Figure 1.
Comparision of efficient frontier of MAD portfolio models
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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 |
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.
Mathematics Subject Classification:
Primary: 90C30, 90C90; Secondary: 65k05.
Citation:
Zhifeng Dai, Huan Zhu, Fenghua Wen.
Two nonparametric approaches to mean absolute deviation portfolio selection model. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019054
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References:
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R. Alemany, C. Bolancé and M. Guillén,
A nonparametric approach to calculating value-at-risk, Insurance Math. Econ., 52 (2013), 255-262.
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G. Athayde, Building a Mean-Downside Risk Portfolio Frontier, Developments in Forecast Combination and Portfolio Choice, John Wiley and Sons. 2001. Google Scholar |
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G. Athayde, The Mean-downside Risk Portfolio Frontier: A Non-parametric Approach, Published in Advances in portfolio construction and implementation. 2003. Google Scholar |
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A. Berlinet, B. Cadre and A. Gannoun,
On the conditional $L_1$
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A. Berlinet, A. Gannoun and E. Matzner,
Asymptotic normality of convergent estimates of conditional quantiles, Stats., 35 (2001b), 139-169.
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A semi-parametric approach to risk management, Quant. Financ., 3 (2003), 426-441.
doi: 10.1088/1469-7688/3/6/302. |
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Z. Cai and X. Wang,
Nonparametric estimation of conditional VaR and expected shortfall, J. Econom., 147 (2008), 120-130.
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S. Chen and C. Y. Tang, Nonparametric inference of value-atrisk for dependent financial returns, J. Financ. Econ., 3 (2005), 227-255. Google Scholar |
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S. Chen,
Nonparametric estimation of expected shortfall, J. Financ. Econ., 6 (2008), 87-107.
doi: 10.1093/jjfinec/nbm019. |
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L. Chiodi, R. Mansini and M. G. Speranza,
Semi-absolute deviation rule for mutual funds portfolio selection, Ann. Oper. Res., 124 (2003), 245-265.
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T. E. Conine and and M. J. Tamarkin, On diversiffication given asymmetry in returns, J. Financ., 36 (1981), 1143-1155. Google Scholar |
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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. |
| [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. Gannoun, J. 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. |
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J. G. Gooijer and A. Gannoun,
Tr multivariate conditional median estimation, Commun. Stat. Simul. C., 36 (2007), 165-176.
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B. E. Hansen, Bandwidth selection for nonparametric distribution estimation., Working Paper, 2004. Google Scholar |
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C. Huang, Z. Yang, T. Yi and X. Zou,
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doi: 10.1016/j.jde.2013.12.015. |
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Nonparametric estimation of valueat-risk, J. Appl. Stat., 36 (2009), 1225-1238.
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P. Jorion, Value at Risk: The New Benchmark for controlling Derivatives Risk, New York: McGraw-Hill. 1997. Google Scholar |
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H. Kellerer, R. Mansini and M. G. Speranza,
Selecting portfolios with fixed costs and minimum transaction lots, Ann. Oper. Res., 99 (2000), 287-304.
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R. Koenker, Quantile Regression, Cambridge Books, Cambridge University Press, Issue. 38, 2005.
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H. Konno and H. Yamazaki,
Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Manage. Sci., 37 (1991), 501-623.
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Q. Li and J. S. Racine, Nonparametric Econometrics: Theory and Practice, Princeton University Press, 2007.
Google Scholar
|
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R. Mansini, W. 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. |
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H. Markowitz, Portfolio Selection, J. Financ., 7 (1952), 77-91. Google Scholar |
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A. R. Pagan and A. Ullah, Nonparametric Econometrics, Cambridge University Press. 1999.
doi: 10.1017/CBO9780511612503. |
| [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. |
| [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. |
| [31] |
A. Perold,
Large scale portfolio selections, Manage. Sci., 30 (1984), 1143-1160.
doi: 10.1287/mnsc.30.10.1143. |
| [32] |
T. Rockfeller and S. Uryasev,
Optimization of conditional value-at-risk, J. Risk., 2 (2000), 21-34.
