Figure 1.
The schematic representation of a scenario tree for T periods
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A type of new consensus protocol for two-dimension multi-agent systems
September
2017, 7(3): 359-377.
doi: 10.3934/naco.2017023
A multistage stochastic programming framework for cardinality constrained portfolio optimization
| 1. | Department of Systems Engineering, IHU University, Tehran, Iran |
| 2. | Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran |
This paper presents a multistage stochastic programming model to deal with multi-period, cardinality constrained portfolio optimization. The presented model aims to minimize investor's expected regret, while ensuring achievement of a minimum expected return. To generate scenarios of market index returns, a random walk model based on the empirical distribution of market-representative index returns is proposed. Then, a single index model is used to estimate stock returns based on market index returns. Afterward, historical returns of a number of stocks, selected from Frankfurt Stock Exchange (FSE), are used to implement the presented scenario generation method, and solve the stochastic programming model. In addition, the impact of cardinality constraints, transaction costs, minimum expected return and predetermined investor's target wealth are investigated. Results show that the inclusion of cardinality constraints and transaction costs significantly influences the investors risk-return tradeoffs. This is also the case for investors target wealth.
Keywords:
Multistage stochastic programming,
multi-period portfolio optimization,
cardinality constraint,
random walk,
single index model.
Mathematics Subject Classification:
90C11, 90C15, 90C90.
Citation:
Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization,
2017, 7
(3)
: 359-377.
doi: 10.3934/naco.2017023
![]()
References:
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D. Barro and E. Canestrelli,
Tracking error: a multistage portfolio model, Ann. Oper. Res., 165 (2009), 47-66.
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M. R. Borges, Efficient market hypothesis in European stock markets, Eur. J. Financ., 16 (2010), 711-726. Google Scholar |
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W. Chen,
Artificial bee colony algorithm for constrained possibilistic portfolio optimization problem, Physica A., 429 (2015), 125-139.
doi: 10.1016/j.physa.2015.02.060. |
| [4] |
Z. Chen, Multiperiod consumption and portfolio decisions under the multivariate GARCH model with transaction costs and CVaR-based risk control, OR Spectrum., 27 (2005), 603-632. Google Scholar |
| [5] |
Z. Chen and D. Xu,
Knowledge-based scenario tree generation methods and application in multiperiod portfolio selection problem, Appl. Stoch. Model. Bus., 30 (2014), 240-257.
doi: 10.1002/asmb.1970. |
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Y. W. Cheung and K. S. Lai, A search for long memory in international stock market returns, J. Int. Money. Financ., 14 (1995), 597-615. Google Scholar |
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A. Consiglio and A. Staino,
A stochastic programming model for the optimal issuance of government bonds, Ann. Oper. Res., 193 (2012), 159-172.
doi: 10.1007/s10479-010-0755-5. |
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G. B. Dantzig and G. Infanger,
Multi-stage stochastic linear programs for portfolio optimization, Ann. Oper. Res., 45 (1993), 59-76.
doi: 10.1007/BF02282041. |
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H. Davari-Ardakani, M. Aminnayeri and A. Seifi,
A study on modeling the dynamics of statistically dependent returns, Physica A., 405 (2014), 35-51.
doi: 10.1016/j.physa.2014.02.077. |
| [10] |
H. Davari-Ardakani, M. Aminnayeri and A. Seifi, Hedging strategies for multi-period portfolio optimization, Sci. Iran., 22 (2015), 2644-2663. Google Scholar |
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H. Davari-Ardakani, M. Aminnayeri and A. Seifi,
Multistage portfolio optimization with stocks and options, Int. Trans. Oper. Res., 23 (2016), 593-622.
doi: 10.1111/itor.12174. |
| [12] |
R. Ferstl and A. Weissensteiner,
Cash management using multi-stage stochastic programming, Quant. Financ., 10 (2010), 209-219.
