HD型VaR估计的研究
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摘要
VaR(Value at Risk)是目前比较流行的一种风险度量方法.由于VaR概念简单、易懂、操作方便,能成功地将风险量化为一个数,因而受到全球各主要银行、投资公司、证券公司及金融监管机构的青睐.大部分计算VaR的方法都需要假定收益的分布,再进一步建立模型.实际上金融回报序列的分布通常呈现尖峰厚尾性,分布形式更加复杂,而分布往往是未知的,因此,在计算VaR的过程中,常常因模型选择的不恰当而产生很大的误差.而非参数方法不需要假定回报序列的统计分布,是一种分布自由的方法,可有效处理回报序列的非对称和厚尾性问题.
我们先讨论了Harrell-Davis估计量(1982)在样本满足α-混合及其他条件下的渐近性质,给出了它的方差速度、偏差速度和均方误差速度,并通过一些引理证明性质.
Koji Inui, Masaaki Kijima, Atsushi Kitano(2005)研究了利用Harrll-Davis估计量作为VaR估计量的一些特征,指出当k = [np]时, Harrell-Davis估计出现明显的正偏差,即产生高估现象.
接着,通过数值模拟我们发现:当k = np时,Harrell-Davis估计作为VaR估计同样出现明显的正偏差;而当k = np + 1时, Harrell-Davis估计则出现明显的负偏差.由于它们出现正偏差和负偏差,所以我们建议使用它们的加权估计作为VaR估计,即HDpM = 12(HDpn p+ HDpn p+1)
这种加权估计对HDpn p的正偏差和HDpn p+1的负偏差得到了较好的修正.以相对偏差的绝对值为方法估计精确度的参考标准,分析三种Harrell-Davis估计量.在同一样本和相同的概率p下,三种HD型VaR估计值中, HDpn p值最大,最小的是HDpn p+1, HDpM介于前两者.HDpM的精度是显著优于HDpn p和HDpn p+1的,无论是在薄尾还是厚尾分布下,在样本比较大时,估计的精度都得到较大的改进.在相同的样本容量和同一分布下, HDpM的相对偏差有正有负,并且相对偏差的绝对值都是比较小的,即精确性较高,这也说明HDpM估计量相对稳定,可以相对较好的估计VaR.
本文就上证指数和深成指数、金融指数、地产指数、机械指数和制造指数在2007年1月4日到2008年12月31日期间进行实证分析,多角度考察了他们的日对数收益率的分布特征,都拒绝了日对数收益率服从正态分布的假设.通过Harrell-Davis估计量计算VaR,并进行沪深两市风险比较及四种行业间风险分析.分析结果说明了深市的风险大于沪市,不同行业因行业的特性具有不同的风险程度.
VaR (Value at Risk) is a popular method of risk measurement at present. VaR has attractedmany global banks, investment companies, securities companies and financial regulators for itssimple concept, easy-to-understand, easy-to-operate and ability of quantifying the risk into a num-ber successfully. Most methods for calculating VaR require assumpting the distribution of returns,and then set up the further model. In fact, the distribution of financial return series is usually peak,fat-tail. The form of the distribution is usually complex and unknown, therefore, in the process ofcalculating the VaR, there is usually a serious bias caused by the inappropriate model selection.However, Non-parametric method is a free distribution method which require no assumption of thestatistical distribution of return series, it can deal with non-symmetric and fat-tail problems of thereturns effectively.
We first discuss the asymptotic properties of the Harrell-Davis estimator (1982)under sam-ples satisfy the alpha - the mixed and other conditions, give the speed of variance, the speed ofdeviation and the speed of mean-square error, and prove the properties by some lemmas.Koji Inui, Masaaki Kijima, Atsushi Kitano (2005) have studied some characteristics that useHarrll-Davis estimator as VaR estimates, and pointed out that when k = [np], there is an obviousbias in Harrell-Davis estimate -namely, it is over-estimated.
Then,we have found by numerical simulation: when k = np, there is also an obvious positivebias using Harrell-Davis estimator as VaR estimator; when k = np+1, there is an obvious negativebias.
