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September  2020, 16(5): 2283-2303. doi: 10.3934/jimo.2019054

Two nonparametric approaches to mean absolute deviation portfolio selection model

1. 

College of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, China

2. 

Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha 410114, China

3. 

Department of finance, College of business, Central South University, Hunan, 410083, China

* Corresponding author: wfh@amss.ac.cn

Received  June 2018 Revised  December 2018 Published  May 2019

In this paper, we apply two nonparametric approaches to mean absolute deviation (MAD) portfolio selection model. The first one is to use the nonparametric kernel mean estimation to replace the returns of assets with five different kernel functions. Then, we construct the nonparametric kernel mean estimation-based MAD portfolio model. The second one is to utilize the nonparametric kernel median estimation to replace the returns of assets with five different kernel functions. Then, we construct the nonparametric kernel median estimation-based MAD portfolio model. We also extend the two kinds of nonparametric approach to mean-Conditional Value-at-Risk portfolio model. Finally, we give the in-sample and out-of-sample analysis of the proposed strategies and compare the performance of the proposed models by using actual stock returns in Shanghai stock exchange of China. The experimental results show the nonparametric estimation-based portfolio models are more efficient than the original portfolio model.

Citation: Zhifeng Dai, Huan Zhu, Fenghua Wen. Two nonparametric approaches to mean absolute deviation portfolio selection model. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2283-2303. doi: 10.3934/jimo.2019054
References:
[1]

R. AlemanyC. Bolancé and M. Guillén, A nonparametric approach to calculating value-at-risk, Insurance Math. Econ., 52 (2013), 255-262.  doi: 10.1016/j.insmatheco.2012.12.008.  Google Scholar

[2]

G. Athayde, Building a Mean-Downside Risk Portfolio Frontier, Developments in Forecast Combination and Portfolio Choice, John Wiley and Sons. 2001. Google Scholar

[3]

G. Athayde, The Mean-downside Risk Portfolio Frontier: A Non-parametric Approach, Published in Advances in portfolio construction and implementation. 2003. Google Scholar

[4]

A. BerlinetB. Cadre and A. Gannoun, On the conditional $L_1$ -median and its estimation, J. Nonparametr. Stat., 13 (2001a), 631-645.  doi: 10.1080/10485250108832869.  Google Scholar

[5]

A. BerlinetA. Gannoun and E. Matzner, Asymptotic normality of convergent estimates of conditional quantiles, Stats., 35 (2001b), 139-169.  doi: 10.1080/02331880108802728.  Google Scholar

[6]

N. BinghamR. Kiesel and R. Schmidt, A semi-parametric approach to risk management, Quant. Financ., 3 (2003), 426-441.  doi: 10.1088/1469-7688/3/6/302.  Google Scholar

[7]

Z. Cai and X. Wang, Nonparametric estimation of conditional VaR and expected shortfall, J. Econom., 147 (2008), 120-130.  doi: 10.1016/j.jeconom.2008.09.005.  Google Scholar

[8]

S. Chen and C. Y. Tang, Nonparametric inference of value-atrisk for dependent financial returns, J. Financ. Econ., 3 (2005), 227-255.   Google Scholar

[9]

S. Chen, Nonparametric estimation of expected shortfall, J. Financ. Econ., 6 (2008), 87-107.  doi: 10.1093/jjfinec/nbm019.  Google Scholar

[10]

L. ChiodiR. Mansini and M. G. Speranza, Semi-absolute deviation rule for mutual funds portfolio selection, Ann. Oper. Res., 124 (2003), 245-265.  doi: 10.1023/B:ANOR.0000004772.15447.5a.  Google Scholar

[11]

T. E. Conine and and M. J. Tamarkin, On diversiffication given asymmetry in returns, J. Financ., 36 (1981), 1143-1155.   Google Scholar

[12]

Z. Dai and F. Wen, Some improved sparse and stable portfolio optimization problems, Finan. Res. Lett., 27 (2018), 46-52.  doi: 10.1016/j.frl.2018.02.026.  Google Scholar

[13]

Z. Dai and F. Wen, A generalized approach to sparse and stable portfolio optimization problem, J. Ind. Manag. Optim., 14 (2018), 1651-1666.   Google Scholar

[14]

Z. Dai and F. Wang, Sparse and robust mean-variance portfolio optimization problems., Physica A, 2019, accepted. Google Scholar

[15]

