多项条件下收益率的分布及其应用研究
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摘要
股票收益率的分布是现代金融理论中的一个极其重要概念。一般认为,大多数股票的收益率服从正态分布,这些正态分布假设收益率的分布与价格无关,证券市场是一个有效的市场。然而,在文献[6]中,肖春来等认为由于市场内外部条件的变化,往往使收益率的统计分布特征发生变化,从而使传统的风险理论在实际应用中受到限制。因此,目前假定某种证券的收益率的统计分布特征在一定时期内基本稳定不符合实际的市场情况。他们以股票市场为例,提出价格条件收益率的思想,即在一定价格水平上的收益率。实际上,在文献[7]中,柴文义等通过对国内外股票市场的研究,结果表明股票收益率与股票价格存在弱的负相关关系。在此基础上,文献[8]在股票收益率和价格对数服从二元联合正态分布的假设下,推导出了价格条件下的股票收益率分布特征表达式和VaR_t的计算方法。
然而,即便如此,由于股票市场瞬息万变,其收益率受到多种因素的影响。价格条件下收益率分布特征中的价格均值μ、方差σ~2和自相关系数ρ就再不能被看作是常数,而应该将其当成变量,即μ_t、σ_t~2和ρ_t,以使收益率的分布特征更符合市场实际。因此,研究多项条件下收益率的分布就显得尤为重要和紧迫。
本文首先详细分析了价格条件下收益率分布在VaR中的应用情况,得出了应将价格均值μ、方差σ~2和自相关系数ρ看成变量而不是常量的结论,而且可用移动平均法达到这一目的;其次,经过对20支股票大量的实证研究,初步确定了移动平均期数的范围,且初步表明收益率分布不仅仅决定于价格,还决定于价格均值μ、方差σ_t~2和自相关系数ρ_t等变量,即收益率服从多项条件分布;再次,为了使价格均值μ、方差σ_t~2和相关系数ρ_t更具预测性,更好的应用于实际,本文对它们分别建立了预测模型;最后,将预测模型应用于VaR_t的计算,并对其预测效果进行了检验。结果表明,用预测值计算的VaR_t值较用移动平均值计算的VaR_t值更准确,这也进一步表明收益率服从多项条件分布。
The distribution of the stock yield is an extremely important concept of modern financial theory.It is generally believed that most stocks possess normal distribution,the stock yield distribution has nothing to do with price,and the stock market is effective.But in paper[6],Xiao Chun-lai,etc,assumed that the conditional distribution theory will encounter difficulties in application of the risk theory because of the changed statistical characters of the distribution which is changing as the external and internal conditions of stock market are changing. Therefore,the hypothesis that the statistical characters of stock yield distribution are stable in a certain period of time is not fitted with the actual market situation. In fact,in paper[7],CHAI Wen-yi points that the stock price has some weakly negative relation with stock yield by studying native and foreign stock markets. Xiao Chun-lai,CHAI Wen-yi have gotten the formulas of the statistical characters of stock yield distribution and VaR_t based on the hypothesis that the stock yield and price possess the joint normal distribution in paper[8].
However,as the stock market is changing time in time and full of complexities,the stock yield is affected by many factors.The mean priceμ, varianceσ~2 and the autocorrelation coefficientρin the stock yield distribution determined by price can no longer be seen as constant,but variables,that isμ~2,σ_t~2 andρ_t,which makes the stock yield distribution more fitted with the actual market.Therefore,studying the stock yield distribution determined by multi- conditions is particularly important and urgent.
In this paper the inaccuracy of VaR brought by the stock yield distribution determined by the stock price is firstly analyzed.The result is that the mean priceμ,varianceσ~2 and autocorrelation coefficientρshould be seen as variables rather than the constant which can be achieved by using moving average method.Secondly,through studying 20 stocks,we initially specified the moving-average range,and it initially demonstrates that the stock yield distribution is not only determined by the price,but also by the mean priceμ_t, varianceσ_t~2 and autocorrelation coefficientsρ_t.That is,the stock yield distribution possesses multi-conditional distribution.Thirdly in order to make the price meanμ_t,varianceσ_t~2 and autocorrelation coefficientsρ_t predictable and practical,their predicted models are set up in this paper.Lastly the predicted models are used to calculate VaR_t,and their predicted effects had been tested.The results show that,the VaR_t calculated by the predicted value is more accurate than that of calculated by the moving average value,which further shows that the stock yield possesses the multi-conditional distribution.