Because of the positive and negative bias, we suggest to use the weighted estimators as VaRestimate-namely,HDpM = 21(HDpn p+ HDpn p+1)this weighted estimator can adjust the positive bias of the HDpn pestimator and the negativebias of HDpn p+1soundly .We take the absolute value of relative bias as the reference standards forthe estimation accuracy,and analysis three Harrell-Davis estimators.Under the same sample andprobability p, HDpn pvalue is the largest among the three types of HD-estimator, and HDpn p+1thesmallest, HDpM between the two above.
The accuracy of HDpM is significantly better than HDpn pand HDpn p+1, no matter under thin- tailed distribution or thick-tailed distribution.when samples size is relatively large, the precision ofestimator has been improved. Under the same sample size and the same distribution, the relativebias of HDpM is maybe negative or positive, and the absolute value of the relative bias is relativelysmall, that means higher accuracy, which also shows that the HDpM estimator is relatively stable,and can estimate VaR more accrutely .
In this paper,we make a demonstration analysis with Shanghai Composite Index ,Shenzhencomponent index, financial index, real-estate index, mechanical index and the manufacturing indexfrom January 4, 2007 to December 31, 2008. We study the distribution characteristics of their dailylog-return-rate from Multi-angle and refuse the assumption that the daily log-return-rate followsnormal distribution. We calculate VaR by Harrell-Davis estimator, and compare the risk of Shang-hai and Shenzhen stock markets and risk analysis of the four industries. The results of the analysisshow that the risk of Shenzhen stock markets is greater than that of Shanghai stock markets, andthat due to different industry sectors ,different industries have different degree of risk.
我们先讨论了Harrell-Davis估计量(1982)在样本满足α-混合及其他条件下的渐近性质,给出了它的方差速度、偏差速度和均方误差速度,并通过一些引理证明性质.
Koji Inui, Masaaki Kijima, Atsushi Kitano(2005)研究了利用Harrll-Davis估计量作为VaR估计量的一些特征,指出当k = [np]时, Harrell-Davis估计出现明显的正偏差,即产生高估现象.
接着,通过数值模拟我们发现:当k = np时,Harrell-Davis估计作为VaR估计同样出现明显的正偏差;而当k = np + 1时, Harrell-Davis估计则出现明显的负偏差.由于它们出现正偏差和负偏差,所以我们建议使用它们的加权估计作为VaR估计,即HDpM = 12(HDpn p+ HDpn p+1)
这种加权估计对HDpn p的正偏差和HDpn p+1的负偏差得到了较好的修正.以相对偏差的绝对值为方法估计精确度的参考标准,分析三种Harrell-Davis估计量.在同一样本和相同的概率p下,三种HD型VaR估计值中, HDpn p值最大,最小的是HDpn p+1, HDpM介于前两者.HDpM的精度是显著优于HDpn p和HDpn p+1的,无论是在薄尾还是厚尾分布下,在样本比较大时,估计的精度都得到较大的改进.在相同的样本容量和同一分布下, HDpM的相对偏差有正有负,并且相对偏差的绝对值都是比较小的,即精确性较高,这也说明HDpM估计量相对稳定,可以相对较好的估计VaR.
本文就上证指数和深成指数、金融指数、地产指数、机械指数和制造指数在2007年1月4日到2008年12月31日期间进行实证分析,多角度考察了他们的日对数收益率的分布特征,都拒绝了日对数收益率服从正态分布的假设.通过Harrell-Davis估计量计算VaR,并进行沪深两市风险比较及四种行业间风险分析.分析结果说明了深市的风险大于沪市,不同行业因行业的特性具有不同的风险程度.
VaR (Value at Risk) is a popular method of risk measurement at present. VaR has attractedmany global banks, investment companies, securities companies and financial regulators for itssimple concept, easy-to-understand, easy-to-operate and ability of quantifying the risk into a num-ber successfully. Most methods for calculating VaR require assumpting the distribution of returns,and then set up the further model. In fact, the distribution of financial return series is usually peak,fat-tail. The form of the distribution is usually complex and unknown, therefore, in the process ofcalculating the VaR, there is usually a serious bias caused by the inappropriate model selection.However, Non-parametric method is a free distribution method which require no assumption of thestatistical distribution of return series, it can deal with non-symmetric and fat-tail problems of thereturns effectively.