doi: 10.1080/14697680802637908. |
| [13] |
S. E. Fleten, K. Hoyland and S. W. Wallace,
The performance of stochastic dynamic and fixed mix portfolio models, Eur. J. Oper. Res., 140 (2002), 37-49.
doi: 10.1016/S0377-2217(01)00195-3. |
| [14] |
A. Geyer, M. Hanke and A. Weissensteiner,
Scenario tree generation and multi-asset financial optimization problems, Oper. Res. lett., 41 (2013), 494-498.
doi: 10.1016/j.orl.2013.06.003. |
| [15] |
N. Gülpinar, B. Rustem and R. Settergren,
Simulation and optimization approaches to scenario tree generation, J. Econ. Dyn. Control., 28 (2004), 1291-1315.
doi: 10.1016/S0165-1889(03)00113-1. |
| [16] |
P. Gupta, G. Mittal and M. K. Mehlawat,
Multiobjective expected value model for portfolio selection in fuzzy environment, Optim. Lett., 7 (2013), 1765-1791.
doi: 10.1007/s11590-012-0521-5. |
| [17] |
P. Gupta, G. Mittal and M. K. Mehlawat,
A multi-period fuzzy portfolio optimization model with minimum transaction lots, Eur. J. Oper. Res., 242 (2015), 933-941.
doi: 10.1016/j.ejor.2014.10.061. |
| [18] |
K. Hoyland and S. W. Wallace, Generating scenario trees for multistage decision problems, Manage. Sci., 47 (2001), 295-307. Google Scholar |
| [19] |
K. Hoyland, M. Kaut and S. W. Wallace,
A heuristic for moment-matching scenario generation, Comput. Optim. Appl., 24 (2003), 169-185.
doi: 10.1023/A:1021853807313. |
| [20] |
B. Jacobsen, Long term dependence in stock returns, J. Eimpir. Financ., 3 (1996), 393-417. Google Scholar |
| [21] |
X. Ji, Sh. Zhu, Sh. Wang and Sh. Zhang, A stochastic linear goal programming approach to multistage portfolio management based on scenario generation via linear programming, IIE. Trans., 37 (2005), 957-969. Google Scholar |
| [22] |
T. Lux, Long term stochastic dependence in financial prices: evidence from German stock market, Appl. Econ. Lett., 3 (1996), 701-706. Google Scholar |
| [23] |
R. Mansini, W. Ogryczak and M. G. Speranza,
Twenty years of linear programming based portfolio optimization, Eur. J. Oper. Res., 234 (2014), 518-535.
doi: 10.1016/j.ejor.2013.08.035. |
| [24] |
H. Markowitz, Advantages of multiperiod portfolio models, J. Portfolio. Manage., 29 (2003), 35-45. Google Scholar |
| [25] |
J. M. Mulvey, W. R. Pauling and R. E. Madey, Portfolio selection, J. Financ., 7 (1952), 77-91. Google Scholar |
| [26] |
P. Rocha and D. Kuhn,
Multistage stochastic portfolio optimisation in deregulated electricity markets using linear decision rules, Eur. J. Oper. Res., 216 (2012), 397-408.
doi: 10.1016/j.ejor.2011.08.001. |
| [27] |
C. T. Şakar and M. Köksalan,
A stochastic programming approach to multicriteria portfolio optimization, J. Global. Optim., 57 (2013), 299-314.
doi: 10.1007/s10898-012-0005-2. |
| [28] |
A. Sensoy and B. M. Tabak, Time-varying long term memory in the European Union stock markets, Physica A., 436 (2015), 147-158. Google Scholar |
| [29] |
J. F. Slifker and S. S. Shapiro, The Johnson system: selection and parameter estimation, Technometrics., 22 (1980), 239-246. Google Scholar |
| [30] |
N. Topaloglou, H. Vladimirou and S. A. Zenios,
A dynamic stochastic programming model for international portfolio management, J. Bank. Financ., 26 (2008), 1501-1524.
