversification in a three moment world, J. Financial Quantitative Anal., 13 (1978), 927-941. doi: 10.2307/2330635. Google Scholar
[35]	H. Soleimani, H. R. Golmakani and M. H. Salimi, Markowitz-based portfolio selection with minimum transaction lots, cardinality constraints and regarding sector capitalization using genetic algorithm, Expert Systems with Appl., 36 (2009), 5058-5063. doi: 10.1016/j.eswa.2008.06.007. Google Scholar
[36]	X. L. Sun, X. J. Zheng and D. Li, Recent advances in mathematical programming with semi-continuous variables and cardinality constraint, J. Oper. Res. Soc. of China, 1 (2013), 55-77. doi: 10.1007/s40305-013-0004-0. Google Scholar
[37]	E. Vercher and J. D. Bermúdez, A possibilistic mean-downside risk-skewness model for efficient portfolio selection, IEEE Transactions on Fuzzy Systems, 3 (2013), 585-595. doi: 10.1109/TFUZZ.2012.2227487. Google Scholar
[38]	J. Wang and P. A. Forsyth, Continuous time mean variance asset allocation: A time-consistent strategy, European J. Oper. Res., 209 (2011), 184-201. doi: 10.1016/j.ejor.2010.09.038. Google Scholar
[39]	J. Wei, K. C. Wong, S. C. P. Yam and S. P. Yung, Markowitz's mean-variance asset-liability management with regime switching: A time-consistent approach, Insurance Math. Econom., 53 (2013), 281-291. doi: 10.1016/j.insmatheco.2013.05.008. Google Scholar
[40]	M. Woodside-Oriakhi, C. Lucas and J. E. Beasley, Heuristic algorithms for the cardinality constrained efficient frontier, European J. Oper. Res., 213 (2011), 538-550. doi: 10.1016/j.ejor.2011.03.030. Google Scholar
[41]	H. Wu and H. Chen, Nash equilibrium strategy for a multi-period mean-variance portfolio selection problem with regime switching, Economic Modell., 46 (2015), 79-90. doi: 10.1016/j.econmod.2014.12.024. 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 Math. Econom., 64 (2015), 396-408. doi: 10.1016/j.insmatheco.2015.07.007. Google Scholar
[43]	W. Yan and S.R. Li, A class of multi-period semi-variance portfolio selection with a four-factor futures price model, J. Appl. Math. Comput., 29 (2009), 19-34. doi: 10.1007/s12190-008-0086-8. Google Scholar
[44]	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
[45]	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 Math. Econom., 68 (2016), 187-202. doi: 10.1016/j.insmatheco.2016.03.002. Google Scholar
[46]	S. S. Zhu, D. Li and S. Y. Wang, Risk control over bankruptcy in dynamic portfolio selection: A generalized mean-variance formulation, IEEE Trans. Automat. 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 solutions when $ K = 8 $, $ u_{ft}^b $ = -500000 dollars, $ \eta _t $ = 0.000001

	The optimal investment proportions						$ X_t $
1	Asset3	Asset 6	Asset 7	Asset 8	Asset 9	Asset 10	1044290
	200000.0	200000.0	200000.0	200000.0	200000.0	200000.0
	Asset11	Asset28	other risk asset
	100000.0	200000.0	0
2	Asset6	Asset 7	Asset 8	Asset 9	Asset 10	Asset 11	1091060
	200000.0	200000.0	200000.0	200000.0	200000.0	200000.0
	Asset 12	Asset 28	other risk asset
	144290.0	200000.0	0
3	Asset3	Asset 6	Asset 7	Asset 8	Asset 9	Asset 10	1140219
	191060.0	200000.0	200000.0	200000.0	200000.0	200000.0
	Asset11	Asset28	other risk asset
	200000.0	200000.0	0
4	Asset3	Asset 6	Asset 7	Asset 8	Asset 9	Asset 10	1188760
	200000.0	200000.0	200000.0	200000.0	200000.0	200000.0
	Asset11	Asset28	other risk asset
	200000.0	200000.0	0
5	Asset3	Asset 6	Asset 7	Asset 8	Asset 9	Asset 10	1235266
	200000.0	200000.0	200000.0	200000.0	200000.0	200000.0
	Asset11	Asset28	other risk asset
	200000.0	200000.0	0

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	The optimal investment proportions						$ X_t $
1	Asset3	Asset 6	Asset 7	Asset 8	Asset 9	Asset 10	1044290
	200000.0	200000.0	200000.0	200000.0	200000.0	200000.0
	Asset11	Asset28	other risk asset
	100000.0	200000.0	0
2	Asset6	Asset 7	Asset 8	Asset 9	Asset 10	Asset 11	1091060
	200000.0	200000.0	200000.0	200000.0	200000.0	200000.0
	Asset 12	Asset 28	other risk asset
	144290.0	200000.0	0
3	Asset3	Asset 6	Asset 7	Asset 8	Asset 9	Asset 10	1140219
	191060.0	200000.0	200000.0	200000.0	200000.0	200000.0
	Asset11	Asset28	other risk asset
	200000.0	200000.0	0
4	Asset3	Asset 6	Asset 7	Asset 8	Asset 9	Asset 10	1188760
	200000.0	200000.0	200000.0	200000.0	200000.0	200000.0
	Asset11	Asset28	other risk asset
	200000.0	200000.0	0
5	Asset3	Asset 6	Asset 7	Asset 8	Asset 9	Asset 10	1235266
	200000.0	200000.0	200000.0	200000.0	200000.0	200000.0
	Asset11	Asset28	other risk asset
	200000.0	200000.0	0

