Figure 1.
Returns of market index S & P Asia 50 (DIII) in one specific 27 weeks in-sample period from 13/10/2017 to 13/04/2018 to indicate different distributions followed by returns
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July
2021, 17(4): 1613-1637.
doi: 10.3934/jimo.2020037
Worst-case analysis of Gini mean difference safety measure
Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India |
The paper introduces the worst-case portfolio optimization models within the robust optimization framework for maximizing return through either the mean or median metrics. The risk in the portfolio is quantified by Gini mean difference. We put forward the worst-case models under the mixed and interval+polyhedral uncertainty sets. The proposed models turn out to be linear and mixed integer linear programs under the mixed uncertainty set, and semidefinite program under interval+polyhedral uncertainty set. The performance comparison of the proposed models on the listed stocks of Euro Stoxx 50, Dow Jones Global Titans 50, S & P Asia 50, consistently exhibit advantage over their conventional non-robust counterpart models on various risk parameters including the standard deviation, worst return, value at risk, conditional value at risk and maximum drawdown of the portfolio.
Keywords:
Portfolio selection robust optimization,
Gini mean difference,
mixed uncertainty,
interval+ polyhedral uncertainty.
Mathematics Subject Classification:
Primary: 91G10, 91B30, 62G35; Secondary: 90C90, 90C05, 90C11.
Citation:
Ruchika Sehgal, Aparna Mehra. Worst-case analysis of Gini mean difference safety measure. Journal of Industrial and Management Optimization,
2021, 17
(4)
: 1613-1637.
doi: 10.3934/jimo.2020037
![]()
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Robust portfolio selection for index tracking, Comput. Oper. Res., 39 (2012), 829-837.
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Robust portfolio selection based on asymmetric measures of variability of stock returns, J. Comput. Appl. Math., 232 (2009), 295-304.
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Worst-case value-at-risk and robust portfolio optimization: A conic programming approach, Oper. Res., 51 (2003), 543-556.
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Practical algorithms for value-at-risk portfolio optimization problems, Quantitative Finance Lett., 3 (2015), 1-9.
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Properties, formulations, and algorithms for portfolio optimization using mean-Gini criteria, Ann. Oper. Res., 248 (2017), 305-343.
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M. Kapsos, N. Christofides and B. Rustem,
Worst-case robust Omega ratio, European J. Oper. Res., 234 (2014), 499-507.
doi: 10.1016/j.ejor.2013.04.025. |
| [21] |
G. Kara, A. Özmen and G.-W. Weber,
Stability advances in robust portfolio optimization under parallelepiped uncertainty, CEJOR Cent. Eur. J. Oper. Res., 27 (2019), 241-261.
doi: 10.1007/s10100-017-0508-5. |
| [22] |
C. Keating and W. F. Shadwick,
A universal performance measure, J. Performance Measurement, 6 (2002), 59-84.
|
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H. Konno and H. Yamazaki,
Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Management Sci., 37 (1991), 519-531.
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| [24] |
R. H. Kwon and D. Wu,
Factor-based robust index tracking, Optim. Eng., 18 (2017), 443-466.
doi: 10.1007/s11081-016-9314-5. |
| [25] |
P. Li, Y. Han and Y. Xia,
Portfolio optimization using asymmetry robust mean absolute deviation model, Finance Res. Lett., 18 (2016), 353-362.
doi: 10.1016/j.frl.2016.05.014. |
| [26] |
B. G. Lindsay, Mixture models: Theory, geometry and applications, in NSF-CBMS Regional Conference Series in Probability and Statistics, (1995), 1–163. |
| [27] |
S.-T. Liu,
The mean-absolute deviation portfolio selection problem with interval-valued returns, J. Comput. Appl. Math., 235 (2011), 4149-4157.
doi: 10.1016/j.cam.2011.03.008. |
| [28] |
R. Mansini, W. Ogryczak and M. G. Speranza,
Conditional value at risk and related linear programming models for portfolio optimization, Ann. Oper. Res., 152 (2007), 227-256.
doi: 10.1007/s10479-006-0142-4. |
| [29] |
R. Mansini, W. Ogryczak and M. G. Speranza, Tail Gini's risk measures and related linear programming models for portfolio optimization, in HERCMA Conference Proceedings, CD, LEA Publishers, Athens, 2007. |
| [30] |
H. Markowitz,
Portfolio selection, J. Finance, 7 (1952), 77-91.
doi: 10.1111/j.1540-6261.1952.tb01525.x. |
| [31] |
Y. Moon and T. Yao,
A robust mean absolute deviation model for portfolio optimization, Comput. Oper. Res., 38 (2011), 1251-1258.
doi: 10.1016/j.cor.2010.10.020. |
| [32] |
K. Natarajan, D. Pachamanova and M. Sim,
Constructing risk measures from uncertainty sets, Oper. Res., 57 (2009), 1129-1141.
doi: 10.1287/opre.1080.0683. |
| [33] |
W. Ogryczak, Risk measurement: Mean absolute deviation versus Gini's mean difference, in Decision Theory and Optimization in Theory and Practice–Proc. 9th Workshop GOR WG Chemnitz, 1999, 33–51. |
| [34] |
D. Peel and G. J. McLachlan,
Robust mixture modelling using the t distribution, Statistics Comput., 10 (2000), 339-348.
|
| [35] |
K. Postek, D. den Hertog and B. Melenberg,
Computationally tractable counterparts of distributionally robust constraints on risk measures, SIAM Rev., 58 (2016), 603-650.
doi: 10.1137/151005221. |
| [36] |
R. T. Rockafellar and S. Uryasev,
Optimization of conditional value-at-risk, J. Risk, 2 (2000), 21-42.
doi: 10.1007/978-1-4757-6594-6_17. |
| [37] |
M. Rudolf, H.-J. Wolter and H. Zimmermann,
A linear model for tracking error minimization, J. Banking Finance, 23 (1999), 85-103.
doi: 10.1016/S0378-4266(98)00076-4. |
| [38] |
R. Sehgal and A. Mehra, Robust reward–risk ratio portfolio optimization, Internat. Transactions Oper. Res., (2019).
doi: 10.1111/itor.12652. |
| [39] |
R. N. Sengupta and R. Kumar,
Robust and reliable portfolio optimization formulation of a chance constrained problem, Foundations Comput. Decision Sci., 42 (2017), 83-117.