doi: 10.1016/j.ejor.2005.07.035. |
| [31] |
N. Topaloglou, H. Vladimirou and S. A. Zenios, Optimizing international portfolios with options and forwards, J. Bank. Financ., 35 (2011), 3188-3201. Google Scholar |
| [32] |
A. C. Worthington and H. Higgs, Random walks and market efficiency in European equity markets, Global. J. Financ. Econ., 1 (2004), 59-78. Google Scholar |
| [33] |
L. Yin and L. Han,
International assets allocation with risk management via multi-stage stochastic programming, Comput. Econ., (2013).
doi: 10.1007/s10614-013-9365-z. |
| [34] |
L. Yin and L. Han,
Options strategies for international portfolios with overall risk management via multi-stage stochastic programming, Ann. Oper. Res., 206 (2013), 557-576.
doi: 10.1007/s10479-013-1375-7. |
| [35] |
P. Zhang, An interval mean-average absolute deviation model for multiperiod portfolio selection with risk control and cardinality constraints, Soft. Comput., 20 (2016), 1203-1212. Google Scholar |
show all references
References:
| [1] |
D. Barro and E. Canestrelli,
Tracking error: a multistage portfolio model, Ann. Oper. Res., 165 (2009), 47-66.
doi: 10.1007/s10479-007-0308-8. |
| [2] |
M. R. Borges, Efficient market hypothesis in European stock markets, Eur. J. Financ., 16 (2010), 711-726. Google Scholar |
| [3] |
W. Chen,
Artificial bee colony algorithm for constrained possibilistic portfolio optimization problem, Physica A., 429 (2015), 125-139.
doi: 10.1016/j.physa.2015.02.060. |
| [4] |
Z. Chen, Multiperiod consumption and portfolio decisions under the multivariate GARCH model with transaction costs and CVaR-based risk control, OR Spectrum., 27 (2005), 603-632. Google Scholar |
| [5] |
Z. Chen and D. Xu,
Knowledge-based scenario tree generation methods and application in multiperiod portfolio selection problem, Appl. Stoch. Model. Bus., 30 (2014), 240-257.
doi: 10.1002/asmb.1970. |
| [6] |
Y. W. Cheung and K. S. Lai, A search for long memory in international stock market returns, J. Int. Money. Financ., 14 (1995), 597-615. Google Scholar |
| [7] |
A. Consiglio and A. Staino,
A stochastic programming model for the optimal issuance of government bonds, Ann. Oper. Res., 193 (2012), 159-172.
doi: 10.1007/s10479-010-0755-5. |
| [8] |
G. B. Dantzig and G. Infanger,
Multi-stage stochastic linear programs for portfolio optimization, Ann. Oper. Res., 45 (1993), 59-76.
doi: 10.1007/BF02282041. |
| [9] |
H. Davari-Ardakani, M. Aminnayeri and A. Seifi,
A study on modeling the dynamics of statistically dependent returns, Physica A., 405 (2014), 35-51.
doi: 10.1016/j.physa.2014.02.077. |
| [10] |
H. Davari-Ardakani, M. Aminnayeri and A. Seifi, Hedging strategies for multi-period portfolio optimization, Sci. Iran., 22 (2015), 2644-2663. Google Scholar |
| [11] |
H. Davari-Ardakani, M. Aminnayeri and A. Seifi,
Multistage portfolio optimization with stocks and options, Int. Trans. Oper. Res., 23 (2016), 593-622.
doi: 10.1111/itor.12174. |
| [12] |
R. Ferstl and A. Weissensteiner,
Cash management using multi-stage stochastic programming, Quant. Financ., 10 (2010), 209-219.
doi: 10.1080/14697680802637908. |
| [13] |
S. E. Fleten, K. Hoyland and S. W. Wallace,
The performance of stochastic dynamic and fixed mix portfolio models, Eur. J. Oper. Res., 140 (2002), 37-49.
