基于Copula函数的投资组合风险价值估计研究
摘要
近年来,世界上一些知名金融机构如巴林银行、日本大和银行等,虽具有较高管理水平,但都因对金融资产的管理或监控力度不足而遭受巨大损失乃至破产。这说明随着世界金融一体化进程的加快和各种衍生金融工具的创新和发展,在强化金融服务多元化、专业化、科学化的同时,风险的波动性和隐蔽性也增加了。因此,针对金融市场风险,如何较为准确地加以度量,建立完善的风险预警体制机制,成为金融机构和监管当局研究的热点问题。
由J P Morgan集团开发的VaR模型,由于该方法简单有效,而且能全面度量整个机构的综合风险,于是很快得到了广泛的应用。
Copula理论在金融领域上的应用是近几年的事情,它刻画了随机变量因子之间的相依关系。有研究表明,应用Copula技术描述风险之间的相依性更加符合金融市场的实际。Copula是利用样本数据和各种风险资产收益率的边缘分布来确定其联合分布的数学方法,是在构造多元联合分布以及随机变量之间相关结构分析中常用的工具,而VaR是基于统计分析基础上的风险度量技术,它的核心在于描述金融时间序列的统计分布或概率密度函数。因此copula技术在研究风险价值VaR时大有用武之地。
本文介绍了VaR的定义及VaR的主要计算方法,然后简要介绍Copula相关理论,并应用Copula函数理论推导出投资组合收益率的概率密度函数,最后应用蒙特卡罗模拟方法和积分变换理论给出资产组合的风险价值VaR。以往国内对资产组合的风险价值VaR的实证研究,多数文献对收益率均采用正态性或对数正态性假设,而事实上,金融资产相关指标的统计特性是尖峰厚尾的,并非正态,文中拟构建模型、方法突破正态性限制。本文采用Copula函数构建了反映随机变量实际分布与相关性的联合分布函数及其联合概率密度函数,具有一般理论意义,克服了描述资产组合收益率分布的困难;同时以蒙特卡罗模拟方法和积分变换方法给出了VaR求解步骤,使得根据更接近现实的收益率联合分布函数来度量金融资产组合风险成为可能。模型能够反映收益率的时变相关性和波动性。
In resent years, many well-known financial Institutions such as the Daiwa Bank, the Bahrain Bank, though sophisticated in administration, all suffered great loss as to the extent of collapse, due to inadequate administration or monitor. All this adds to the fact that with the acceleration of global finance integration and the innovation and development of the derivative financial instruments, while we are strengthening the diversification, specialization and scientificalization of financial services, the volatility and concealment of risks are also increased. So, how to accurately measure financial risks, establish perfect risk warning systems and mechanisms, has become a hot issue of financial institutions and regulatory authorities.
The VaR (Value at risk) model, developed by the J P Morgan Group, due to its simplicity, efficiency and capacity to measure the integrated risks of financial institutions, has got widespread application.
The application of copula theory in the financial sector happened just a few years ago. It describes the dependence of random variables. Research shows that it meets the real financial market when using the copula theory to describe risk dependences. Copula is a theory which uses sample data and marginal distributions of the returns’ratio to determine the joint distribution of the portfolio’s return, it is one of the tools that are mostly used in both the construction of joint distributions and the analysis of the dependence between random variables. Based on the statistical analysis, the VaR model, as a risk measure technique, has a core issue which lies in the description of the statistical distribution or the probabilistic density of financial time series. So, it comes in handy when using the copula theory to study VaR.