A. GannounJ. Saracco and K. Yu, Nonparametric time series prediction by conditional median and quantiles, J. stat. Plan. inferenc., 117 (2003), 207-223.  doi: 10.1016/S0378-3758(02)00384-1.  Google Scholar

[16]

J. G. Gooijer and A. Gannoun, Tr multivariate conditional median estimation, Commun. Stat. Simul. C., 36 (2007), 165-176.  doi: 10.1080/03610910601096270.  Google Scholar

[17]

B. E. Hansen, Bandwidth selection for nonparametric distribution estimation., Working Paper, 2004. Google Scholar

[18]

Z. HeL. He and F. Wen, Risk compensation and market returns: The role of investor sentiment in the stock market, Emerg. Mark. Financ. Tr., 55 (2019), 704-718.   Google Scholar

[19]

C. HuangZ. YangT. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Differential Equations, 256 (2014), 2101-2114.  doi: 10.1016/j.jde.2013.12.015.  Google Scholar

[20]

S. O. Jeong and K. H. Kang, Nonparametric estimation of valueat-risk, J. Appl. Stat., 36 (2009), 1225-1238.  doi: 10.1080/02664760802607517.  Google Scholar

[21]

P. Jorion, Value at Risk: The New Benchmark for controlling Derivatives Risk, New York: McGraw-Hill. 1997. Google Scholar

[22]

H. KellererR. Mansini and M. G. Speranza, Selecting portfolios with fixed costs and minimum transaction lots, Ann. Oper. Res., 99 (2000), 287-304.  doi: 10.1023/A:1019279918596.  Google Scholar

[23] R. Koenker, Quantile Regression, Cambridge Books, Cambridge University Press, Issue. 38, 2005.  doi: 10.1017/CBO9780511754098.  Google Scholar
[24]

H. Konno and H. Yamazaki, Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Manage. Sci., 37 (1991), 501-623.  doi: 10.1287/mnsc.37.5.519.  Google Scholar

[25] Q. Li and J. S. Racine, Nonparametric Econometrics: Theory and Practice, Princeton University Press, 2007.   Google Scholar
[26]

R. MansiniW. Ogryczak and M. G. Speranza, LP solvable models for portfolio optimization: A classification and computational comparison, IMA J. Manag. Math., 14 (2003), 187-220.  doi: 10.1093/imaman/14.3.187.  Google Scholar

[27]

H. Markowitz, Portfolio Selection, J. Financ., 7 (1952), 77-91.   Google Scholar

[28]

A. R. Pagan and A. Ullah, Nonparametric Econometrics, Cambridge University Press. 1999. doi: 10.1017/CBO9780511612503.  Google Scholar

[29]

C. Papahristodoulou and E. Dotzauer, Optimal portfolios using linear programming models, J. Oper. Res. Soc., 55 (2004), 1169-1177.  doi: 10.1057/palgrave.jors.2601765.  Google Scholar

[30]

J. S. Pang, A new efficient algorithm for a class of portfolio selection problems, Oper. Res., 28 (1980), 754-767.  doi: 10.1287/opre.28.3.754.  Google Scholar

[31]

A. Perold, Large scale portfolio selections, Manage. Sci., 30 (1984), 1143-1160.  doi: 10.1287/mnsc.30.10.1143.  Google Scholar

[32]

T. Rockfeller and S. Uryasev, Optimization of conditional value-at-risk, J. Risk., 2 (2000), 21-34.  doi: 10.21314/JOR.2000.038.  Google Scholar

[33]

H. B. SalahM. ChaouchA. GannounC. D. Peretti and A. Trabelsi, Mean and median-based nonparametric estimation of returns in mean-downside risk portfolio frontier, Ann. Oper. Res., 262 (2018), 653-681.  doi: 10.1007/s10479-016-2235-z.  Google Scholar

[34]

O. Scaillet, Nonparametric estimation and sensitivity analysis of expected shortfall, Math. Financ., 14 (2004), 115-129.  doi: 10.1111/j.0960-1627.2004.00184.x.  Google Scholar

[35]

W. F. Sharpe, Capital asset prices: A theory of market equilibrium under conditions of risk, J. Financ., 19 (1964), 425-442.   Google Scholar

[36]

P. Shen, Median regression model with left truncated and right censored data, J. Stat. Plan. Infer., 142 (2012), 1757-1766.  doi: 10.1016/j.jspi.2012.02.014.  Google Scholar

[37]