doi: 10.1515/fcds-2017-0004. |
| [40] |
H. Shalit and S. Yitzhaki,
Mean-Gini, portfolio theory, and the pricing of risky assets, J. Finance, 39 (1984), 1449-1468.
doi: 10.1111/j.1540-6261.1984.tb04917.x. |
| [41] |
H. Shalit and S. Yitzhaki,
The mean-Gini efficient portfolio frontier, J. Financial Res., 28 (2005), 59-75.
doi: 10.1111/j.1475-6803.2005.00114.x. |
| [42] |
A. Sharma, S. Agrawal and A. Mehra,
Enhanced indexing for risk averse investors using relaxed second order stochastic dominance, Optim. Eng., 18 (2017), 407-442.
doi: 10.1007/s11081-016-9329-y. |
| [43] |
A. Sharma, S. Utz and A. Mehra,
Omega-CVaR portfolio optimization and its worst case analysis, OR Spectrum, 39 (2017), 505-539.
doi: 10.1007/s00291-016-0462-y. |
| [44] |
W. F. Sharpe, Mean-absolute-deviation characteristic lines for securities and portfolios, Management Sci., 18 (1971), B–1.
doi: 10.1287/mnsc.18.2.B1. |
| [45] |
S. Yitzhaki,
Stochastic dominance, mean variance, and Gini's mean difference, American Economic Review, 72 (1982), 178-185.
|
| [46] |
M. R. Young,
A minimax portfolio selection rule with linear programming solution, Management Sci., 44 (1998), 673-683.
doi: 10.1287/mnsc.44.5.673. |
| [47] |
X. Zheng, X. Sun, D. Li and Y. Xu,
On zero duality gap in nonconvex quadratic programming problems, J. Global Optim., 52 (2012), 229-242.
doi: 10.1007/s10898-011-9660-y. |
| [48] |
S. Zhu and M. Fukushima,
Worst-case conditional value-at-risk with application to robust portfolio management, Oper. Res., 57 (2009), 1155-1168.
doi: 10.1287/opre.1080.0684. |
| [49] |
S. Zhu, D. Li and S. Wang,
Robust portfolio selection under downside risk measures, Quant. Finance, 9 (2009), 869-885.
doi: 10.1080/14697680902852746. |
show all references
References:
| [1] |
P. Artzner, F. Delbaen, J.-M. Eber and D. Heath,
Coherent measures of risk, Math. Finance, 9 (1999), 203-228.
doi: 10.1111/1467-9965.00068. |
| [2] |
A. Ben-Tal, D. Den Hertog and J.-P. Vial,
Deriving robust counterparts of nonlinear uncertain inequalities, Math. Program., 149 (2015), 265-299.
doi: 10.1007/s10107-014-0750-8. |
| [3] |
S. Benati,
Using medians in portfolio optimization, J. Oper. Res. Soc., 66 (2015), 720-731.
doi: 10.1057/jors.2014.57. |
| [4] |
M. Berkhouch, G. Lakhnati and M. B. Righi,
Extended gini-type measures of risk and variability, Appl. Math. Finance, 25 (2018), 295-314.
doi: 10.1080/1350486X.2018.1538806. |
| [5] |
D. Bertsimas and M. Sim,
The price of robustness, Oper. Res., 52 (2004), 35-53.
doi: 10.1287/opre.1030.0065. |
| [6] |
M. J. Best and R. R. Grauer,
On the sensitivity of mean-variance-efficient portfolios to changes in asset means: Some analytical and computational results, Rev. Financial Studies, 4 (1991), 315-342.
doi: 10.1093/rfs/4.2.315. |
| [7] |
F. Black and R. Litterman,
Global portfolio optimization, Financial Analysts Journal, 48 (1992), 28-43.
doi: 10.2469/faj.v48.n5.28. |
| [8] |
B. Bower and P. Wentz, Portfolio optimization: MAD vs. Markowitz, Rose-Hulman Undergraduate Mathematics Journal, 6 (2005), 3. |
| [9] |
C. Chen and R. H. Kwon,
Robust portfolio selection for index tracking, Comput. Oper. Res., 39 (2012), 829-837.
doi: 10.1016/j.cor.2010.08.019. |
| [10] |
W. Chen and S. Tan,
Robust portfolio selection based on asymmetric measures of variability of stock returns, J. Comput. Appl. Math., 232 (2009), 295-304.
doi: 10.1016/j.cam.2009.06.010. |
| [11] |
L. El Ghaoui, M. Oks and F. Oustry,
Worst-case value-at-risk and robust portfolio optimization: A conic programming approach, Oper. Res., 51 (2003), 543-556.
doi: 10.1287/opre.51.4.543.16101. |
| [12] |
M. Feng, A. Wächter and J. Staum,
Practical algorithms for value-at-risk portfolio optimization problems, Quantitative Finance Lett., 3 (2015), 1-9.
doi: 10.1080/21649502.2014.995214. |
| [13] |
E. Furman, R. Wang and R. Zitikis,
Gini-type measures of risk and variability: Gini shortfall, capital allocations, and heavy-tailed risks, J. Banking Finance, 83 (2017), 70-84.
doi: 10.2139/ssrn.2836281. |
| [14] |
C. Gerstenberger and D. Vogel,
On the efficiency of Gini's mean difference, Stat. Methods Appl., 24 (2015), 569-596.
doi: 10.1007/s10260-015-0315-x. |
| [15] |
M. Gharakhani, F. Zarea Fazlelahi and S. Sadjadi,
A robust optimization approach for index tracking problem, J. Computer Sci., 10 (2014), 2450-2463.
doi: 10.3844/jcssp.2014.2450.2463. |
| [16] |
D. Goldfarb and G. Iyengar,
Robust portfolio selection problems, Math. Oper. Res., 28 (2003), 1-38.