doi: 10.1016/S0377-2217(01)00195-3. |
| [14] |
A. Geyer, M. Hanke and A. Weissensteiner,
Scenario tree generation and multi-asset financial optimization problems, Oper. Res. lett., 41 (2013), 494-498.
doi: 10.1016/j.orl.2013.06.003. |
| [15] |
N. Gülpinar, B. Rustem and R. Settergren,
Simulation and optimization approaches to scenario tree generation, J. Econ. Dyn. Control., 28 (2004), 1291-1315.
doi: 10.1016/S0165-1889(03)00113-1. |
| [16] |
P. Gupta, G. Mittal and M. K. Mehlawat,
Multiobjective expected value model for portfolio selection in fuzzy environment, Optim. Lett., 7 (2013), 1765-1791.
doi: 10.1007/s11590-012-0521-5. |
| [17] |
P. Gupta, G. Mittal and M. K. Mehlawat,
A multi-period fuzzy portfolio optimization model with minimum transaction lots, Eur. J. Oper. Res., 242 (2015), 933-941.
doi: 10.1016/j.ejor.2014.10.061. |
| [18] |
K. Hoyland and S. W. Wallace, Generating scenario trees for multistage decision problems, Manage. Sci., 47 (2001), 295-307. Google Scholar |
| [19] |
K. Hoyland, M. Kaut and S. W. Wallace,
A heuristic for moment-matching scenario generation, Comput. Optim. Appl., 24 (2003), 169-185.
doi: 10.1023/A:1021853807313. |
| [20] |
B. Jacobsen, Long term dependence in stock returns, J. Eimpir. Financ., 3 (1996), 393-417. Google Scholar |
| [21] |
X. Ji, Sh. Zhu, Sh. Wang and Sh. Zhang, A stochastic linear goal programming approach to multistage portfolio management based on scenario generation via linear programming, IIE. Trans., 37 (2005), 957-969. Google Scholar |
| [22] |
T. Lux, Long term stochastic dependence in financial prices: evidence from German stock market, Appl. Econ. Lett., 3 (1996), 701-706. Google Scholar |
| [23] |
R. Mansini, W. Ogryczak and M. G. Speranza,
Twenty years of linear programming based portfolio optimization, Eur. J. Oper. Res., 234 (2014), 518-535.
doi: 10.1016/j.ejor.2013.08.035. |
| [24] |
H. Markowitz, Advantages of multiperiod portfolio models, J. Portfolio. Manage., 29 (2003), 35-45. Google Scholar |
| [25] |
J. M. Mulvey, W. R. Pauling and R. E. Madey, Portfolio selection, J. Financ., 7 (1952), 77-91. Google Scholar |
| [26] |
P. Rocha and D. Kuhn,
Multistage stochastic portfolio optimisation in deregulated electricity markets using linear decision rules, Eur. J. Oper. Res., 216 (2012), 397-408.
doi: 10.1016/j.ejor.2011.08.001. |
| [27] |
C. T. Şakar and M. Köksalan,
A stochastic programming approach to multicriteria portfolio optimization, J. Global. Optim., 57 (2013), 299-314.
doi: 10.1007/s10898-012-0005-2. |
| [28] |
A. Sensoy and B. M. Tabak, Time-varying long term memory in the European Union stock markets, Physica A., 436 (2015), 147-158. Google Scholar |
| [29] |
J. F. Slifker and S. S. Shapiro, The Johnson system: selection and parameter estimation, Technometrics., 22 (1980), 239-246. Google Scholar |
| [30] |
N. Topaloglou, H. Vladimirou and S. A. Zenios,
A dynamic stochastic programming model for international portfolio management, J. Bank. Financ., 26 (2008), 1501-1524.