Table 2. The terminal wealth when $ K $ = 6, and $ K = 8 $, $ u_{ft}^b $ = -500000 dollars, $ \eta_{t} = 0, 0.000001, \dots, 0.08 $

$ \eta_{t} $	0	0.000001	0.000002	0.000003	0.000004	0.000005	0.000006	0.000007	0.000008
$ X_5 $	1203337	1199885	1192648	1187153	1181143	1171130	1155254	1146196	1128999
$ X_5^{'} $	1237168	1233682	1224935	1215756	1201734	1185194	1169038	1150985	1137892
$ \eta_{t} $	0.000009	0.00001	0.00002	0.00003	0.00004	0.00005	0.00006	0.00007	0.00008
$ X_5 $	1119367	1110519	1065707	1048836	1040399	1035338	1031963	1029552	1027745
$ X_5^{'} $	1127173	1122523	1067509	1050035	1041300	1036057	1032563	1030067	1028194
$ \eta_{t} $	0.00009	0.0001	0.0002	0.0003	0.0004	0.0005	0.0006	0.0007	0.0008
$ X_5 $	1026339	1025214	1020151	1018466	1017621	1017139	1016767	1016526	1016347
$ X_5^{'} $	1027030	1025575	1020331	1018585	1017711	1017187	1016837	1016588	1016400
$ \eta_{t} $	0.0009	0.001	0.002	0.003	0.004	0.005	0.006	0.007	0.008
$ X_5 $	1016208	1016095	1015593	1015425	1015343	1015291	1015257	1015234	1015216
$ X_5^{'} $	1016256	1016139	1015614	1015441	1015352	1015300	1015266	1015241	1015221
$ \eta_{t} $	0.009	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08
$ X_5 $	1015201	1015191	1015141	1015123	1015115	1015110	1015106	1015106	1015106
$ X_5^{'} $	1015207	1015195	1015142	1015126	1015117	1015111	1015106	1015106	1015106

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$ \eta_{t} $	0	0.000001	0.000002	0.000003	0.000004	0.000005	0.000006	0.000007	0.000008
$ X_5 $	1203337	1199885	1192648	1187153	1181143	1171130	1155254	1146196	1128999
$ X_5^{'} $	1237168	1233682	1224935	1215756	1201734	1185194	1169038	1150985	1137892
$ \eta_{t} $	0.000009	0.00001	0.00002	0.00003	0.00004	0.00005	0.00006	0.00007	0.00008
$ X_5 $	1119367	1110519	1065707	1048836	1040399	1035338	1031963	1029552	1027745
$ X_5^{'} $	1127173	1122523	1067509	1050035	1041300	1036057	1032563	1030067	1028194
$ \eta_{t} $	0.00009	0.0001	0.0002	0.0003	0.0004	0.0005	0.0006	0.0007	0.0008
$ X_5 $	1026339	1025214	1020151	1018466	1017621	1017139	1016767	1016526	1016347
$ X_5^{'} $	1027030	1025575	1020331	1018585	1017711	1017187	1016837	1016588	1016400
$ \eta_{t} $	0.0009	0.001	0.002	0.003	0.004	0.005	0.006	0.007	0.008
$ X_5 $	1016208	1016095	1015593	1015425	1015343	1015291	1015257	1015234	1015216
$ X_5^{'} $	1016256	1016139	1015614	1015441	1015352	1015300	1015266	1015241	1015221
$ \eta_{t} $	0.009	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08
$ X_5 $	1015201	1015191	1015141	1015123	1015115	1015110	1015106	1015106	1015106
$ X_5^{'} $	1015207	1015195	1015142	1015126	1015117	1015111	1015106	1015106	1015106

Table 3. The terminal wealth when $ K = 8 $, $ u_{ft}^b $ = -500000 and $ u_{ft}^b $ = -1000000, $ \eta_{t} = 0, 0.000001, \dots, 0.000009 $

$ \eta_{t} $	0	0.0000001	0.0000002	0.0000003	0.0000004
$ X_5 $	1237168	1237167	1237031	1236828	1235819
$ X_5^{'} $	1240294	240294	1240154	1240154	1239692
$ \eta_{t} $	0.0000005	0.0000006	0.0000007	0.0000008	0.0000009
$ X_5 $	1235560	1234737	1234664	1234642	1233682
$ X_5^{'} $	1239692	1237130	1237130	1237130	1237130

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$ \eta_{t} $	0	0.0000001	0.0000002	0.0000003	0.0000004
$ X_5 $	1237168	1237167	1237031	1236828	1235819
$ X_5^{'} $	1240294	240294	1240154	1240154	1239692
$ \eta_{t} $	0.0000005	0.0000006	0.0000007	0.0000008	0.0000009
$ X_5 $	1235560	1234737	1234664	1234642	1233682
$ X_5^{'} $	1239692	1237130	1237130	1237130	1237130

Table 4. The terminal wealth when $ K = 0, 1, \dots, 13 $, $ u_{ft}^b $ = -500000 dollars, $ \eta_{t} $ = 0.0000001

$ K $	0	1	2	3	4	5	6
$ X_5 $	1015090	1065669	1102726	1136562	1160553	1181499	1202122
$ K $	7	8	9	10	11	12	13
$ X_5 $	1221479	1240294	1258686	1275590	1284427	1284833	1284833

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$ K $	0	1	2	3	4	5	6
$ X_5 $	1015090	1065669	1102726	1136562	1160553	1181499	1202122
$ K $	7	8	9	10	11	12	13
$ X_5 $	1221479	1240294	1258686	1275590	1284427	1284833	1284833

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American Institute of Mathematical Sciences

Time-consistent multiperiod mean semivariance portfolio selection with the real constraints

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Time-consistent multiperiod mean semivariance portfolio selection with the real constraints

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