doi: 10.1287/moor.28.1.1.14260. |
| [17] |
J.-Y. Gotoh, K. Shinozaki and A. Takeda,
Robust portfolio techniques for mitigating the fragility of CVaR minimization and generalization to coherent risk measures, Quant. Finance, 13 (2013), 1621-1635.
doi: 10.1080/14697688.2012.738930. |
| [18] |
J. A. Hall, B. W. Brorsen and S. H. Irwin,
The distribution of futures prices: A test of the stable paretian and mixture of normals hypotheses, J. Financial Quantitative Anal., 24 (1989), 105-116.
doi: 10.2307/2330751. |
| [19] |
R. Ji, M. A. Lejeune and S. Y. Prasad,
Properties, formulations, and algorithms for portfolio optimization using mean-Gini criteria, Ann. Oper. Res., 248 (2017), 305-343.
doi: 10.1007/s10479-016-2230-4. |
| [20] |
M. Kapsos, N. Christofides and B. Rustem,
Worst-case robust Omega ratio, European J. Oper. Res., 234 (2014), 499-507.
doi: 10.1016/j.ejor.2013.04.025. |
| [21] |
G. Kara, A. Özmen and G.-W. Weber,
Stability advances in robust portfolio optimization under parallelepiped uncertainty, CEJOR Cent. Eur. J. Oper. Res., 27 (2019), 241-261.
doi: 10.1007/s10100-017-0508-5. |
| [22] |
C. Keating and W. F. Shadwick,
A universal performance measure, J. Performance Measurement, 6 (2002), 59-84.
|
| [23] |
H. Konno and H. Yamazaki,
Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Management Sci., 37 (1991), 519-531.
doi: 10.1287/mnsc.37.5.519. |
| [24] |
R. H. Kwon and D. Wu,
Factor-based robust index tracking, Optim. Eng., 18 (2017), 443-466.
doi: 10.1007/s11081-016-9314-5. |
| [25] |
P. Li, Y. Han and Y. Xia,
Portfolio optimization using asymmetry robust mean absolute deviation model, Finance Res. Lett., 18 (2016), 353-362.
doi: 10.1016/j.frl.2016.05.014. |
| [26] |
B. G. Lindsay, Mixture models: Theory, geometry and applications, in NSF-CBMS Regional Conference Series in Probability and Statistics, (1995), 1–163. |
| [27] |
S.-T. Liu,
The mean-absolute deviation portfolio selection problem with interval-valued returns, J. Comput. Appl. Math., 235 (2011), 4149-4157.
doi: 10.1016/j.cam.2011.03.008. |
| [28] |
R. Mansini, W. Ogryczak and M. G. Speranza,
Conditional value at risk and related linear programming models for portfolio optimization, Ann. Oper. Res., 152 (2007), 227-256.
doi: 10.1007/s10479-006-0142-4. |
| [29] |
R. Mansini, W. Ogryczak and M. G. Speranza, Tail Gini's risk measures and related linear programming models for portfolio optimization, in HERCMA Conference Proceedings, CD, LEA Publishers, Athens, 2007. |
| [30] |
H. Markowitz,
Portfolio selection, J. Finance, 7 (1952), 77-91.
doi: 10.1111/j.1540-6261.1952.tb01525.x. |
| [31] |
Y. Moon and T. Yao,
A robust mean absolute deviation model for portfolio optimization, Comput. Oper. Res., 38 (2011), 1251-1258.
doi: 10.1016/j.cor.2010.10.020. |
| [32] |
K. Natarajan, D. Pachamanova and M. Sim,
Constructing risk measures from uncertainty sets, Oper. Res., 57 (2009), 1129-1141.
doi: 10.1287/opre.1080.0683. |
| [33] |
W. Ogryczak, Risk measurement: Mean absolute deviation versus Gini's mean difference, in Decision Theory and Optimization in Theory and Practice–Proc. 9th Workshop GOR WG Chemnitz, 1999, 33–51. |
| [34] |
D. Peel and G. J. McLachlan,
Robust mixture modelling using the t distribution, Statistics Comput., 10 (2000), 339-348.
|
| [35] |
K. Postek, D. den Hertog and B. Melenberg,
Computationally tractable counterparts of distributionally robust constraints on risk measures, SIAM Rev., 58 (2016), 603-650.
doi: 10.1137/151005221. |
| [36] |
R. T. Rockafellar and S. Uryasev,
Optimization of conditional value-at-risk, J. Risk, 2 (2000), 21-42.
doi: 10.1007/978-1-4757-6594-6_17. |
| [37] |
M. Rudolf, H.-J. Wolter and H. Zimmermann,
A linear model for tracking error minimization, J. Banking Finance, 23 (1999), 85-103.
doi: 10.1016/S0378-4266(98)00076-4. |
| [38] |
R. Sehgal and A. Mehra, Robust reward–risk ratio portfolio optimization, Internat. Transactions Oper. Res., (2019).
doi: 10.1111/itor.12652. |
| [39] |
R. N. Sengupta and R. Kumar,
Robust and reliable portfolio optimization formulation of a chance constrained problem, Foundations Comput. Decision Sci., 42 (2017), 83-117.
doi: 10.1515/fcds-2017-0004. |
| [40] |
H. Shalit and S. Yitzhaki,
Mean-Gini, portfolio theory, and the pricing of risky assets, J. Finance, 39 (1984), 1449-1468.
doi: 10.1111/j.1540-6261.1984.tb04917.x. |
| [41] |
H. Shalit and S. Yitzhaki,
The mean-Gini efficient portfolio frontier, J. Financial Res., 28 (2005), 59-75.
doi: 10.1111/j.1475-6803.2005.00114.x. |
| [42] |
A. Sharma, S. Agrawal and A. Mehra,
Enhanced indexing for risk averse investors using relaxed second order stochastic dominance, Optim. Eng., 18 (2017), 407-442.
doi: 10.1007/s11081-016-9329-y. |
| [43] |
A. Sharma, S. Utz and A. Mehra,
Omega-CVaR portfolio optimization and its worst case analysis, OR Spectrum, 39 (2017), 505-539.
doi: 10.1007/s00291-016-0462-y. |
| [44] |
W. F. Sharpe, Mean-absolute-deviation characteristic lines for securities and portfolios, Management Sci., 18 (1971), B–1.