然而,即便如此,由于股票市场瞬息万变,其收益率受到多种因素的影响。价格条件下收益率分布特征中的价格均值μ、方差σ~2和自相关系数ρ就再不能被看作是常数,而应该将其当成变量,即μ_t、σ_t~2和ρ_t,以使收益率的分布特征更符合市场实际。因此,研究多项条件下收益率的分布就显得尤为重要和紧迫。
本文首先详细分析了价格条件下收益率分布在VaR中的应用情况,得出了应将价格均值μ、方差σ~2和自相关系数ρ看成变量而不是常量的结论,而且可用移动平均法达到这一目的;其次,经过对20支股票大量的实证研究,初步确定了移动平均期数的范围,且初步表明收益率分布不仅仅决定于价格,还决定于价格均值μ、方差σ_t~2和自相关系数ρ_t等变量,即收益率服从多项条件分布;再次,为了使价格均值μ、方差σ_t~2和相关系数ρ_t更具预测性,更好的应用于实际,本文对它们分别建立了预测模型;最后,将预测模型应用于VaR_t的计算,并对其预测效果进行了检验。结果表明,用预测值计算的VaR_t值较用移动平均值计算的VaR_t值更准确,这也进一步表明收益率服从多项条件分布。
The distribution of the stock yield is an extremely important concept of modern financial theory.It is generally believed that most stocks possess normal distribution,the stock yield distribution has nothing to do with price,and the stock market is effective.But in paper[6],Xiao Chun-lai,etc,assumed that the conditional distribution theory will encounter difficulties in application of the risk theory because of the changed statistical characters of the distribution which is changing as the external and internal conditions of stock market are changing. Therefore,the hypothesis that the statistical characters of stock yield distribution are stable in a certain period of time is not fitted with the actual market situation. In fact,in paper[7],CHAI Wen-yi points that the stock price has some weakly negative relation with stock yield by studying native and foreign stock markets. Xiao Chun-lai,CHAI Wen-yi have gotten the formulas of the statistical characters of stock yield distribution and VaR_t based on the hypothesis that the stock yield and price possess the joint normal distribution in paper[8].
However,as the stock market is changing time in time and full of complexities,the stock yield is affected by many factors.The mean priceμ, varianceσ~2 and the autocorrelation coefficientρin the stock yield distribution determined by price can no longer be seen as constant,but variables,that isμ~2,σ_t~2 andρ_t,which makes the stock yield distribution more fitted with the actual market.Therefore,studying the stock yield distribution determined by multi- conditions is particularly important and urgent.
In this paper the inaccuracy of VaR brought by the stock yield distribution determined by the stock price is firstly analyzed.The result is that the mean priceμ,varianceσ~2 and autocorrelation coefficientρshould be seen as variables rather than the constant which can be achieved by using moving average method.Secondly,through studying 20 stocks,we initially specified the moving-average range,and it initially demonstrates that the stock yield distribution is not only determined by the price,but also by the mean priceμ_t, varianceσ_t~2 and autocorrelation coefficientsρ_t.That is,the stock yield distribution possesses multi-conditional distribution.Thirdly in order to make the price meanμ_t,varianceσ_t~2 and autocorrelation coefficientsρ_t predictable and practical,their predicted models are set up in this paper.Lastly the predicted models are used to calculate VaR_t,and their predicted effects had been tested.The results show that,the VaR_t calculated by the predicted value is more accurate than that of calculated by the moving average value,which further shows that the stock yield possesses the multi-conditional distribution.
引文
[1]Bachelier,L.Theory of Speculaliou,in P.Cootner,ed.The Random Character of Stock Price[M].Cambridge,MA:M.I.T Press,1964.
[2]Robert C,Blattberg and Nicholas J,Gonedes.A comparison of Student and Stable Distribution as Statistical Models Stock Prices[J].Journal of Business.1974,47.
[3]Osborne.M.F.M.Brownian Motion in the Stock Market,in P.Cootner,ed.The Random Character of Stock Market Price[M]Cambridge,MA:M.I.T Press,1964.Box GEP,TiaoG.C.,Afurther look at robustness via Bayes'stheorem[J].Biometreka,1962,(49):419-432.
[4]JP.A.Samuelson Efficient portfolio selection for pareto-levy investment[J].The Journal of Financial and Quantitative Analysis,1967,12(3):107-122.
[5]陶亚民,蔡明超,杨朝军.上海股票市场收益率分布特征的研究[J].预测,1999,(2):57-58.