doi: 10.21314/JOR.2000.038. |
| [33] |
H. B. Salah, M. Chaouch, A. Gannoun, C. 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [39] |
S. Subramanian,
Median regression using nonparametric kernel estimation, J. Nonparametr. Stat., 14 (2002), 583-605.
doi: 10.1080/10485250213907. |
| [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. |
| [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. |
| [43] |
F. Wen, J. Xiao, C. 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. |
| [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. |
| [45] |
F. Wen, X. Yang and W. Zhou,
Tail dependence networks of global stock markets, Int. J. Financ. Econ., 24 (2019), 558-567.
doi: 10.1002/ijfe.1679. |
| [46] |
F. Wen, J. 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. |
| [47] |
J. Xiao, M. 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. |
| [48] |
H. Yao, Z. 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. |
| [49] |
H. Yao, Y. 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. |
| [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. Yuan, Z. 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. |
| [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. |
| [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. |
show all references
References:
| [1] |
R. Alemany, C. 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. |
| [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. Berlinet, B. Cadre and A. Gannoun,
On the conditional $L_1$
-median and its estimation, J. Nonparametr. Stat., 13 (2001a), 631-645.
doi: 10.1080/10485250108832869. |
| [5] |
A. Berlinet, A. Gannoun and E. Matzner,
Asymptotic normality of convergent estimates of conditional quantiles, Stats., 35 (2001b), 139-169.
doi: 10.1080/02331880108802728. |
| [6] |
N. Bingham, R. 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. |
| [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. |
| [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. |
| [10] |
L. Chiodi, R. 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. |
| [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. |
| [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. Gannoun, J. 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. |
| [16] |
J. G. Gooijer and A. Gannoun,
Tr multivariate conditional median estimation, Commun. Stat. Simul. C., 36 (2007), 165-176.
doi: 10.1080/03610910601096270. |
| [17] |
B. E. Hansen, Bandwidth selection for nonparametric distribution estimation., Working Paper, 2004. Google Scholar |
| [18] |
Z. He, L. 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. Huang, Z. Yang, T. 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. |
| [20] |
S. O. Jeong and K. H. Kang,
Nonparametric estimation of valueat-risk, J. Appl. Stat., 36 (2009), 1225-1238.
doi: 10.1080/02664760802607517. |
| [21] |
P. Jorion, Value at Risk: The New Benchmark for controlling Derivatives Risk, New York: McGraw-Hill. 1997. Google Scholar |
| [22] |
H. Kellerer, R. 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. |
| [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. |
| [25] |
Q. Li and J. S. Racine, Nonparametric Econometrics: Theory and Practice, Princeton University Press, 2007.
Google Scholar
|
| [26] |
R. Mansini, W. 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. |
| [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. |
| [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. |
| [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. |
| [31] |
A. Perold,
Large scale portfolio selections, Manage. Sci., 30 (1984), 1143-1160.
doi: 10.1287/mnsc.30.10.1143. |
| [32] |
T. Rockfeller and S. Uryasev,
Optimization of conditional value-at-risk, J. Risk., 2 (2000), 21-34.
doi: 10.21314/JOR.2000.038. |
| [33] |
H. B. Salah, M. Chaouch, A. Gannoun, C. 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [39] |
S. Subramanian,
Median regression using nonparametric kernel estimation, J. Nonparametr. Stat., 14 (2002), 583-605.
doi: 10.1080/10485250213907. |
| [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. |
| [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. |
| [43] |
F. Wen, J. Xiao, C. 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. |
| [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. |
| [45] |
F. Wen, X. Yang and W. Zhou,
Tail dependence networks of global stock markets, Int. J. Financ. Econ., 24 (2019), 558-567.
doi: 10.1002/ijfe.1679. |
| [46] |
F. Wen, J. 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. |
| [47] |
J. Xiao, M. 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. |
| [48] |
H. Yao, Z. 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. |
| [49] |
H. Yao, Y. 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. |
| [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. Yuan, Z. 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. |
| [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. |
| [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. |
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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