We first discuss the asymptotic properties of the Harrell-Davis estimator (1982)under sam-ples satisfy the alpha - the mixed and other conditions, give the speed of variance, the speed ofdeviation and the speed of mean-square error, and prove the properties by some lemmas.Koji Inui, Masaaki Kijima, Atsushi Kitano (2005) have studied some characteristics that useHarrll-Davis estimator as VaR estimates, and pointed out that when k = [np], there is an obviousbias in Harrell-Davis estimate -namely, it is over-estimated.
Then,we have found by numerical simulation: when k = np, there is also an obvious positivebias using Harrell-Davis estimator as VaR estimator; when k = np+1, there is an obvious negativebias.
Because of the positive and negative bias, we suggest to use the weighted estimators as VaRestimate-namely,HDpM = 21(HDpn p+ HDpn p+1)this weighted estimator can adjust the positive bias of the HDpn pestimator and the negativebias of HDpn p+1soundly .We take the absolute value of relative bias as the reference standards forthe estimation accuracy,and analysis three Harrell-Davis estimators.Under the same sample andprobability p, HDpn pvalue is the largest among the three types of HD-estimator, and HDpn p+1thesmallest, HDpM between the two above.
The accuracy of HDpM is significantly better than HDpn pand HDpn p+1, no matter under thin- tailed distribution or thick-tailed distribution.when samples size is relatively large, the precision ofestimator has been improved. Under the same sample size and the same distribution, the relativebias of HDpM is maybe negative or positive, and the absolute value of the relative bias is relativelysmall, that means higher accuracy, which also shows that the HDpM estimator is relatively stable,and can estimate VaR more accrutely .
In this paper,we make a demonstration analysis with Shanghai Composite Index ,Shenzhencomponent index, financial index, real-estate index, mechanical index and the manufacturing indexfrom January 4, 2007 to December 31, 2008. We study the distribution characteristics of their dailylog-return-rate from Multi-angle and refuse the assumption that the daily log-return-rate followsnormal distribution. We calculate VaR by Harrell-Davis estimator, and compare the risk of Shang-hai and Shenzhen stock markets and risk analysis of the four industries. The results of the analysisshow that the risk of Shenzhen stock markets is greater than that of Shanghai stock markets, andthat due to different industry sectors ,different industries have different degree of risk.
引文
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[21]黄雄艳.上证指数的VaR风险测量及有效性分析[J].长沙大学学报, 2005, 19(6): 38-41.
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[26] Yu H. A Note on Strong Approximation for Quantile Processes of Strong Mixing Se-quence[J]. Statistics and Probability Letters, 1995, 30: 1-7.
[27] Xianglan Wei , Shanchao Yang,Keming Yu.Bahadur Representation of Linear Kernel Quan-tile Estimator of VaR under Mixing Assumptions.待发表
[28] Shao Q. Exponential Inequalities for Dependent Random Variables[J]. Acta Mathematicaeapplicate Sinica, 1990, 4, 338-350.
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[32] Genon-Catalot, V., Jeantheau, T. and Laredo, C.. Stochastic Volatility models as hiddenMarkov models and statistical applications[J]. Bernoulli, 2000, 6: 1051-1079.
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[38] Siegl, T and West, A.. Statistical Bootstrappling Methods in VaR Calculation [J]. AppliedMathematical Finance ,2001, 8: 167-181.
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[2] Danielsson, J., Jorgensen, B.N., Sarma, M. and Vries,C.G. Sub-additivity Re-examined: theCase for Value-at-Risk[J]. Working paper, 2005.
[3] Smithson, C. and Minton, L. Value at Risk[M]. 1996, 9: 25-27.
[4] Duffie D, Pan J. An overview of value at risk[J]. Journal of Derivatives, 1997, 4(3): 7-49.
[5] Jorion P. Value at Risk, 2nd Edition[M]. New York : McGraw-Hill, 2001.
[6] Boudoukh J, Richardson M, Whitelaw R. The best of both worlds[J]. Risk, 1998, 11(5):64-66.
[7] Hull J C, White AW. Incorporating Volatility up-dating into the Historical Simulationmethod for Value at Risk [J]. Journal of Risk, 1998, 1(1): 5-19.
[8] Monica B, Loriana P. Value at Risk:a multivariate switching regime approach[J]. Journal ofEmpirical Finance, 2000,7(5):531-554.
[9]王春锋.金融市场风险管理[M].天津:天津大学出版社,2001.