B. W. Silverman, Density Estimation for Statistics and Data Analysis, Chapman and Hall, New York. 1986. doi: 10.1007/978-1-4899-3324-9.  Google Scholar

[38]

J. Schaumburg, Predicting extreme value at risk: Nonparametric quantile regression with refinements from extreme value theory, Comput. Stat. Data Anal., 56 (2012), 4081-4096.  doi: 10.1016/j.csda.2012.03.016.  Google Scholar

[39]

S. Subramanian, Median regression using nonparametric kernel estimation, J. Nonparametr. Stat., 14 (2002), 583-605.  doi: 10.1080/10485250213907.  Google Scholar

[40]

S. Subramanian, Median regression analysis from data with left and right censored observations, Stat. Methodol., 4 (2007), 121-131.  doi: 10.1016/j.stamet.2006.03.001.  Google Scholar

[41]

M. G. Speranza, Linear programming models for portfolio optimization, Finance., 14 (1993), 107-123.   Google Scholar

[42]

M. G. Speranza, A heuristic algorithm for a portfolio optimization model applied to the Milan stock market, Comput. Oper. Res., 23 (1996), 433-441.  doi: 10.1016/0305-0548(95)00030-5.  Google Scholar

[43]

F. WenJ. XiaoC. Huang and X. Xia, Interaction between oil and US dollar exchange rate: Nonlinear causality, time-varying influence and structural breaks in volatility, Appl. Econ., 50 (2018), 319-334.  doi: 10.1080/00036846.2017.1321838.  Google Scholar

[44]

F. Wen, F. Min and Y. J. Zhang et al., Crude oil price shocks, monetary policy, and China's economy, Int. J. Financ. Econ., (2018) online. doi: 10.1002/ijfe.1692.  Google Scholar

[45]

F. WenX. Yang and W. Zhou, Tail dependence networks of global stock markets, Int. J. Financ. Econ., 24 (2019), 558-567.  doi: 10.1002/ijfe.1679.  Google Scholar

[46]

F. WenJ. Xiao and X. Xia et al., Oil prices and chinese stock market: Nonlinear causality and volatility persistence, Emerg. Mark. Financ. Tr., 55 (2019), 1247-1263.  doi: 10.1080/1540496X.2018.1496078.  Google Scholar

[47]

J. XiaoM. Zhou and F. Wen et al., Asymmetric impacts of oil price uncertainty on Chinese stock returns under different market conditions: Evidence from oil volatility index, Energ. Econ., 74 (2018), 777-786.  doi: 10.1016/j.eneco.2018.07.026.  Google Scholar

[48]

H. YaoZ. Li and Y. Lai, Mean-CVaR portfolio selection: A nonparametric estimation framework, Comput. Oper. Res., 40 (2013), 1014-1022.  doi: 10.1016/j.cor.2012.11.007.  Google Scholar

[49]

H. YaoY. Li and K. Benson, A smooth non-parametric estimation framework for safety-first portfolio optimization, Quant. Financ., 15 (2015), 1865-1884.  doi: 10.1080/14697688.2014.971857.  Google Scholar

[50]

K. Yu, A. Allay, S. Yang and D. J. Hand, Kernel quantile based estimation of expected shortfall, J. Risk., 12 (2010), 15-32. Google Scholar

[51]

G. YuanZ. H. Meng and Y. Li, A modified Hestenes and Stiefel conjugate gradient algorithm for large-scale nonsmooth minimizations and nonlinear equations, J. Optim. Theory. Appl., 168 (2016), 129-152.   Google Scholar

[52]

G. Zhao and Y. Y. Ma, Robust nonparametric kernel regression estimator, Stat. Probabil. Lett., 116 (2016), 72-79.  doi: 10.1016/j.spl.2016.04.010.  Google Scholar

[53]

Y. Zhao and F. Chen, Empirical likelihood inference for censored median regression model via nonparametric kernel estimation, J. Multivariate Anal., 99 (2008), 215-231.  doi: 10.1016/j.jmva.2007.05.002.  Google Scholar

[54]

Y. Zhao and H. Cui, Empirical likelihood for median regression model with designed censoring variables, J. Multivariate Anal., 101 (2010), 240-251.  doi: 10.1016/j.jmva.2009.07.008.  Google Scholar

show all references

References:
[1]

R. AlemanyC. Bolancé and M. Guillén, A nonparametric approach to calculating value-at-risk, Insurance Math. Econ., 52 (2013), 255-262.  doi: 10.1016/j.insmatheco.2012.12.008.  Google Scholar

