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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

* Corresponding author: Ruchika Sehgal

Received  June 2019 Revised  October 2019 Published  February 2020

Fund Project: The first author is supported by IIT Delhi, India, for financial grant through GATE scholarship

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.

Citation: Ruchika Sehgal, Aparna Mehra. Worst-case analysis of Gini mean difference safety measure. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020037
References:
[1]

P. ArtznerF. DelbaenJ.-M. Eber and D. Heath, Coherent measures of risk, Math. Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068.  Google Scholar

[2]

A. Ben-TalD. 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.  Google Scholar

[3]

S. Benati, Using medians in portfolio optimization, J. Oper. Res. Soc., 66 (2015), 720-731.  doi: 10.1057/jors.2014.57.  Google Scholar

[4]

M. BerkhouchG. 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.  Google Scholar

[5]

D. Bertsimas and M. Sim, The price of robustness, Oper. Res., 52 (2004), 35-53.  doi: 10.1287/opre.1030.0065.  Google Scholar

[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.  Google Scholar

[7]

F. Black and R. Litterman, Global portfolio optimization, Financial Analysts Journal, 48 (1992), 28-43.  doi: 10.2469/faj.v48.n5.28.  Google Scholar

[8]

B. Bower and P. Wentz, Portfolio optimization: MAD vs. Markowitz, Rose-Hulman Undergraduate Mathematics Journal, 6 (2005), 3. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

L. El GhaouiM. 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.  Google Scholar

[12]

M. FengA. 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.  Google Scholar

[13]

E. FurmanR. 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.  Google Scholar

[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.  Google Scholar

[15]

M. GharakhaniF. 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.  Google Scholar

[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.  Google Scholar

[17]

J.-Y. GotohK. 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.  Google Scholar

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J. A. HallB. 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.  Google Scholar

[19]

R. JiM. 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.  Google Scholar

[20]

M. KapsosN. 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.  Google Scholar

[21]

G. KaraA. 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.  Google Scholar

[22]

C. Keating and W. F. Shadwick, A universal performance measure, J. Performance Measurement, 6 (2002), 59-84.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

P. LiY. 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.  Google Scholar

[26]

B. G. Lindsay, Mixture models: Theory, geometry and applications, in NSF-CBMS Regional Conference Series in Probability and Statistics, (1995), 1–163. Google Scholar

[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.  Google Scholar

[28]

R. MansiniW. 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.  Google Scholar

[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. Google Scholar

[30]

H. Markowitz, Portfolio selection, J. Finance, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x.  Google Scholar

[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.  Google Scholar

[32]

K. NatarajanD. Pachamanova and M. Sim, Constructing risk measures from uncertainty sets, Oper. Res., 57 (2009), 1129-1141.  doi: 10.1287/opre.1080.0683.  Google Scholar

[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. Google Scholar

[34]

D. Peel and G. J. McLachlan, Robust mixture modelling using the t distribution, Statistics Comput., 10 (2000), 339-348.   Google Scholar

[35]

K. PostekD. 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.  Google Scholar

[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.  Google Scholar

[37]

M. RudolfH.-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.  Google Scholar

[38]

R. Sehgal and A. Mehra, Robust reward–risk ratio portfolio optimization, Internat. Transactions Oper. Res., (2019). doi: 10.1111/itor.12652.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

A. SharmaS. 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.  Google Scholar

[43]

A. SharmaS. 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.  Google Scholar

[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.  Google Scholar

[45]

S. Yitzhaki, Stochastic dominance, mean variance, and Gini's mean difference, American Economic Review, 72 (1982), 178-185.   Google Scholar

[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.  Google Scholar

[47]

X. ZhengX. SunD. 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.  Google Scholar

[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.  Google Scholar

[49]

S. ZhuD. Li and S. Wang, Robust portfolio selection under downside risk measures, Quant. Finance, 9 (2009), 869-885.  doi: 10.1080/14697680902852746.  Google Scholar

show all references

References:
[1]

P. ArtznerF. DelbaenJ.-M. Eber and D. Heath, Coherent measures of risk, Math. Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068.  Google Scholar

[2]

A. Ben-TalD. 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.  Google Scholar

[3]

S. Benati, Using medians in portfolio optimization, J. Oper. Res. Soc., 66 (2015), 720-731.  doi: 10.1057/jors.2014.57.  Google Scholar

[4]

M. BerkhouchG. 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.  Google Scholar

[5]

D. Bertsimas and M. Sim, The price of robustness, Oper. Res., 52 (2004), 35-53.  doi: 10.1287/opre.1030.0065.  Google Scholar

[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.  Google Scholar

[7]

F. Black and R. Litterman, Global portfolio optimization, Financial Analysts Journal, 48 (1992), 28-43.  doi: 10.2469/faj.v48.n5.28.  Google Scholar

[8]

B. Bower and P. Wentz, Portfolio optimization: MAD vs. Markowitz, Rose-Hulman Undergraduate Mathematics Journal, 6 (2005), 3. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

L. El GhaouiM. 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.  Google Scholar

[12]

M. FengA. 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.  Google Scholar

[13]

E. FurmanR. 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.  Google Scholar

[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.  Google Scholar

[15]

M. GharakhaniF. 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.  Google Scholar

[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.  Google Scholar

[17]

J.-Y. GotohK. 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.  Google Scholar

[18]

J. A. HallB. 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.  Google Scholar

[19]

R. JiM. 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.  Google Scholar

[20]

M. KapsosN. 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.  Google Scholar

[21]

G. KaraA. 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.  Google Scholar

[22]

C. Keating and W. F. Shadwick, A universal performance measure, J. Performance Measurement, 6 (2002), 59-84.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