doi: 10.1287/mnsc.18.2.B1. |
| [45] |
S. Yitzhaki,
Stochastic dominance, mean variance, and Gini's mean difference, American Economic Review, 72 (1982), 178-185.
|
| [46] |
M. R. Young,
A minimax portfolio selection rule with linear programming solution, Management Sci., 44 (1998), 673-683.
doi: 10.1287/mnsc.44.5.673. |
| [47] |
X. Zheng, X. Sun, D. Li and Y. Xu,
On zero duality gap in nonconvex quadratic programming problems, J. Global Optim., 52 (2012), 229-242.
doi: 10.1007/s10898-011-9660-y. |
| [48] |
S. Zhu and M. Fukushima,
Worst-case conditional value-at-risk with application to robust portfolio management, Oper. Res., 57 (2009), 1155-1168.
doi: 10.1287/opre.1080.0684. |
| [49] |
S. Zhu, D. Li and S. Wang,
Robust portfolio selection under downside risk measures, Quant. Finance, 9 (2009), 869-885.
doi: 10.1080/14697680902852746. |
Figure 2.
Out-of-sample downside risk of $ (DIII) $ for (a) $ (\mu GMD) $ and $ (R\mu GMD) $ model and (b) $ (MGMD) $ and $ (RMGMD) $ model
Figure 3.
Cumulative returns of market indices corresponding to $ (DI) $ , $ (DII) $ , $ D(III) $ and $ (DIV) $ depicting different phases of markets
Figure 4.
Out-of-sample cumulative returns of UP-DOWN phase of $ (DIII) $ for (a) $ (\mu GMD) $ and $ (R\mu GMD) $ model and (b) $ (MGMD) $ and $ (RMGMD) $ model
Figure 5.
Out-of-sample downside risk of UP-DOWN phase of $ (DIII) $ for (a) $ (\mu GMD) $ and $ (R\mu GMD) $ model and (b) $ (MGMD) $ and $ (RMGMD) $ model
Table 1.
Statistics of returns of market index S & P Asia 50 from 13/10/2017 to 13/04/2018
| period | weekly average |
weakly sd |
| 1-9 weeks | 3.606 | 19.559 |
| 10-18 weeks | 7.111 | 42.395 |
| 19-27 weeks | -2.078 | 23.281 |
| period | weekly average |
weakly sd |
| 1-9 weeks | 3.606 | 19.559 |
| 10-18 weeks | 7.111 | 42.395 |
| 19-27 weeks | -2.078 | 23.281 |
Table 2.
Out-of-sample statistics($ \times 10^{-3} $ ) of $ (DI), (DII), (DIII) \; \text{and}\; (DIV) $ on rolling window analysis under mixed uncertainty set $ Q_M $
| average | 0.458 | 1.541 | -0.143 | 0.127 | |
| sd | 21.973 | 22.432 | 21.183 | 22.164 | |
| median | -0.404 | 3.035 | 1.032 | -0.717 | |
| min | -59.257 | -67.184 | -57.742 | -51.915 | |
| (DI) | max | 55.669 | 63.717 | 57.2492 | 65.108 |
| neg returns | 80 | 72 | 77 | 82 | |
| 53.369 | 55.862 | 50.934 | 49.827 | ||
| 50.236 | 48.982 | 46.816 | 47.84 | ||
| 49.945 | 44.695 | 43.091 | 46.16 | ||
| 40.909 | 35.241 | 38.74 | 43.921 | ||
| MD | 213.407 | 156.96 | 173.27 | 224.3 | |
| average | 5.348 | 5.315 | 3.652 | 3.934 | |
| sd | 24.384 | 23.277 | 19.905 | 21.320 | |
| median | 7.812 | 5.212 | 5.081 | 5.634 | |
| min | -69.647 | -82.617 | -71.400 | -52.323 | |
| (DII) | max | 145.054 | 104.310 | 63.899 | 81.426 |
| neg returns | 56 | 57 | 55 | 61 | |
| 57.914 | 53.802 | 57.633 | 47.689 | ||
| 49.836 | 47.188 | 48.391 | 43.867 | ||
| 43.523 | 38.888 | 37.368 | 41.209 | ||
| 33.153 | 34.584 | 31.053 | 37.39 | ||
| MD | 118.11 | 146.64 | 111.057 | 149.49 | |
| average | 4.188 | 5.561 | 3.968 | 2.365 | |
| sd | 23.907 | 30.314 | 19.597 | 24.278 | |
| median | 2.817 | 3.380 | 4.128 | 2.592 | |
| min | -93.670 | -108.797 | -82.422 | -88.448 | |
| (DIII) | max | 87.815 | 79.020 | 62.667 | 74.162 |
| neg returns | 68 | 72 | 62 | 70 | |
| 61.937 | 73.741 | 52.024 | 63.487 | ||
| 51.816 | 61.994 | 40.559 | 52.514 | ||
| 46.458 | 47.761 | 28.247 | 51.358 | ||
| 28.933 | 41.034 | 22.291 | 32.79 | ||
| MD | 157.024 | 106.34 | 190.67 | 188.84 | |
| average | 1.926 | 1.852 | 1.120 | 0.232 | |
| sd | 23.337 | 25.528 | 21.178 | 23.817 | |
| median | 3.853 | 2.276 | 1.655 | 1.307 | |
| min | -65.330 | -80.298 | -67.686 | -68.472 | |
| (DIV) | max | 54.355 | 72.800 | 50.498 | 60.039 |
| neg returns | 71 | 70 | 73 | 76 | |
| 57.727 | 61.945 | 58.193 | 58.013 | ||