doi: 10.1016/j.ejor.2005.07.035. |
| [31] |
N. Topaloglou, H. Vladimirou and S. A. Zenios, Optimizing international portfolios with options and forwards, J. Bank. Financ., 35 (2011), 3188-3201. Google Scholar |
| [32] |
A. C. Worthington and H. Higgs, Random walks and market efficiency in European equity markets, Global. J. Financ. Econ., 1 (2004), 59-78. Google Scholar |
| [33] |
L. Yin and L. Han,
International assets allocation with risk management via multi-stage stochastic programming, Comput. Econ., (2013).
doi: 10.1007/s10614-013-9365-z. |
| [34] |
L. Yin and L. Han,
Options strategies for international portfolios with overall risk management via multi-stage stochastic programming, Ann. Oper. Res., 206 (2013), 557-576.
doi: 10.1007/s10479-013-1375-7. |
| [35] |
P. Zhang, An interval mean-average absolute deviation model for multiperiod portfolio selection with risk control and cardinality constraints, Soft. Comput., 20 (2016), 1203-1212. Google Scholar |
Figure 2.
A schematic representation of the proposed scenario tree generation method
Figure 3.
Risk vs. expected return for portfolios with and without cardinality constraints (Target wealth = ＄1000000)
Figure 4.
Risk vs. expected return obtained by setting different levels of target wealth (＄1000000 and ＄1050000) for portfolios with and without cardinality constraints
Figure 5.
Investor's risk for different levels of proportional transaction costs
Table 1.
Descriptive statistics of historical CDAX returns
| Mean | Standard Deviation | Median | Minimum | Maximum | Skewness | Kurtosis |
| 0.0060 | 0.0569 | 0.0103 | -0.1795 | 0.1745 | -0.5381 | 1.7814 |
| Mean | Standard Deviation | Median | Minimum | Maximum | Skewness | Kurtosis |
| 0.0060 | 0.0569 | 0.0103 | -0.1795 | 0.1745 | -0.5381 | 1.7814 |
Table 2.
αi and βi values of the single index model for all stocks
| Stock | B & A | LR81 | LTEC | MZA | NEC1 | N2X | OTP |
| Intercept | 0.015231 | 0.0008692 | -0.0028 | 0.039533 | -3.1E-05 | 0.001772 | -0.01099 |
| Slope | 0.756845 | 1.211379 | 0.889253 | 1.837928 | 0.644086 | 0.971493 | 1.961487 |
| Stock | SIE | TAH | BMW | XCY | O4B | ZYT | - |
| Intercept | -.00063 | 0.006095 | 0.0098 | 0.024254 | 0.001565 | 0.003712 | - |
| Slope | 1.091311 | 0.292933 | 1.186136 | 0.592039 | 0.564903 | 1.498048 | - |
| Stock | B & A | LR81 | LTEC | MZA | NEC1 | N2X | OTP |
| Intercept | 0.015231 | 0.0008692 | -0.0028 | 0.039533 | -3.1E-05 | 0.001772 | -0.01099 |
| Slope | 0.756845 | 1.211379 | 0.889253 | 1.837928 | 0.644086 | 0.971493 | 1.961487 |
| Stock | SIE | TAH | BMW | XCY | O4B | ZYT | - |
| Intercept | -.00063 | 0.006095 | 0.0098 | 0.024254 | 0.001565 | 0.003712 | - |
| Slope | 1.091311 | 0.292933 | 1.186136 | 0.592039 | 0.564903 | 1.498048 | - |
Table 3.
