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

![]() |
The optimal investment proportions | ||||||
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 |
![]() |
The optimal investment proportions | ||||||
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 |
0 | 0.000001 | 0.000002 | 0.000003 | 0.000004 | 0.000005 | 0.000006 | 0.000007 | 0.000008 | |
1203337 | 1199885 | 1192648 | 1187153 | 1181143 | 1171130 | 1155254 | 1146196 | 1128999 | |
1237168 | 1233682 | 1224935 | 1215756 | 1201734 | 1185194 | 1169038 | 1150985 | 1137892 | |
0.000009 | 0.00001 | 0.00002 | 0.00003 | 0.00004 | 0.00005 | 0.00006 | 0.00007 | 0.00008 | |
1119367 | 1110519 | 1065707 | 1048836 | 1040399 | 1035338 | 1031963 | 1029552 | 1027745 | |
1127173 | 1122523 | 1067509 | 1050035 | 1041300 | 1036057 | 1032563 | 1030067 | 1028194 | |
0.00009 | 0.0001 | 0.0002 | 0.0003 | 0.0004 | 0.0005 | 0.0006 | 0.0007 | 0.0008 | |
1026339 | 1025214 | 1020151 | 1018466 | 1017621 | 1017139 | 1016767 | 1016526 | 1016347 | |
1027030 | 1025575 | 1020331 | 1018585 | 1017711 | 1017187 | 1016837 | 1016588 | 1016400 | |
0.0009 | 0.001 | 0.002 | 0.003 | 0.004 | 0.005 | 0.006 | 0.007 | 0.008 | |
1016208 | 1016095 | 1015593 | 1015425 | 1015343 | 1015291 | 1015257 | 1015234 | 1015216 | |
1016256 | 1016139 | 1015614 | 1015441 | 1015352 | 1015300 | 1015266 | 1015241 | 1015221 | |
0.009 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | |
1015201 | 1015191 | 1015141 | 1015123 | 1015115 | 1015110 | 1015106 | 1015106 | 1015106 | |
1015207 | 1015195 | 1015142 | 1015126 | 1015117 | 1015111 | 1015106 | 1015106 | 1015106 |
0 | 0.000001 | 0.000002 | 0.000003 | 0.000004 | 0.000005 | 0.000006 | 0.000007 | 0.000008 | |
1203337 | 1199885 | 1192648 | 1187153 | 1181143 | 1171130 | 1155254 | 1146196 | 1128999 | |
1237168 | 1233682 | 1224935 | 1215756 | 1201734 | 1185194 | 1169038 | 1150985 | 1137892 | |
0.000009 | 0.00001 | 0.00002 | 0.00003 | 0.00004 | 0.00005 | 0.00006 | 0.00007 | 0.00008 | |
1119367 | 1110519 | 1065707 | 1048836 | 1040399 | 1035338 | 1031963 | 1029552 | 1027745 | |
1127173 | 1122523 | 1067509 | 1050035 | 1041300 | 1036057 | 1032563 | 1030067 | 1028194 | |
0.00009 | 0.0001 | 0.0002 | 0.0003 | 0.0004 | 0.0005 | 0.0006 | 0.0007 | 0.0008 | |
1026339 | 1025214 | 1020151 | 1018466 | 1017621 | 1017139 | 1016767 | 1016526 | 1016347 | |
1027030 | 1025575 | 1020331 | 1018585 | 1017711 | 1017187 | 1016837 | 1016588 | 1016400 | |
0.0009 | 0.001 | 0.002 | 0.003 | 0.004 | 0.005 | 0.006 | 0.007 | 0.008 | |
1016208 | 1016095 | 1015593 | 1015425 | 1015343 | 1015291 | 1015257 | 1015234 | 1015216 | |
1016256 | 1016139 | 1015614 | 1015441 | 1015352 | 1015300 | 1015266 | 1015241 | 1015221 | |
0.009 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | |
1015201 | 1015191 | 1015141 | 1015123 | 1015115 | 1015110 | 1015106 | 1015106 | 1015106 | |
1015207 | 1015195 | 1015142 | 1015126 | 1015117 | 1015111 | 1015106 | 1015106 | 1015106 |
0 | 0.0000001 | 0.0000002 | 0.0000003 | 0.0000004 | |
1237168 | 1237167 | 1237031 | 1236828 | 1235819 | |
1240294 | 240294 | 1240154 | 1240154 | 1239692 | |
0.0000005 | 0.0000006 | 0.0000007 | 0.0000008 | 0.0000009 | |
1235560 | 1234737 | 1234664 | 1234642 | 1233682 | |
1239692 | 1237130 | 1237130 | 1237130 | 1237130 |
0 | 0.0000001 | 0.0000002 | 0.0000003 | 0.0000004 | |
1237168 | 1237167 | 1237031 | 1236828 | 1235819 | |
1240294 | 240294 | 1240154 | 1240154 | 1239692 | |
0.0000005 | 0.0000006 | 0.0000007 | 0.0000008 | 0.0000009 | |
1235560 | 1234737 | 1234664 | 1234642 | 1233682 | |
1239692 | 1237130 | 1237130 | 1237130 | 1237130 |
0 | 1 | 2 | 3 | 4 | 5 | 6 | |
1015090 | 1065669 | 1102726 | 1136562 | 1160553 | 1181499 | 1202122 | |
7 | 8 | 9 | 10 | 11 | 12 | 13 | |
1221479 | 1240294 | 1258686 | 1275590 | 1284427 | 1284833 | 1284833 |
0 | 1 | 2 | 3 | 4 | 5 | 6 | |
1015090 | 1065669 | 1102726 | 1136562 | 1160553 | 1181499 | 1202122 | |
7 | 8 | 9 | 10 | 11 | 12 | 13 | |
1221479 | 1240294 | 1258686 | 1275590 | 1284427 | 1284833 | 1284833 |
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