[2]

G. Athayde, Building a Mean-Downside Risk Portfolio Frontier, Developments in Forecast Combination and Portfolio Choice, John Wiley and Sons. 2001. Google Scholar

[3]

G. Athayde, The Mean-downside Risk Portfolio Frontier: A Non-parametric Approach, Published in Advances in portfolio construction and implementation. 2003. Google Scholar

[4]

A. BerlinetB. Cadre and A. Gannoun, On the conditional $L_1$ -median and its estimation, J. Nonparametr. Stat., 13 (2001a), 631-645.  doi: 10.1080/10485250108832869.  Google Scholar

[5]

A. BerlinetA. Gannoun and E. Matzner, Asymptotic normality of convergent estimates of conditional quantiles, Stats., 35 (2001b), 139-169.  doi: 10.1080/02331880108802728.  Google Scholar

[6]

N. BinghamR. Kiesel and R. Schmidt, A semi-parametric approach to risk management, Quant. Financ., 3 (2003), 426-441.  doi: 10.1088/1469-7688/3/6/302.  Google Scholar

[7]

Z. Cai and X. Wang, Nonparametric estimation of conditional VaR and expected shortfall, J. Econom., 147 (2008), 120-130.  doi: 10.1016/j.jeconom.2008.09.005.  Google Scholar

[8]

S. Chen and C. Y. Tang, Nonparametric inference of value-atrisk for dependent financial returns, J. Financ. Econ., 3 (2005), 227-255.   Google Scholar

[9]

S. Chen, Nonparametric estimation of expected shortfall, J. Financ. Econ., 6 (2008), 87-107.  doi: 10.1093/jjfinec/nbm019.  Google Scholar

[10]

L. ChiodiR. Mansini and M. G. Speranza, Semi-absolute deviation rule for mutual funds portfolio selection, Ann. Oper. Res., 124 (2003), 245-265.  doi: 10.1023/B:ANOR.0000004772.15447.5a.  Google Scholar

[11]

T. E. Conine and and M. J. Tamarkin, On diversiffication given asymmetry in returns, J. Financ., 36 (1981), 1143-1155.   Google Scholar

[12]

Z. Dai and F. Wen, Some improved sparse and stable portfolio optimization problems, Finan. Res. Lett., 27 (2018), 46-52.  doi: 10.1016/j.frl.2018.02.026.  Google Scholar

[13]

Z. Dai and F. Wen, A generalized approach to sparse and stable portfolio optimization problem, J. Ind. Manag. Optim., 14 (2018), 1651-1666.   Google Scholar

[14]

Z. Dai and F. Wang, Sparse and robust mean-variance portfolio optimization problems., Physica A, 2019, accepted. Google Scholar

[15]

A. GannounJ. Saracco and K. Yu, Nonparametric time series prediction by conditional median and quantiles, J. stat. Plan. inferenc., 117 (2003), 207-223.  doi: 10.1016/S0378-3758(02)00384-1.  Google Scholar

[16]

J. G. Gooijer and A. Gannoun, Tr multivariate conditional median estimation, Commun. Stat. Simul. C., 36 (2007), 165-176.  doi: 10.1080/03610910601096270.  Google Scholar

[17]

B. E. Hansen, Bandwidth selection for nonparametric distribution estimation., Working Paper, 2004. Google Scholar

[18]

Z. HeL. He and F. Wen, Risk compensation and market returns: The role of investor sentiment in the stock market, Emerg. Mark. Financ. Tr., 55 (2019), 704-718.   Google Scholar

[19]

C. HuangZ. YangT. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Differential Equations, 256 (2014), 2101-2114.  doi: 10.1016/j.jde.2013.12.015.  Google Scholar

[20]

S. O. Jeong and K. H. Kang, Nonparametric estimation of valueat-risk, J. Appl. Stat., 36 (2009), 1225-1238.  doi: 10.1080/02664760802607517.  Google Scholar

[21]

P. Jorion, Value at Risk: The New Benchmark for controlling Derivatives Risk, New York: McGraw-Hill. 1997. Google Scholar

[22]

H. KellererR. Mansini and M. G. Speranza, Selecting portfolios with fixed costs and minimum transaction lots, Ann. Oper. Res., 99 (2000), 287-304.  doi: 10.1023/A:1019279918596.  Google Scholar