Investor's expected regret considering different target wealth, minimum expected return and proportional transaction costs
| Target wealth | 1000000 | 1050000 | 1100000 | ||||||
| Proportional transaction cost | 0 | 0.01 | 0.02 | 0 | 0.01 | 0.02 | 0 | 0.01 | 0.02 |
| 0.95 | 0 | 0 | 0 | 63429.6 | 85112.7 | 103678.9 | 157646.9 | 196749.4 | 223658.1 |
| 0.99 | 0 | 0 | 0 | 63429.6 | 85112.7 | 103678.9 | 157646.9 | 196749.4 | 223658.1 |
| 1.01 | 0 | 1290.1 | 2987.9 | 63429.6 | 85112.7 | 103678.9 | 157646.9 | 196749.4 | 223658.1 |
| 1.03 | 36.1 | 3953.9 | 9018.7 | 63429.6 | 85112.7 | 103678.9 | 157646.9 | 196749.4 | 223658.1 |
| 1.04 | 400.0 | 5567.1 | 12654.5 | 63429.6 | 85112.7 | 103678.9 | 157646.9 | 196749.4 | 223658.1 |
| 1.05 | 1142.6 | 7705.6 | 17739.6 | 63429.6 | 85112.7 | 104334.1 | 157646.9 | 196749.4 | 223658.1 |
| 1.06 | 2350.1 | 11121.8 | 26179.7 | 63429.6 | 85134.9 | 109571.7 | 157646.9 | 196749.4 | 223838.3 |
| 1.07 | 3904.2 | 15907.5 | 37152.7 | 63429.6 | 87879.1 | 118868.3 | 157646.9 | 198064.3 | 226873.5 |
| 1.08 | 5954.9 | 22562.9 | 49694.6 | 63429.6 | 95349.1 | 129104.5 | 157646.9 | 202106.9 | 232405.6 |
| 1.09 | 8525.3 | 36300.4 | 66271.2 | 63694.1 | 106415.1 | 140267.7 | 157646.9 | 208788.3 | 240209 |
| 1.10 | 12086.2 | 54243.7 | 88133.5 | 64838.1 | 120626.0 | 157811.5 | 157675.5 | 217600.2 | 251621.1 |
| 1.11 | 17279.9 | 74827.6 | - | 67119.8 | 138148.1 | - | 158591.5 | 228534.1 | - |
| 1.12 | 24358.6 | 98255.0 | - | 74520.5 | 159434.9 | - | 163337.6 | 242911.9 | - |
| 1.13 | 52656.5 | - | - | 104774.3 | - | - | 185917.5 | - | - |
| 1.14 | - | - | - | - | - | - | - | - | - |
| Target wealth | 1000000 | 1050000 | 1100000 | ||||||
| Proportional transaction cost | 0 | 0.01 | 0.02 | 0 | 0.01 | 0.02 | 0 | 0.01 | 0.02 |
| 0.95 | 0 | 0 | 0 | 63429.6 | 85112.7 | 103678.9 | 157646.9 | 196749.4 | 223658.1 |
| 0.99 | 0 | 0 | 0 | 63429.6 | 85112.7 | 103678.9 | 157646.9 | 196749.4 | 223658.1 |
| 1.01 | 0 | 1290.1 | 2987.9 | 63429.6 | 85112.7 | 103678.9 | 157646.9 | 196749.4 | 223658.1 |
| 1.03 | 36.1 | 3953.9 | 9018.7 | 63429.6 | 85112.7 | 103678.9 | 157646.9 | 196749.4 | 223658.1 |
| 1.04 | 400.0 | 5567.1 | 12654.5 | 63429.6 | 85112.7 | 103678.9 | 157646.9 | 196749.4 | 223658.1 |
| 1.05 | 1142.6 | 7705.6 | 17739.6 | 63429.6 | 85112.7 | 104334.1 | 157646.9 | 196749.4 | 223658.1 |
| 1.06 | 2350.1 | 11121.8 | 26179.7 | 63429.6 | 85134.9 | 109571.7 | 157646.9 | 196749.4 | 223838.3 |
| 1.07 | 3904.2 | 15907.5 | 37152.7 | 63429.6 | 87879.1 | 118868.3 | 157646.9 | 198064.3 | 226873.5 |
| 1.08 | 5954.9 | 22562.9 | 49694.6 | 63429.6 | 95349.1 | 129104.5 | 157646.9 | 202106.9 | 232405.6 |
| 1.09 | 8525.3 | 36300.4 | 66271.2 | 63694.1 | 106415.1 | 140267.7 | 157646.9 | 208788.3 | 240209 |
| 1.10 | 12086.2 | 54243.7 | 88133.5 | 64838.1 | 120626.0 | 157811.5 | 157675.5 | 217600.2 | 251621.1 |
| 1.11 | 17279.9 | 74827.6 | - | 67119.8 | 138148.1 | - | 158591.5 | 228534.1 | - |
| 1.12 | 24358.6 | 98255.0 | - | 74520.5 | 159434.9 | - | 163337.6 | 242911.9 | - |
| 1.13 | 52656.5 | - | - | 104774.3 | - | - | 185917.5 | - | - |
| 1.14 | - | - | - | - | - | - | - | - | - |
Table 4.