| 50.202 | 56.738 | 50.081 | 52.9 | ||
| 52.377 | 52.549 | 48.15 | 48.715 | ||
| 36.036 | 48.283 | 31.815 | 43.657 | ||
| MD | 180.98 | 233.87 | 170.43 | 191.327 |
| average | 0.458 | 1.541 | -0.143 | 0.127 | |
| sd | 21.973 | 22.432 | 21.183 | 22.164 | |
| median | -0.404 | 3.035 | 1.032 | -0.717 | |
| min | -59.257 | -67.184 | -57.742 | -51.915 | |
| (DI) | max | 55.669 | 63.717 | 57.2492 | 65.108 |
| neg returns | 80 | 72 | 77 | 82 | |
| 53.369 | 55.862 | 50.934 | 49.827 | ||
| 50.236 | 48.982 | 46.816 | 47.84 | ||
| 49.945 | 44.695 | 43.091 | 46.16 | ||
| 40.909 | 35.241 | 38.74 | 43.921 | ||
| MD | 213.407 | 156.96 | 173.27 | 224.3 | |
| average | 5.348 | 5.315 | 3.652 | 3.934 | |
| sd | 24.384 | 23.277 | 19.905 | 21.320 | |
| median | 7.812 | 5.212 | 5.081 | 5.634 | |
| min | -69.647 | -82.617 | -71.400 | -52.323 | |
| (DII) | max | 145.054 | 104.310 | 63.899 | 81.426 |
| neg returns | 56 | 57 | 55 | 61 | |
| 57.914 | 53.802 | 57.633 | 47.689 | ||
| 49.836 | 47.188 | 48.391 | 43.867 | ||
| 43.523 | 38.888 | 37.368 | 41.209 | ||
| 33.153 | 34.584 | 31.053 | 37.39 | ||
| MD | 118.11 | 146.64 | 111.057 | 149.49 | |
| average | 4.188 | 5.561 | 3.968 | 2.365 | |
| sd | 23.907 | 30.314 | 19.597 | 24.278 | |
| median | 2.817 | 3.380 | 4.128 | 2.592 | |
| min | -93.670 | -108.797 | -82.422 | -88.448 | |
| (DIII) | max | 87.815 | 79.020 | 62.667 | 74.162 |
| neg returns | 68 | 72 | 62 | 70 | |
| 61.937 | 73.741 | 52.024 | 63.487 | ||
| 51.816 | 61.994 | 40.559 | 52.514 | ||
| 46.458 | 47.761 | 28.247 | 51.358 | ||
| 28.933 | 41.034 | 22.291 | 32.79 | ||
| MD | 157.024 | 106.34 | 190.67 | 188.84 | |
| average | 1.926 | 1.852 | 1.120 | 0.232 | |
| sd | 23.337 | 25.528 | 21.178 | 23.817 | |
| median | 3.853 | 2.276 | 1.655 | 1.307 | |
| min | -65.330 | -80.298 | -67.686 | -68.472 | |
| (DIV) | max | 54.355 | 72.800 | 50.498 | 60.039 |
| neg returns | 71 | 70 | 73 | 76 | |
| 57.727 | 61.945 | 58.193 | 58.013 | ||
| 50.202 | 56.738 | 50.081 | 52.9 | ||
| 52.377 | 52.549 | 48.15 | 48.715 | ||
| 36.036 | 48.283 | 31.815 | 43.657 | ||
| MD | 180.98 | 233.87 | 170.43 | 191.327 |
Table 3.
Out-of-sample statistics(×10−3) of (DI) and (DII) under mixed uncertainty set QM for market directions based four phases data
| (DI) | |||||
| average | -6.039 | -6.859 | 1.1184 | -4.035 | |
| sd | 43.196 | 52.886 | 24.373 | 40.885 | |
| median | -19.16 | -18.64 | -2.775 | -10.72 | |
| UP-DOWN | min | -80.7 | -87.88 | -41.22 | -64.31 |
| max | 112.33 | 139.21 | 67.997 | 109.74 | |
| neg returns | 12 | 13 | 11 | 12 | |
| MD | 151.538 | 151.107 | 52.69 | 95.87 | |
| average | -3.582 | -1.58 | -3.62 | -2.333 | |
| sd | 39.36 | 44.553 | 38.817 | 39.861 | |
| median | -8.466 | -3.124 | -7.898 | -7.301 | |
| DOWN-DOWN | min | -76.31 | -122.3 | -77.25 | -93.32 |
| max | 116.87 | 73.054 | 113.99 | 80.892 | |
| neg returns | 12 | 12 | 12 | 11 | |
| MD | 209.357 | 241.872 | 206.61 | 240.81 | |
| average | 4.5948 | 6.5533 | 6.3614 | 9.0281 | |
| sd | 17.897 | 24.9 | 18.545 | 21.693 | |
| median | 0.7725 | 4.0728 | 6.5631 | 11.843 | |
| DOWN-UP | min | -26.7 | -32.28 | -17.9 | -29.67 |
| max | 42.982 | 60.549 | 50.299 | 37.346 | |
| neg returns | 9 | 7 | 9 | 7 | |
| MD | 36.5 | 41.1155 | 31.584 | 29.673 | |
| average | 4.5768 | 4.0462 | 5.1412 | 5.5771 | |
| sd | 19.647 | 28.427 | 18.428 | 16.52 | |
| median | 2.6147 | 5.4864 | 0.6005 | 6.9434 | |
| UP-UP | min | -28.86 | -39.84 | -25.75 | -35.76 |
| max | 36.136 | 50.381 | 37.729 | 29.958 | |
| neg returns | 10 | 8 | 10 | 6 | |
| MD | 48.181 | 87.913 | 35.078 | 40.38 | |
| (DII) | |||||
| average | -5.029 | -3.828 | -5.007 | 1.4072 | |
| sd | 66.172 | 72.696 | 61.369 | 58.82 | |
| median | -5.895 | -0.19 | 8.9474 | 3.7326 | |
| UP-DOWN | min | -179.4 | -193.8 | -154.4 | -150.6 |