P. LiY. 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.  Google Scholar

[26]

B. G. Lindsay, Mixture models: Theory, geometry and applications, in NSF-CBMS Regional Conference Series in Probability and Statistics, (1995), 1–163. Google Scholar

[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.  Google Scholar

[28]

R. MansiniW. 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.  Google Scholar

[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. Google Scholar

[30]

H. Markowitz, Portfolio selection, J. Finance, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x.  Google Scholar

[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.  Google Scholar

[32]

K. NatarajanD. Pachamanova and M. Sim, Constructing risk measures from uncertainty sets, Oper. Res., 57 (2009), 1129-1141.  doi: 10.1287/opre.1080.0683.  Google Scholar

[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. Google Scholar

[34]

D. Peel and G. J. McLachlan, Robust mixture modelling using the t distribution, Statistics Comput., 10 (2000), 339-348.   Google Scholar

[35]

K. PostekD. 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.  Google Scholar

[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.  Google Scholar

[37]

M. RudolfH.-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.  Google Scholar

[38]

R. Sehgal and A. Mehra, Robust reward–risk ratio portfolio optimization, Internat. Transactions Oper. Res., (2019). doi: 10.1111/itor.12652.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

A. SharmaS. 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.  Google Scholar

[43]

A. SharmaS. 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.  Google Scholar

[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.  Google Scholar

[45]

S. Yitzhaki, Stochastic dominance, mean variance, and Gini's mean difference, American Economic Review, 72 (1982), 178-185.   Google Scholar

[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.  Google Scholar

[47]

X. ZhengX. SunD. 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.  Google Scholar

[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.  Google Scholar

[49]

S. ZhuD. Li and S. Wang, Robust portfolio selection under downside risk measures, Quant. Finance, 9 (2009), 869-885.  doi: 10.1080/14697680902852746.  Google Scholar

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
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 $ \times 10^{-3} $ weakly sd $ \times 10^{-3} $
1-9 weeks 3.606 19.559
10-18 weeks 7.111 42.395
19-27 weeks -2.078 23.281
period weekly average $ \times 10^{-3} $ weakly sd $ \times 10^{-3} $
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 $
$ \mu GMD $ $ MGMD $ $ R\mu GMD $ $ RMGMD $
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
$ CVaR_{0.03} $ 53.369 55.862 50.934 49.827
$ CVaR_{0.05} $ 50.236 48.982 46.816 47.84
$ VaR_{0.03} $ 49.945 44.695 43.091 46.16
$ VaR_{0.05} $ 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
$ CVaR_{0.03} $ 57.914 53.802 57.633 47.689
$ CVaR_{0.05} $ 49.836 47.188 48.391 43.867
$ VaR_{0.03} $ 43.523 38.888 37.368 41.209
$ VaR_{0.05} $ 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
$ CVaR_{0.03} $ 61.937 73.741 52.024 63.487
$ CVaR_{0.05} $ 51.816 61.994 40.559 52.514
$ VaR_{0.03} $ 46.458 47.761 28.247 51.358
$ VaR_{0.05} $ 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
$ CVaR_{0.03} $ 57.727 61.945 58.193 58.013
$ CVaR_{0.05} $ 50.202 56.738 50.081 52.9
$ VaR_{0.03} $ 52.377 52.549 48.15 48.715
$ VaR_{0.05} $ 36.036 48.283 31.815 43.657
MD 180.98 233.87 170.43 191.327
$ \mu GMD $ $ MGMD $ $ R\mu GMD $ $ RMGMD $
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
$ CVaR_{0.03} $ 53.369 55.862 50.934 49.827
$ CVaR_{0.05} $ 50.236 48.982 46.816 47.84
$ VaR_{0.03} $ 49.945 44.695 43.091 46.16
$ VaR_{0.05} $ 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
$ CVaR_{0.03} $ 57.914 53.802 57.633 47.689
$ CVaR_{0.05} $ 49.836 47.188 48.391 43.867
$ VaR_{0.03} $ 43.523 38.888 37.368 41.209
$ VaR_{0.05} $ 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
$ CVaR_{0.03} $ 61.937 73.741 52.024 63.487
$ CVaR_{0.05} $ 51.816 61.994 40.559 52.514
$ VaR_{0.03} $ 46.458 47.761 28.247 51.358
$ VaR_{0.05} $ 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
$ CVaR_{0.03} $ 57.727 61.945 58.193 58.013
$ CVaR_{0.05} $ 50.202 56.738 50.081 52.9
$ VaR_{0.03} $ 52.377 52.549 48.15 48.715
$ VaR_{0.05} $ 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) $ \mu GMD $ $ MGMD $ $ R\mu GMD $ $ RMGMD $
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) $ \mu GMD $ $ MGMD $ $ R\mu GMD $ $ RMGMD $
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) $ \mu GMD $ $ MGMD $ $ R\mu GMD $ $ RMGMD $
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) $ \mu GMD $ $ MGMD $ $ R\mu GMD $ $ RMGMD $
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) $ \mu GMD $ $ MGMD $ $ R\mu GMD $ $ RMGMD $
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) $ \mu GMD $ $ MGMD $ $ R\mu GMD $ $ RMGMD $
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) $ \mu GMD $ $ MGMD $ $ R\mu GMD $ $ RMGMD $
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) $ \mu GMD $ $ MGMD $ $ R\mu GMD $ $ RMGMD $
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
$ \mu GMD $ $ MGMD $ $ R\mu GMD $ $ RMGMD $ $ RB \mu GMD $
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
$ \mu GMD $ $ MGMD $ $ R\mu GMD $ $ RMGMD $ $ RB \mu GMD $
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
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