Investor's expected regret considering different target wealth and minimum expected return with and without cardinality
| Cardinality Constraints | No Cardinality Constraints | |||||
| Target wealth | 1000000 | 1050000 | 1100000 | 1000000 | 1050000 | 1100000 |
| 0.95 | 0 | 81742.31 | 192944.5 | 0 | 63429.62 | 157646.9 |
| 0.99 | 0 | 81742.31 | 192944.5 | 0 | 63429.62 | 157646.9 |
| 1 | 0 | 81742.31 | 192944.5 | 0 | 63429.62 | 157646.9 |
| 1.01 | 0 | 81742.31 | 192944.5 | 0 | 63429.62 | 157646.9 |
| 1.02 | 197.428 | 81742.31 | 192944.5 | 0 | 63429.62 | 157646.9 |
| 1.03 | 893.094 | 81742.31 | 192944.5 | 36.097 | 63429.62 | 157646.9 |
| 1.04 | 2574.389 | 81742.31 | 192944.5 | 399.947 | 63429.62 | 157646.9 |
| 1.05 | 5261.672 | 81879.22 | 192944.5 | 1142.637 | 63429.62 | 157646.9 |
| 1.06 | 8917.349 | 82803.35 | 192944.5 | 2350.142 | 63429.62 | 157646.9 |
| 1.07 | 18358.44 | 87336.35 | 193443 | 3904.241 | 63429.62 | 157646.9 |
| 1.08 | 35077.99 | 96174.55 | 198126.4 | 5954.918 | 63429.62 | 157646.9 |
| 1.09 | - | - | - | 8525.318 | 63694.05 | 157646.9 |
| 1.10 | - | - | - | 12086.15 | 64838.09 | 157675.5 |
| 1.11 | - | - | - | 17279.88 | 67119.82 | 158591.5 |
| 1.12 | - | - | - | 24358.63 | 74520.5 | 163337.6 |
| 1.13 | - | - | - | 52656.51 | 104774.3 | 185917.5 |
| 1.14 | - | - | - | - | - | - |
| Cardinality Constraints | No Cardinality Constraints | |||||
| Target wealth | 1000000 | 1050000 | 1100000 | 1000000 | 1050000 | 1100000 |
| 0.95 | 0 | 81742.31 | 192944.5 | 0 | 63429.62 | 157646.9 |
| 0.99 | 0 | 81742.31 | 192944.5 | 0 | 63429.62 | 157646.9 |
| 1 | 0 | 81742.31 | 192944.5 | 0 | 63429.62 | 157646.9 |
| 1.01 | 0 | 81742.31 | 192944.5 | 0 | 63429.62 | 157646.9 |
| 1.02 | 197.428 | 81742.31 | 192944.5 | 0 | 63429.62 | 157646.9 |
| 1.03 | 893.094 | 81742.31 | 192944.5 | 36.097 | 63429.62 | 157646.9 |
| 1.04 | 2574.389 | 81742.31 | 192944.5 | 399.947 | 63429.62 | 157646.9 |
| 1.05 | 5261.672 | 81879.22 | 192944.5 | 1142.637 | 63429.62 | 157646.9 |
| 1.06 | 8917.349 | 82803.35 | 192944.5 | 2350.142 | 63429.62 | 157646.9 |
| 1.07 | 18358.44 | 87336.35 | 193443 | 3904.241 | 63429.62 | 157646.9 |
| 1.08 | 35077.99 | 96174.55 | 198126.4 | 5954.918 | 63429.62 | 157646.9 |
| 1.09 | - | - | - | 8525.318 | 63694.05 | 157646.9 |
| 1.10 | - | - | - | 12086.15 | 64838.09 | 157675.5 |
| 1.11 | - | - | - | 17279.88 | 67119.82 | 158591.5 |
| 1.12 | - | - | - | 24358.63 | 74520.5 | 163337.6 |
| 1.13 | - | - | - | 52656.51 | 104774.3 | 185917.5 |
| 1.14 | - | - | - | - | - | - |
Table 5.