[23] R. Koenker, Quantile Regression, Cambridge Books, Cambridge University Press, Issue. 38, 2005.  doi: 10.1017/CBO9780511754098.  Google Scholar
[24]

H. Konno and H. Yamazaki, Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Manage. Sci., 37 (1991), 501-623.  doi: 10.1287/mnsc.37.5.519.  Google Scholar

[25] Q. Li and J. S. Racine, Nonparametric Econometrics: Theory and Practice, Princeton University Press, 2007.   Google Scholar
[26]

R. MansiniW. Ogryczak and M. G. Speranza, LP solvable models for portfolio optimization: A classification and computational comparison, IMA J. Manag. Math., 14 (2003), 187-220.  doi: 10.1093/imaman/14.3.187.  Google Scholar

[27]

H. Markowitz, Portfolio Selection, J. Financ., 7 (1952), 77-91.   Google Scholar

[28]

A. R. Pagan and A. Ullah, Nonparametric Econometrics, Cambridge University Press. 1999. doi: 10.1017/CBO9780511612503.  Google Scholar

[29]

C. Papahristodoulou and E. Dotzauer, Optimal portfolios using linear programming models, J. Oper. Res. Soc., 55 (2004), 1169-1177.  doi: 10.1057/palgrave.jors.2601765.  Google Scholar

[30]

J. S. Pang, A new efficient algorithm for a class of portfolio selection problems, Oper. Res., 28 (1980), 754-767.  doi: 10.1287/opre.28.3.754.  Google Scholar

[31]

A. Perold, Large scale portfolio selections, Manage. Sci., 30 (1984), 1143-1160.  doi: 10.1287/mnsc.30.10.1143.  Google Scholar

[32]

T. Rockfeller and S. Uryasev, Optimization of conditional value-at-risk, J. Risk., 2 (2000), 21-34.  doi: 10.21314/JOR.2000.038.  Google Scholar

[33]

H. B. SalahM. ChaouchA. GannounC. D. Peretti and A. Trabelsi, Mean and median-based nonparametric estimation of returns in mean-downside risk portfolio frontier, Ann. Oper. Res., 262 (2018), 653-681.  doi: 10.1007/s10479-016-2235-z.  Google Scholar

[34]

O. Scaillet, Nonparametric estimation and sensitivity analysis of expected shortfall, Math. Financ., 14 (2004), 115-129.  doi: 10.1111/j.0960-1627.2004.00184.x.  Google Scholar

[35]

W. F. Sharpe, Capital asset prices: A theory of market equilibrium under conditions of risk, J. Financ., 19 (1964), 425-442.   Google Scholar

[36]

P. Shen, Median regression model with left truncated and right censored data, J. Stat. Plan. Infer., 142 (2012), 1757-1766.  doi: 10.1016/j.jspi.2012.02.014.  Google Scholar

[37]

B. W. Silverman, Density Estimation for Statistics and Data Analysis, Chapman and Hall, New York. 1986. doi: 10.1007/978-1-4899-3324-9.  Google Scholar

[38]

J. Schaumburg, Predicting extreme value at risk: Nonparametric quantile regression with refinements from extreme value theory, Comput. Stat. Data Anal., 56 (2012), 4081-4096.  doi: 10.1016/j.csda.2012.03.016.  Google Scholar

[39]

S. Subramanian, Median regression using nonparametric kernel estimation, J. Nonparametr. Stat., 14 (2002), 583-605.  doi: 10.1080/10485250213907.  Google Scholar

[40]

S. Subramanian, Median regression analysis from data with left and right censored observations, Stat. Methodol., 4 (2007), 121-131.  doi: 10.1016/j.stamet.2006.03.001.  Google Scholar

[41]

M. G. Speranza, Linear programming models for portfolio optimization, Finance., 14 (1993), 107-123.   Google Scholar

[42]

M. G. Speranza, A heuristic algorithm for a portfolio optimization model applied to the Milan stock market, Comput. Oper. Res., 23 (1996), 433-441.  doi: 10.1016/0305-0548(95)00030-5.  Google Scholar

[43]

F. WenJ. XiaoC. Huang and X. Xia, Interaction between oil and US dollar exchange rate: Nonlinear causality, time-varying influence and structural breaks in volatility, Appl. Econ., 50 (2018), 319-334.  doi: 10.1080/00036846.2017.1321838.  Google Scholar

[44]

