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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 and 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.

[2]

S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Rev. Financial Studies, 23 (2010), 2970-3016.  doi: 10.1093/rfs/hhq028.

[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.

[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.

[5]

D. Bienstock, Computational study of a family of mixed-integer quadratic programming problems, Math. Programming, 74 (1996), 121-140.  doi: 10.1007/BF02592208.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.
[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.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, Monograph, 16, John Wiley & Sons, Inc., New York, 1959.

[28]

H. M. Markowitz, Portfolio selection analysis, J. Finance, 7 (1952), 77-91. 

[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.

[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.

[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.

[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.

[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.

[34]

M. Simkowitz and W. Beedles, Diversification in a three moment world, J. Financial Quantitative Anal., 13 (1978), 927-941.  doi: 10.2307/2330635.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[2]

S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Rev. Financial Studies, 23 (2010), 2970-3016.  doi: 10.1093/rfs/hhq028.

[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.

[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.

[5]

D. Bienstock, Computational study of a family of mixed-integer quadratic programming problems, Math. Programming, 74 (1996), 121-140.  doi: 10.1007/BF02592208.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.
[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.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, Monograph, 16, John Wiley & Sons, Inc., New York, 1959.

[28]

H. M. Markowitz, Portfolio selection analysis, J. Finance, 7 (1952), 77-91. 

[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.

[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.

[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.

[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.

[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.

[34]

M. Simkowitz and W. Beedles, Diversification in a three moment world, J. Financial Quantitative Anal., 13 (1978), 927-941.  doi: 10.2307/2330635.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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
Table Options

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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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