| max | 129.46 | 146.11 | 126.09 | 140.28 | |
| neg returns | 10 | 10 | 9 | 9 | |
| MD | 235.221 | 236.219 | 222.13 | 150.582 | |
| average | -1.545 | 0.006 | -2.109 | -2.249 | |
| sd | 34.09 | 30.876 | 35.189 | 33.007 | |
| median | -4.819 | -3.358 | -4.806 | -4.546 | |
| DOWN-DOWN | min | -67.06 | -48.79 | -61.39 | -90.7 |
| max | 115.75 | 72.059 | 122.57 | 76.009 | |
| neg returns | 11 | 11 | 12 | 13 | |
| MD | 162.256 | 113.9822 | 173.88 | 159.283 | |
| average | -6.301 | -6.031 | -2.023 | -4.701 | |
| sd | 17.096 | 24.228 | 15.888 | 19.831 | |
| median | -4.875 | -9.129 | -3.986 | -6.138 | |
| DOWN-UP | min | -37.04 | -47 | -33.08 | -39.68 |
| max | 28.022 | 53.495 | 38.54 | 58.312 | |
| neg returns | 15 | 12 | 13 | 14 | |
| MD | 132.87 | 142.629 | 92.60 | 119.743 | |
| average | 2.7559 | 2.8771 | 5.6723 | 7.6174 | |
| sd | 21.022 | 28.97 | 12.812 | 18.045 | |
| median | 5.1358 | 2.4316 | 4.687 | 4.2454 | |
| UP-UP | min | -38.74 | -46.22 | -18.15 | -31.4 |
| max | 51.007 | 65.855 | 36.726 | 48.432 | |
| neg returns | 9 | 8 | 6 | 6 | |
| MD | 47.811 | 109.512 | 18.517 | 36.649 |
| (DI) | |||||
| average | -6.039 | -6.859 | 1.1184 | -4.035 | |
| sd | 43.196 | 52.886 | 24.373 | 40.885 | |
| median | -19.16 | -18.64 | -2.775 | -10.72 | |
| UP-DOWN | min | -80.7 | -87.88 | -41.22 | -64.31 |
| max | 112.33 | 139.21 | 67.997 | 109.74 | |
| neg returns | 12 | 13 | 11 | 12 | |
| MD | 151.538 | 151.107 | 52.69 | 95.87 | |
| average | -3.582 | -1.58 | -3.62 | -2.333 | |
| sd | 39.36 | 44.553 | 38.817 | 39.861 | |
| median | -8.466 | -3.124 | -7.898 | -7.301 | |
| DOWN-DOWN | min | -76.31 | -122.3 | -77.25 | -93.32 |
| max | 116.87 | 73.054 | 113.99 | 80.892 | |
| neg returns | 12 | 12 | 12 | 11 | |
| MD | 209.357 | 241.872 | 206.61 | 240.81 | |
| average | 4.5948 | 6.5533 | 6.3614 | 9.0281 | |
| sd | 17.897 | 24.9 | 18.545 | 21.693 | |
| median | 0.7725 | 4.0728 | 6.5631 | 11.843 | |
| DOWN-UP | min | -26.7 | -32.28 | -17.9 | -29.67 |
| max | 42.982 | 60.549 | 50.299 | 37.346 | |
| neg returns | 9 | 7 | 9 | 7 | |
| MD | 36.5 | 41.1155 | 31.584 | 29.673 | |
| average | 4.5768 | 4.0462 | 5.1412 | 5.5771 | |
| sd | 19.647 | 28.427 | 18.428 | 16.52 | |
| median | 2.6147 | 5.4864 | 0.6005 | 6.9434 | |
| UP-UP | min | -28.86 | -39.84 | -25.75 | -35.76 |
| max | 36.136 | 50.381 | 37.729 | 29.958 | |
| neg returns | 10 | 8 | 10 | 6 | |
| MD | 48.181 | 87.913 | 35.078 | 40.38 | |
| (DII) | |||||
| average | -5.029 | -3.828 | -5.007 | 1.4072 | |
| sd | 66.172 | 72.696 | 61.369 | 58.82 | |
| median | -5.895 | -0.19 | 8.9474 | 3.7326 | |
| UP-DOWN | min | -179.4 | -193.8 | -154.4 | -150.6 |
| max | 129.46 | 146.11 | 126.09 | 140.28 | |
| neg returns | 10 | 10 | 9 | 9 | |
| MD | 235.221 | 236.219 | 222.13 | 150.582 | |
| average | -1.545 | 0.006 | -2.109 | -2.249 | |
| sd | 34.09 | 30.876 | 35.189 | 33.007 | |
| median | -4.819 | -3.358 | -4.806 | -4.546 | |
| DOWN-DOWN | min | -67.06 | -48.79 | -61.39 | -90.7 |
| max | 115.75 | 72.059 | 122.57 | 76.009 | |
| neg returns | 11 | 11 | 12 | 13 | |
| MD | 162.256 | 113.9822 | 173.88 | 159.283 | |
| average | -6.301 | -6.031 | -2.023 | -4.701 | |
| sd | 17.096 | 24.228 | 15.888 | 19.831 | |
| median | -4.875 | -9.129 | -3.986 | -6.138 | |
| DOWN-UP | min | -37.04 | -47 | -33.08 | -39.68 |
| max | 28.022 | 53.495 | 38.54 | 58.312 | |
| neg returns | 15 | 12 | 13 | 14 | |
| MD | 132.87 | 142.629 | 92.60 | 119.743 | |
| average | 2.7559 | 2.8771 | 5.6723 | 7.6174 | |
| sd | 21.022 | 28.97 | 12.812 | 18.045 | |
| median | 5.1358 | 2.4316 | 4.687 | 4.2454 | |
| UP-UP | min | -38.74 | -46.22 | -18.15 | -31.4 |
| max | 51.007 | 65.855 | 36.726 | 48.432 | |
| neg returns | 9 | 8 | 6 | 6 | |
| MD | 47.811 | 109.512 | 18.517 | 36.649 |
Table 4.