This paper showed the definition and the main calculation methods of VaR and then briefly introduced the related theories of copula, and using the copula theory, this paper deduced the probabilistic density of portfolio’s return ratio. At last, this paper gives the solution of VaR based on both the Monte Carlo simulation method and the integral transformation method. In the past, most empirical literature adopted a normal or lognormal assumption on return’s ratio; however, it turned out to be not the case. The statistical characteristics of some financial assets’indicator are peak and fat tail, not normal. This paper intended to establish a model or method to overcome the normal restrictions. This paper established the random variables’joint distribution and probabilistic density which can reflect their dependence. The result is theoretically significant and overcomes the difficulty when describing the joint distributions of returns’ratio. Meanwhile, this paper gives the solution of VaR based on both the Monte Carlo simulation method and the integral transformation method and makes it possible to calculate the risks of portfolios according to some real distributions. This method could reflect time-varying relevance and volatility of returns’ratio.
由J P Morgan集团开发的VaR模型,由于该方法简单有效,而且能全面度量整个机构的综合风险,于是很快得到了广泛的应用。
Copula理论在金融领域上的应用是近几年的事情,它刻画了随机变量因子之间的相依关系。有研究表明,应用Copula技术描述风险之间的相依性更加符合金融市场的实际。Copula是利用样本数据和各种风险资产收益率的边缘分布来确定其联合分布的数学方法,是在构造多元联合分布以及随机变量之间相关结构分析中常用的工具,而VaR是基于统计分析基础上的风险度量技术,它的核心在于描述金融时间序列的统计分布或概率密度函数。因此copula技术在研究风险价值VaR时大有用武之地。
本文介绍了VaR的定义及VaR的主要计算方法,然后简要介绍Copula相关理论,并应用Copula函数理论推导出投资组合收益率的概率密度函数,最后应用蒙特卡罗模拟方法和积分变换理论给出资产组合的风险价值VaR。以往国内对资产组合的风险价值VaR的实证研究,多数文献对收益率均采用正态性或对数正态性假设,而事实上,金融资产相关指标的统计特性是尖峰厚尾的,并非正态,文中拟构建模型、方法突破正态性限制。本文采用Copula函数构建了反映随机变量实际分布与相关性的联合分布函数及其联合概率密度函数,具有一般理论意义,克服了描述资产组合收益率分布的困难;同时以蒙特卡罗模拟方法和积分变换方法给出了VaR求解步骤,使得根据更接近现实的收益率联合分布函数来度量金融资产组合风险成为可能。模型能够反映收益率的时变相关性和波动性。
In resent years, many well-known financial Institutions such as the Daiwa Bank, the Bahrain Bank, though sophisticated in administration, all suffered great loss as to the extent of collapse, due to inadequate administration or monitor. All this adds to the fact that with the acceleration of global finance integration and the innovation and development of the derivative financial instruments, while we are strengthening the diversification, specialization and scientificalization of financial services, the volatility and concealment of risks are also increased. So, how to accurately measure financial risks, establish perfect risk warning systems and mechanisms, has become a hot issue of financial institutions and regulatory authorities.
The VaR (Value at risk) model, developed by the J P Morgan Group, due to its simplicity, efficiency and capacity to measure the integrated risks of financial institutions, has got widespread application.
The application of copula theory in the financial sector happened just a few years ago. It describes the dependence of random variables. Research shows that it meets the real financial market when using the copula theory to describe risk dependences. Copula is a theory which uses sample data and marginal distributions of the returns’ratio to determine the joint distribution of the portfolio’s return, it is one of the tools that are mostly used in both the construction of joint distributions and the analysis of the dependence between random variables. Based on the statistical analysis, the VaR model, as a risk measure technique, has a core issue which lies in the description of the statistical distribution or the probabilistic density of financial time series. So, it comes in handy when using the copula theory to study VaR.
This paper showed the definition and the main calculation methods of VaR and then briefly introduced the related theories of copula, and using the copula theory, this paper deduced the probabilistic density of portfolio’s return ratio. At last, this paper gives the solution of VaR based on both the Monte Carlo simulation method and the integral transformation method. In the past, most empirical literature adopted a normal or lognormal assumption on return’s ratio; however, it turned out to be not the case. The statistical characteristics of some financial assets’indicator are peak and fat tail, not normal. This paper intended to establish a model or method to overcome the normal restrictions. This paper established the random variables’joint distribution and probabilistic density which can reflect their dependence. The result is theoretically significant and overcomes the difficulty when describing the joint distributions of returns’ratio. Meanwhile, this paper gives the solution of VaR based on both the Monte Carlo simulation method and the integral transformation method and makes it possible to calculate the risks of portfolios according to some real distributions. This method could reflect time-varying relevance and volatility of returns’ratio.