Investor's expected regret considering different proportional transaction costs and number of assets
| Number of assets | 6 | 12 | ||||
| Proportional transaction cost | 0 | 0.01 | 0.02 | 0 | 0.01 | 0.02 |
| 0.95 | 229314.2 | 275191.5 | 318875.4 | 192944.5 | 225372.1 | 258752.1 |
| 0.99 | 229314.2 | 275191.5 | 318875.4 | 192944.5 | 225372.1 | 258752.1 |
| 1.01 | 229314.2 | 275191.5 | 318875.4 | 192944.5 | 225372.1 | 258752.1 |
| 1.03 | 229314.2 | 275191.5 | 318875.4 | 192944.5 | 225372.1 | 258752.1 |
| 1.04 | 229314.2 | 275191.5 | 318875.4 | 192944.5 | 225372.1 | 258752.1 |
| 1.05 | 229314.2 | 275191.5 | 320958.9 | 192944.5 | 225372.1 | 258752.1 |
| 1.06 | 229314.2 | 277364.2 | 338961.2 | 192944.5 | 225372.1 | 258752.1 |
| 1.07 | 229314.2 | 280367.1 | - | 192944.5 | 230553.7 | 263452.1 |
| 1.08 | 229314.3 | - | - | 192944.5 | 235638.9 | - |
| 1.09 | 231175.9 | - | - | 193443.0 | 239987.4 | - |
| 1.10 | - | - | - | 198126.4 | - | - |
| 1.11 | - | - | - | - | - | - |
| Number of assets | 6 | 12 | ||||
| Proportional transaction cost | 0 | 0.01 | 0.02 | 0 | 0.01 | 0.02 |
| 0.95 | 229314.2 | 275191.5 | 318875.4 | 192944.5 | 225372.1 | 258752.1 |
| 0.99 | 229314.2 | 275191.5 | 318875.4 | 192944.5 | 225372.1 | 258752.1 |
| 1.01 | 229314.2 | 275191.5 | 318875.4 | 192944.5 | 225372.1 | 258752.1 |
| 1.03 | 229314.2 | 275191.5 | 318875.4 | 192944.5 | 225372.1 | 258752.1 |
| 1.04 | 229314.2 | 275191.5 | 318875.4 | 192944.5 | 225372.1 | 258752.1 |
| 1.05 | 229314.2 | 275191.5 | 320958.9 | 192944.5 | 225372.1 | 258752.1 |
| 1.06 | 229314.2 | 277364.2 | 338961.2 | 192944.5 | 225372.1 | 258752.1 |
| 1.07 | 229314.2 | 280367.1 | - | 192944.5 | 230553.7 | 263452.1 |
| 1.08 | 229314.3 | - | - | 192944.5 | 235638.9 | - |
| 1.09 | 231175.9 | - | - | 193443.0 | 239987.4 | - |
| 1.10 | - | - | - | 198126.4 | - | - |
| 1.11 | - | - | - | - | - | - |
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