[6]肖春来,丁绍芳,洪媛.条件收益率下的VaR分析[J].北方工业大学学报,2003,15(3):88-94.
[7]柴文义,肖春来,相斐.股票价格(指数)与收益率的相关性分析[J].北方工业大学学报,2005,17(1):72-76.
[8]肖春来,柴文义.基于二维正态分布的条件VaR研究[J].数学的实践与认识,2007,37(9):144-147.
[9]Mandelbrot B.The Variation of Certain Speculative Prices.Journal of Business,1963,36:394-419.
[10]Mittnik S,Rachev S T.Modelling Asset Returns with Alternative Stable Distributions.Econometric Rev.1993,12(3):261-330.
[11]Smith R L.Estimation Tails of Probability Distributions.Ann Statist,1987,15:1174-1207.
[12]石美娟.ARIMA模型在上海市全社会固定资产投资预测中的应用[J].数理统计与管理,2005,24(1):69-74.
[13]万建强,文洲.ARIMA模型与ARCH模型在香港股指预测方面的应用比较[J].数理统计与管理,2001,20(5):1-4.
[14]韩卫国.ARIMA模型在GDP预测中的应用[J].内蒙古科技与经济,2007,(16):242-243.
[15]吴海军.APIMA模型在北京市全社会固定资产投资预测中的应用[J].经济研究导刊,2007,(2):131-133.
[16]尚林,秦伟良.基于ARIMA和EARCH模型的中国入境旅游收汇预测分析[J].统计教育,2007,(4):46-48.
[17]罗佐县.利用ARIMA模型预测2007年油价走势[J].天然气技术,2007,1(1):46-48.
[18]Jorion P,Risk:Measuring the Risk in Value at Risk.Financial Analysts Journal,1996,November/December:47-55.
[19]Nolan J P.Fitting Data and Assessing Goodness-of-fit with Stable Distributions[D].Washington DC:Department of Mathematics and Statistics,American University,1999.
[20]Kozubowski T J,Podnorski K.Asymmetric Laplace Laws and Modeling Financial Data[J].Mathematical and Computer Modeling,2001,34(9-11):1003-1021.
[21]Katerina Simons,"Value at Risk New Approaches to Risk Management New England Economic Review,September/Ocotober 1996.
[22]Fama,E.The Behabior of Stock Market Prices[J].Journal of Business,1965,(38):34-105.
[23]池启水.中国石油消费量增长趋势分析--基于ARIMA模型的预测与分析[J].资源科学,2007,29(5):69-73.
[24]徐峰,王志芳,王宝圣.AR模型应用于振动信号趋势预测的研究[J].清华大学学报(自然科学版),1999,39(4):57-59.
[25]熊建平,吴建华,万国金.AR模型在人口增长预测中的应用[J].计算机与现代化,2005,(10):11-12.
[26]阎长俊.AR模型的建模与预测[J].沈阳建筑工程学院学报,1997,13(4).
[27]熊波,李国林等.AR模型在时间序列误差分析中的应用[J].海军航空工程学院学报,2005,20(2):248-250.
[28]王红军等.非线性时间序列建模的异方差混合双AR模型[J].控制理论与应用,2006,23(6):779-885.
[29]Paret.P.D..The Distribution of Share price Changes[J],Journal of Business.1972,(45):317-335 Peters.E..E..Chaos and Order in the Capital Markets[M].John Wiely &Sons,Inc..New York,1991.
[30]Hosking,J.R.M.(1990).L moments:Analysis and Estimation of Distributions using Linear Combinations of Order Statistics[R].J.R.Statist.Soc.B,52(1):105-124.
[31]Kon S.Midels of stock return-a comparison[J].Journal of Finance,1984,39:147-165.
[32]王振龙,胡永宏.应用时间序列分析.北京:科学出版社,2007
[33]张树京,齐立心.时间序列分析简明教程.北京:清华大学、北方交通大学出版社,2003
[34]伍尤桂,李元等.应用时间序列分析.广西:广西师范大学出版社,1998
[35]易丹辉.数据分析与Eviews应用.北京:中国统计出版社,2005
[2]Robert C,Blattberg and Nicholas J,Gonedes.A comparison of Student and Stable Distribution as Statistical Models Stock Prices[J].Journal of Business.1974,47.
[3]Osborne.M.F.M.Brownian Motion in the Stock Market,in P.Cootner,ed.The Random Character of Stock Market Price[M]Cambridge,MA:M.I.T Press,1964.Box GEP,TiaoG.C.,Afurther look at robustness via Bayes'stheorem[J].Biometreka,1962,(49):419-432.