[10]郑文通.金融风险管理的VaR方法及其应用[J].国际金融研究, 1997, (9): 58-62.
[11]牛昂.银行风险管理的新方法[J].国际金融研究, 1997, 13(4): 61-65.
[12]姚刚.风险值测定法浅析[J].经济科学, 1998, 19(1): 56-61.
[13]马超群,李红权. VaR方法及其在金融风险管理中应用[J].系统工程, 2000. 18(2):56-59.
[14]刘宇飞. VaR模型及其在金融监管中的应用[J].经济科学, 1999, 1.
[15]詹原瑞.市场风险的量度: VaR的计算与应用[J].系统工程理论与实践, 1999. 12: 1-7.
[16]范英. VaR方法及其在股市风险分析中的应用初探.中国管理科学, 2000, 8(3): 26-32.
[17]王春峰,万海辉,李刚.基于MCMC的金融市场风险VaR的估计[J].管理科学学报,2000 ,3(2): 54-61.
[18]陈守东.基于GARCH模型的VAR方法对中国股市的分析[J].吉林大学社会科学学报, 2002, 7(4): 28-33.
[19]邱阳,林勇. VaR模型及其在股票风险评估中的应用[J].重庆大学学报2002, (2):34-36.
[20]姚奎栋. VaR的计算方法[J].沈阳工业航空工业学报, 2002, 19(3).
[21]黄雄艳.上证指数的VaR风险测量及有效性分析[J].长沙大学学报, 2005, 19(6): 38-41.
[22] Gourieroux C. , Scaillet O. and Laurent J P. . Sensitivity Analysis of Values at Risk [J].Journal of Empirical Finance, 2000, 7, 225-245.
[23] Dowd, K.. Estimating VaR with Order Statistics[J]. Journal of Derivatives, 2001: 23-30.
[24] Koji Inui, Masaaki Kijima, Atsushi Kitano. VaR is subject to a significant positive bias[J].Statistics&Probability Letters, 2005, 72(4): 299-311.
[25] Harrell, F.El, Davis,C.E. . A new distribution-free quantile estimator[J]. Biometrika, 1982,69, 635-640.
[26] Yu H. A Note on Strong Approximation for Quantile Processes of Strong Mixing Se-quence[J]. Statistics and Probability Letters, 1995, 30: 1-7.
[27] Xianglan Wei , Shanchao Yang,Keming Yu.Bahadur Representation of Linear Kernel Quan-tile Estimator of VaR under Mixing Assumptions.待发表
[28] Shao Q. Exponential Inequalities for Dependent Random Variables[J]. Acta Mathematicaeapplicate Sinica, 1990, 4, 338-350.
[29] Bosq D. Nonpapametric Statistics for Stochastic Processes[M]. Heidelberg: Spinger-Verlag,1998: 110.
[30] Chen, S. X. and Tang, C. Y. Nonparametric Inference of Value at Risk for Dependent Finan-cial Returns[J]. Journal of Financial Econometics, 2005, 3: 227-255.
[32] Chen, J. and Qin, J.. Empirical likelihood estimation for finite populations and the effectiveusage of auxiliary information[J]. Biometrika, 1993, 80: 107-116.
[32] Genon-Catalot, V., Jeantheau, T. and Laredo, C.. Stochastic Volatility models as hiddenMarkov models and statistical applications[J]. Bernoulli, 2000, 6: 1051-1079.
[33] Moricz, F.. Moment inequalities and the strong laws of large numbers[J]. Z. Wahrschein-lichkeitstheorie verw. Gebiete, 1976, 35: 299-314.
[34] Hwang, S. Y. and Kim, T. Y.. Power transformation and threshold modeling for ARCHinnovations with applications to tests for ARCH structure[J]. Stochastic Processes and TheirApplications, 2004, 110: 295-314.
[35] Inone, A. and Kilian, L.. Bootstrapping Smoothing Functions of Slope Parametres and In-novation Variances in VaR Models [J]. International Economic Review, 2002.
[36] Lealie, A. B.. Measuring Investing Risk : A Review [M]. The Journal of Investing, 1995.
[37] Parzen,E.. Nonparametric Statistical Data Modeling[J]. J. American Statistical Association,1979, 74: 105-121.
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