F. Wen, F. Min and Y. J. Zhang et al., Crude oil price shocks, monetary policy, and China's economy, Int. J. Financ. Econ., (2018) online. doi: 10.1002/ijfe.1692.  Google Scholar

[45]

F. WenX. Yang and W. Zhou, Tail dependence networks of global stock markets, Int. J. Financ. Econ., 24 (2019), 558-567.  doi: 10.1002/ijfe.1679.  Google Scholar

[46]

F. WenJ. Xiao and X. Xia et al., Oil prices and chinese stock market: Nonlinear causality and volatility persistence, Emerg. Mark. Financ. Tr., 55 (2019), 1247-1263.  doi: 10.1080/1540496X.2018.1496078.  Google Scholar

[47]

J. XiaoM. Zhou and F. Wen et al., Asymmetric impacts of oil price uncertainty on Chinese stock returns under different market conditions: Evidence from oil volatility index, Energ. Econ., 74 (2018), 777-786.  doi: 10.1016/j.eneco.2018.07.026.  Google Scholar

[48]

H. YaoZ. Li and Y. Lai, Mean-CVaR portfolio selection: A nonparametric estimation framework, Comput. Oper. Res., 40 (2013), 1014-1022.  doi: 10.1016/j.cor.2012.11.007.  Google Scholar

[49]

H. YaoY. Li and K. Benson, A smooth non-parametric estimation framework for safety-first portfolio optimization, Quant. Financ., 15 (2015), 1865-1884.  doi: 10.1080/14697688.2014.971857.  Google Scholar

[50]

K. Yu, A. Allay, S. Yang and D. J. Hand, Kernel quantile based estimation of expected shortfall, J. Risk., 12 (2010), 15-32. Google Scholar

[51]

G. YuanZ. H. Meng and Y. Li, A modified Hestenes and Stiefel conjugate gradient algorithm for large-scale nonsmooth minimizations and nonlinear equations, J. Optim. Theory. Appl., 168 (2016), 129-152.   Google Scholar

[52]

G. Zhao and Y. Y. Ma, Robust nonparametric kernel regression estimator, Stat. Probabil. Lett., 116 (2016), 72-79.  doi: 10.1016/j.spl.2016.04.010.  Google Scholar

[53]

Y. Zhao and F. Chen, Empirical likelihood inference for censored median regression model via nonparametric kernel estimation, J. Multivariate Anal., 99 (2008), 215-231.  doi: 10.1016/j.jmva.2007.05.002.  Google Scholar

[54]

Y. Zhao and H. Cui, Empirical likelihood for median regression model with designed censoring variables, J. Multivariate Anal., 101 (2010), 240-251.  doi: 10.1016/j.jmva.2009.07.008.  Google Scholar