Out-of-sample statistics($ \times 10^{-3} $ ) of $ (DIII)\; \text{and}\; (DIV) $ under mixed uncertainty set $ Q_M $ for market directions based four phases data
| (DIII) | |||||
| average | -0.2681 | -3.322 | 0.499 | -0.047 | |
| sd | 19.1129 | 35.824 | 18.919 | 18.732 | |
| median | 2.61737 | 1.5277 | 2.3188 | -0.6 | |
| UP-DOWN | min | -44.737 | -55.41 | -40.08 | -45.67 |
| max | 34.6569 | 77.502 | 35.718 | 32.997 | |
| neg returns | 9 | 10 | 9 | 11 | |
| MD | 74.493 | 159.476 | 68.053 | 64.932 | |
| average | -3.1162 | 0.4164 | -4.551 | -4.109 | |
| sd | 21.5749 | 28.782 | 19.827 | 21.41 | |
| median | 2.12342 | -0.945 | -2.471 | -2.008 | |
| DOWN-DOWN | min | -48.014 | -48.23 | -63.18 | -66.7 |
| max | 30.412 | 66.157 | 25.402 | 25.383 | |
| neg returns | 9 | 10 | 11 | 10 | |
| MD | 119.543 | 106.662 | 117.954 | 122.734 | |
| average | 11.2417 | 30.789 | 4.1896 | 8.3275 | |
| sd | 20.3046 | 48.878 | 16.079 | 19.256 | |
| median | 11.6296 | 39.14 | 7.2174 | 8.4051 | |
| DOWN-UP | min | -27.77 | -71 | -33.04 | -40.3 |
| max | 41.2838 | 98.964 | 25.703 | 46.593 | |
| neg returns | 4 | 6 | 7 | 5 | |
| MD | 27.77 | 78.851 | 36.124 | 48.21 | |
| average | 1.05369 | -3.017 | 0.2775 | -0.183 | |
| sd | 38.8387 | 42.249 | 34.946 | 31.665 | |
| median | 2.73823 | 3.0081 | 1.0448 | 3.6441 | |
| UP-UP | min | -77.25 | -79.58 | -69.68 | -59.33 |
| max | 66.1491 | 66.947 | 56.4 | 40.851 | |
| neg returns | 9 | 10 | 10 | 9 | |
| MD | 131.452 | 166.753 | 127.845 | 124.788 | |
| (DIV) | |||||
| average | -23.233 | -28.74 | -24.08 | -23.28 | |
| sd | 94.011 | 107.97 | 87.816 | 88.499 | |
| median | -27.104 | -37.74 | -26.68 | -29.55 | |
| UP-DOWN | min | -197.61 | -222.9 | -192.2 | -170 |
| max | 224.871 | 271.88 | 225.58 | 216.6 | |
| neg returns | 13 | 13 | 13 | 14 | |
| MD | 517.924 | 576.729 | 507.51 | 489.529 | |
| average | 0.58685 | 1.1156 | -0.835 | 0.5133 | |
| sd | 28.4234 | 58.191 | 29.021 | 33.774 | |
| median | -3.4995 | -1.348 | -2.763 | -7.861 | |
| DOWN-DOWN | min | -36.563 | -79.36 | -46.89 | -47.11 |
| max | 65.827 | 152.43 | 70.592 | 96.089 | |
| neg returns | 11 | 10 | 10 | 11 | |
| MD | 137.989 | 285.251 | 160.875 | 164.657 | |
| average | 7.63253 | 3.1941 | 6.4851 | 10.284 | |
| sd | 16.2213 | 15.549 | 15.238 | 19.419 | |
| median | 8.10496 | 2.7367 | 9.7434 | 10.025 | |
| DOWN-UP | min | -27.858 | -24.52 | -24.91 | -30.12 |
| max | 47.6872 | 28.554 | 34.307 | 39.199 | |
| neg returns | 5 | 7 | 7 | 7 | |
| MD | 42.35 | 40.646 | 35.51 | 53.741 | |
| average | 34.0544 | 30.368 | 17.863 | 15.659 | |
| sd | 56.3158 | 63.198 | 47.088 | 67.943 | |
| median | 51.5261 | 43.3 | 20.626 | 19.016 | |
| UP-UP | min | -124.52 | -136.7 | -97.17 | -109.3 |
| max | 100.258 | 152.26 | 111.24 | 174.18 | |
| neg returns | 4 | 7 | 6 | 7 | |
| MD | 124.517 | 144.539 | 97.174 | 139.654 |
| (DIII) | |||||
| average | -0.2681 | -3.322 | 0.499 | -0.047 | |
| sd | 19.1129 | 35.824 | 18.919 | 18.732 | |
| median | 2.61737 | 1.5277 | 2.3188 | -0.6 | |
| UP-DOWN | min | -44.737 | -55.41 | -40.08 | -45.67 |
| max | 34.6569 | 77.502 | 35.718 | 32.997 | |
| neg returns | 9 | 10 | 9 | 11 | |
| MD | 74.493 | 159.476 | 68.053 | 64.932 | |
| average | -3.1162 | 0.4164 | -4.551 | -4.109 | |
| sd | 21.5749 | 28.782 | 19.827 | 21.41 | |
| median | 2.12342 | -0.945 | -2.471 | -2.008 | |
| DOWN-DOWN | min | -48.014 | -48.23 | -63.18 | -66.7 |
| max | 30.412 | 66.157 | 25.402 | 25.383 | |
| neg returns | 9 | 10 | 11 | 10 | |
| MD | 119.543 | 106.662 | 117.954 | 122.734 | |
| average | 11.2417 | 30.789 | 4.1896 | 8.3275 | |
| sd | 20.3046 | 48.878 | 16.079 | 19.256 | |
| median | 11.6296 | 39.14 | 7.2174 | 8.4051 | |
| DOWN-UP | min | -27.77 | -71 | -33.04 | -40.3 |
| max | 41.2838 | 98.964 | 25.703 | 46.593 | |
| neg returns | 4 | 6 | 7 | 5 | |
| MD | 27.77 | 78.851 | 36.124 | 48.21 | |
| average | 1.05369 | -3.017 | 0.2775 | -0.183 | |
| sd | 38.8387 | 42.249 | 34.946 | 31.665 | |
| median | 2.73823 | 3.0081 | 1.0448 | 3.6441 | |
| UP-UP | min | -77.25 | -79.58 | -69.68 | -59.33 |
| max | 66.1491 | 66.947 | 56.4 | 40.851 | |
| neg returns | 9 | 10 | 10 | 9 | |