[4]JP.A.Samuelson Efficient portfolio selection for pareto-levy investment[J].The Journal of Financial and Quantitative Analysis,1967,12(3):107-122.
[5]陶亚民,蔡明超,杨朝军.上海股票市场收益率分布特征的研究[J].预测,1999,(2):57-58.
[6]肖春来,丁绍芳,洪媛.条件收益率下的VaR分析[J].北方工业大学学报,2003,15(3):88-94.
[7]柴文义,肖春来,相斐.股票价格(指数)与收益率的相关性分析[J].北方工业大学学报,2005,17(1):72-76.
[8]肖春来,柴文义.基于二维正态分布的条件VaR研究[J].数学的实践与认识,2007,37(9):144-147.
[9]Mandelbrot B.The Variation of Certain Speculative Prices.Journal of Business,1963,36:394-419.
[10]Mittnik S,Rachev S T.Modelling Asset Returns with Alternative Stable Distributions.Econometric Rev.1993,12(3):261-330.
[11]Smith R L.Estimation Tails of Probability Distributions.Ann Statist,1987,15:1174-1207.
[12]石美娟.ARIMA模型在上海市全社会固定资产投资预测中的应用[J].数理统计与管理,2005,24(1):69-74.
[13]万建强,文洲.ARIMA模型与ARCH模型在香港股指预测方面的应用比较[J].数理统计与管理,2001,20(5):1-4.
[14]韩卫国.ARIMA模型在GDP预测中的应用[J].内蒙古科技与经济,2007,(16):242-243.
[15]吴海军.APIMA模型在北京市全社会固定资产投资预测中的应用[J].经济研究导刊,2007,(2):131-133.
[16]尚林,秦伟良.基于ARIMA和EARCH模型的中国入境旅游收汇预测分析[J].统计教育,2007,(4):46-48.
[17]罗佐县.利用ARIMA模型预测2007年油价走势[J].天然气技术,2007,1(1):46-48.
[18]Jorion P,Risk:Measuring the Risk in Value at Risk.Financial Analysts Journal,1996,November/December:47-55.
[19]Nolan J P.Fitting Data and Assessing Goodness-of-fit with Stable Distributions[D].Washington DC:Department of Mathematics and Statistics,American University,1999.
[20]Kozubowski T J,Podnorski K.Asymmetric Laplace Laws and Modeling Financial Data[J].Mathematical and Computer Modeling,2001,34(9-11):1003-1021.
[21]Katerina Simons,"Value at Risk New Approaches to Risk Management New England Economic Review,September/Ocotober 1996.
[22]Fama,E.The Behabior of Stock Market Prices[J].Journal of Business,1965,(38):34-105.
[23]池启水.中国石油消费量增长趋势分析--基于ARIMA模型的预测与分析[J].资源科学,2007,29(5):69-73.
[24]徐峰,王志芳,王宝圣.AR模型应用于振动信号趋势预测的研究[J].清华大学学报(自然科学版),1999,39(4):57-59.
[25]熊建平,吴建华,万国金.AR模型在人口增长预测中的应用[J].计算机与现代化,2005,(10):11-12.
[26]阎长俊.AR模型的建模与预测[J].沈阳建筑工程学院学报,1997,13(4).
[27]熊波,李国林等.AR模型在时间序列误差分析中的应用[J].海军航空工程学院学报,2005,20(2):248-250.
[28]王红军等.非线性时间序列建模的异方差混合双AR模型[J].控制理论与应用,2006,23(6):779-885.
[29]Paret.P.D..The Distribution of Share price Changes[J],Journal of Business.1972,(45):317-335 Peters.E..E..Chaos and Order in the Capital Markets[M].John Wiely &Sons,Inc..New York,1991.
[30]Hosking,J.R.M.(1990).L moments:Analysis and Estimation of Distributions using Linear Combinations of Order Statistics[R].J.R.Statist.Soc.B,52(1):105-124.
[31]Kon S.Midels of stock return-a comparison[J].Journal of Finance,1984,39:147-165.
[32]王振龙,胡永宏.应用时间序列分析.北京:科学出版社,2007
[33]张树京,齐立心.时间序列分析简明教程.北京:清华大学、北方交通大学出版社,2003
[34]伍尤桂,李元等.应用时间序列分析.广西:广西师范大学出版社,1998
[35]易丹辉.数据分析与Eviews应用.北京:中国统计出版社,2005