Figure 1.  Comparision of efficient frontier of MAD portfolio models
Figure 2.  Efficient frontier of the MAD model based on the kernel median estimation
Figure 3.  Efficient frontier of the mean-CVaR portfolio models
Figure 4.  Efficient frontier of MAD models of kernel mean estimation under different kernel functions
Figure 5.  Efficient frontier of the mean-CVaR portfolio models of kernel mean estimation under different kernel functions
Figure 6.  Efficient frontier of MAD models of kernel median estimation under different kernel functions
Figure 7.  Efficient frontier of mean-CVaR models of kernel median estimation under different kernel functions
Figure 8.  Comparison betwen the daily return of Shanghai Composite Index and portfolio return of the MAD model
Figure 9.  Comparison betwen the daily return of Shanghai Composite Index and portfolio return of MAD model of kernel mean estimation
Figure 10.  Comparison betwen the daily return of Shanghai Composite Index and portfolio return of MAD of kernel median estimation
Figure 11.  Comparison betwen the daily return of Shanghai Composite Index and portfolio return of the mean-CVaR model
Figure 12.  Comparison betwen the daily return of Shanghai Composite Index and portfolio return of mean-CVaR model of kernel mean estimation
Figure 13.  Comparison betwen the daily return of Shanghai Composite Index and portfolio return of mean-CVaR of kernel median estimation
Table 1.  Descriptive Statistics of daily return of 20 stocks from Shanghai A shares
Stock nam Number Max Min Mean Sd. Skewness Kurtosis
P.R.E. 600048 0.10038 -0.35544 0.00049 0.03308 -1.47313 22.2281
T.J.H. 600717 0.10025 -0.10029 0.00074 0.03040 -0.08191 5.4232
H.E. 600060 0.10035 -0.11024 0.00109 0.03355 -0.00870 5.15926
H.N.I. 600011 0.10050 -0.10036 0.00064 0.02654 -0.28255 7.6319
N.J.H.T. 600064 0.10028 -0.32864 0.00112 0.03146 -1.69324 20.8171
S.H.I. 600031 0.10066 -0.10042 0.00015 0.02843 -0.14598 6.8630
S.H.A. 600009 0.09996 -0.10007 0.00112 0.02464 -0.09756 7.3309
T.R.T. 600085 0.10022 -0.10016 0.00109 0.02834 -0.04908 6.7348
Ch.M.B. 600036 0.09571 -0.09914 0.00087 0.01999 0.44853 8.3975
Ch.S. 600150 0.10016 -0.10011 0.00101 0.03593 -0.04488 4.6552
Ch.U. 600050 0.10103 -0.10056 0.00096 0.02846 0.21470 6.7539
Sino. 600028 0.10035 -0.10040 0.00037 0.02138 -0.15015 8.9252
Ch.S. 600118 0.10027 -0.10009 0.00170 0.03683 -0.03555 4.7026
C.S. 600030 0.10043 -0.10012 0.00091 0.03038 0.23151 5.9787
C.S.M. 601098 0.10027 -0.10017 0.00120 0.02910 -0.14500 5.4879
Ch.L.I. 601628 0.10036 -0.10007 0.00095 0.02689 0.62490 6.7596
O.F.X. 600612 0.10010 -0.10006 0.00109 0.02811 0.29691 5.8163
C.Q.B. 600132 0.10027 -0.10699 0.00042 0.02841 -0.37571 6.9771
J.J.I. 600650 0.10037 -0.10018 0.00202 0.03929 0.20423 4.3077
Q.J.B. 600706 0.10028 -0.10024 0.00125 0.03459 -0.26875 4.7732
Stock nam Number Max Min Mean Sd. Skewness Kurtosis
P.R.E. 600048 0.10038 -0.35544 0.00049 0.03308 -1.47313 22.2281
T.J.H. 600717 0.10025 -0.10029 0.00074 0.03040 -0.08191 5.4232
H.E. 600060 0.10035 -0.11024 0.00109 0.03355 -0.00870 5.15926
H.N.I. 600011 0.10050 -0.10036 0.00064 0.02654 -0.28255 7.6319
N.J.H.T. 600064 0.10028 -0.32864 0.00112 0.03146 -1.69324 20.8171
S.H.I. 600031 0.10066 -0.10042 0.00015 0.02843 -0.14598 6.8630
S.H.A. 600009 0.09996 -0.10007 0.00112 0.02464 -0.09756 7.3309
T.R.T. 600085 0.10022 -0.10016 0.00109 0.02834 -0.04908 6.7348
Ch.M.B. 600036 0.09571 -0.09914 0.00087 0.01999 0.44853 8.3975
Ch.S. 600150 0.10016 -0.10011 0.00101 0.03593 -0.04488 4.6552
Ch.U. 600050 0.10103 -0.10056 0.00096 0.02846 0.21470 6.7539
Sino. 600028 0.10035 -0.10040 0.00037 0.02138 -0.15015 8.9252
Ch.S. 600118 0.10027 -0.10009 0.00170 0.03683 -0.03555 4.7026
C.S. 600030 0.10043 -0.10012 0.00091 0.03038 0.23151 5.9787
C.S.M. 601098 0.10027 -0.10017 0.00120 0.02910 -0.14500 5.4879
Ch.L.I. 601628 0.10036 -0.10007 0.00095 0.02689 0.62490 6.7596
O.F.X. 600612 0.10010 -0.10006 0.00109 0.02811 0.29691 5.8163
C.Q.B. 600132 0.10027 -0.10699 0.00042 0.02841 -0.37571 6.9771
J.J.I. 600650 0.10037 -0.10018 0.00202 0.03929 0.20423 4.3077
Q.J.B. 600706 0.10028 -0.10024 0.00125 0.03459 -0.26875 4.7732
Table 2.  Descriptive statistics of daily return of Shanghai Composite Index
Max Min average sd Skewness Kurtosis
Shanghai S-I-R 0.04310 -0.07045 -0.00016 0.01566 -1.37612 8.38382
Max Min average sd Skewness Kurtosis
Shanghai S-I-R 0.04310 -0.07045 -0.00016 0.01566 -1.37612 8.38382
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