| MD | 131.452 | 166.753 | 127.845 | 124.788 | |
| (DIV) | |||||
| average | -23.233 | -28.74 | -24.08 | -23.28 | |
| sd | 94.011 | 107.97 | 87.816 | 88.499 | |
| median | -27.104 | -37.74 | -26.68 | -29.55 | |
| UP-DOWN | min | -197.61 | -222.9 | -192.2 | -170 |
| max | 224.871 | 271.88 | 225.58 | 216.6 | |
| neg returns | 13 | 13 | 13 | 14 | |
| MD | 517.924 | 576.729 | 507.51 | 489.529 | |
| average | 0.58685 | 1.1156 | -0.835 | 0.5133 | |
| sd | 28.4234 | 58.191 | 29.021 | 33.774 | |
| median | -3.4995 | -1.348 | -2.763 | -7.861 | |
| DOWN-DOWN | min | -36.563 | -79.36 | -46.89 | -47.11 |
| max | 65.827 | 152.43 | 70.592 | 96.089 | |
| neg returns | 11 | 10 | 10 | 11 | |
| MD | 137.989 | 285.251 | 160.875 | 164.657 | |
| average | 7.63253 | 3.1941 | 6.4851 | 10.284 | |
| sd | 16.2213 | 15.549 | 15.238 | 19.419 | |
| median | 8.10496 | 2.7367 | 9.7434 | 10.025 | |
| DOWN-UP | min | -27.858 | -24.52 | -24.91 | -30.12 |
| max | 47.6872 | 28.554 | 34.307 | 39.199 | |
| neg returns | 5 | 7 | 7 | 7 | |
| MD | 42.35 | 40.646 | 35.51 | 53.741 | |
| average | 34.0544 | 30.368 | 17.863 | 15.659 | |
| sd | 56.3158 | 63.198 | 47.088 | 67.943 | |
| median | 51.5261 | 43.3 | 20.626 | 19.016 | |
| UP-UP | min | -124.52 | -136.7 | -97.17 | -109.3 |
| max | 100.258 | 152.26 | 111.24 | 174.18 | |
| neg returns | 4 | 7 | 6 | 7 | |
| MD | 124.517 | 144.539 | 97.174 | 139.654 |
Table 5.
Out-of-sample statistics $ (\times 10^{-3}) $ of $ (DI) $ , $ (DII) $ , $ (DIII) $ and $ (DIV) $ on a single window analysis under mixed and interval+polyhedral uncertainty sets
| average | -9.9697 | -17.83 | -9.51 | -5.5184 | -8.024 | |
| sd | 7.95041 | 12.595 | 5.629 | 19.863 | 12.937 | |
| (DI) | med | -8.499 | -13.149 | -7.412 | -6.5741 | -6.643 |
| min | -20.925 | -36.073 | -17.7 | -27.913 | -25.07 | |
| max | -1.9555 | -8.9473 | -5.514 | 18.987 | 6.2582 | |
| average | -1.505 | -0.2718 | -0.303 | -5.5184 | -1.547 | |
| sd | 15.727 | 16.813 | 15.154 | 19.863 | 17.94 | |
| (DII) | med | 4.52 | 0.3627 | 2.1474 | -6.5741 | 1.4017 |
| min | -24.706 | -20.442 | -20.77 | -27.913 | -26.07 | |
| max | 9.631 | 18.629 | 15.258 | 18.987 | 17.077 | |
| average | 4.60903 | 4.288 | 0.1445 | 0.3184 | 5.2191 | |
| sd | 23.0986 | 28.943 | 17.512 | 25.937 | 17.18 | |
| (DIII) | med | 8.31506 | 9.593 | 3.2971 | 2.3598 | 9.9122 |
| min | -24.436 | -33.33 | -21.39 | -33.356 | -18.75 | |
| max | 26.2417 | 31.299 | 15.372 | 29.91 | 19.8 | |
| average | -12.383 | -18.126 | -11.54 | -18.126 | -7.655 | |
| sd | 14.3273 | 24.613 | 18.997 | 24.613 | 18.297 | |
| (DIV) | med | -13.526 | -6.7408 | -9.971 | -6.7408 | -6.39 |
| min | -28.661 | -54.996 | -35.76 | -54.996 | -30.83 | |
| max | 6.17971 | -4.0262 | 9.5596 | -4.0262 | 12.992 |
| average | -9.9697 | -17.83 | -9.51 | -5.5184 | -8.024 | |
| sd | 7.95041 | 12.595 | 5.629 | 19.863 | 12.937 | |
| (DI) | med | -8.499 | -13.149 | -7.412 | -6.5741 | -6.643 |
| min | -20.925 | -36.073 | -17.7 | -27.913 | -25.07 | |
| max | -1.9555 | -8.9473 | -5.514 | 18.987 | 6.2582 | |
| average | -1.505 | -0.2718 | -0.303 | -5.5184 | -1.547 | |
| sd | 15.727 | 16.813 | 15.154 | 19.863 | 17.94 | |
| (DII) | med | 4.52 | 0.3627 | 2.1474 | -6.5741 | 1.4017 |
| min | -24.706 | -20.442 | -20.77 | -27.913 | -26.07 | |
| max | 9.631 | 18.629 | 15.258 | 18.987 | 17.077 | |
| average | 4.60903 | 4.288 | 0.1445 | 0.3184 | 5.2191 | |
| sd | 23.0986 | 28.943 | 17.512 | 25.937 | 17.18 | |
| (DIII) | med | 8.31506 | 9.593 | 3.2971 | 2.3598 | 9.9122 |
| min | -24.436 | -33.33 | -21.39 | -33.356 | -18.75 | |
| max | 26.2417 | 31.299 | 15.372 | 29.91 | 19.8 | |
| average | -12.383 | -18.126 | -11.54 | -18.126 | -7.655 | |
| sd | 14.3273 | 24.613 | 18.997 | 24.613 | 18.297 | |
| (DIV) | med | -13.526 | -6.7408 | -9.971 | -6.7408 | -6.39 |
| min | -28.661 | -54.996 | -35.76 | -54.996 | -30.83 | |
| max | 6.17971 | -4.0262 | 9.5596 | -4.0262 | 12.992 |
| [1] |
Shihan Di, Dong Ma, Peibiao Zhao. $ \alpha $-robust portfolio optimization problem under the distribution uncertainty. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022054 |
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