引文
曹辉,李忠民.2006.基于Copula理论的VaR算法与实证分析.辽宁工程技术大学学报(社会科学版).8(1):133~35
陈守东,胡铮洋,孔繁利.2006.Copula函数度量风险价值的Monte Carlo模拟.吉林大学社会科学版, 46(2):85~91
陈伟忠.2004.组合投资与投资基金管理.北京:中国金融出版社:1~8
范英,魏一鸣,应尚军.2006.金融复杂系统模型与实证.北京:科学出版社:1~27
方国久.2009.基于Copula方法的投资组合VaR的度量研究.[硕士学位论文].重庆:重庆大学
复旦大学. 1979.概率论:第一册.北京:人民教育出版社:279~294
傅强,邢琳琳.2009.基于极值理论和Copula函数的条件VaR计算.系统工程学报: l24(5):531~537
黄海,卢祖帝. 2003.VaR的主要计算方法述评.管理评论.15(7):31~36
黄健,李国印.2008.风险测量方法VaR的缺陷及其修正模型CVaR.时代经贸:6(102):61~62
孔繁利,段素芬,华志强.2006.Copula度量投资组合VaR的应用研究.内蒙古民族大学学报(自然科学版),21(6):603~606
李笃群.1998.计算多重广义积分的一种有效方法和程序.宁德师专学报(自然科学),10(4):269~ 270
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刘凡,宋福铁.2006.基于VaR方法对开放式基金风险评估的实证研究.华东理工大学学报,2:34~39
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唐丽娜. 2009.股票投资风险管理的数学模型研究.[硕士学位论文].西安:西安建筑科技大学
王新宇,宋学锋.2006.拟合中国股票市场收益的统计分布.系统工程理论与实践.12:40~45
韦艳华,张世英等.2003.Copula理论在金融上的应用.西北农林科技大学学报(社会科学版),3(5):97~101
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徐永春,甘斌. 2008.VaR的不同计算方法研究.统计与决策.12:147~148.
曾霞,何平. 2008.基于g函数的多元Copula的构造.数理统计与管理.27(5):844~849
张国勇,杨宝臣. 2003.VaR计算方法综述.天津理工学院学报.19(4):74~76
张金清,李徐. 2008.资产组合的集成风险度量及其应用-基于最优拟合Copula函数的VaR方法.系统工程理论与实践.6:15~21
张明恒. 2004.多金融资产风险价值的Copula计量方法研究.数量经济技术经济研究.4:67~70
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陈伟忠.2004.组合投资与投资基金管理.北京:中国金融出版社:1~8
范英,魏一鸣,应尚军.2006.金融复杂系统模型与实证.北京:科学出版社:1~27
方国久.2009.基于Copula方法的投资组合VaR的度量研究.[硕士学位论文].重庆:重庆大学
复旦大学. 1979.概率论:第一册.北京:人民教育出版社:279~294
傅强,邢琳琳.2009.基于极值理论和Copula函数的条件VaR计算.系统工程学报: l24(5):531~537
黄海,卢祖帝. 2003.VaR的主要计算方法述评.管理评论.15(7):31~36
黄健,李国印.2008.风险测量方法VaR的缺陷及其修正模型CVaR.时代经贸:6(102):61~62
孔繁利,段素芬,华志强.2006.Copula度量投资组合VaR的应用研究.内蒙古民族大学学报(自然科学版),21(6):603~606
李笃群.1998.计算多重广义积分的一种有效方法和程序.宁德师专学报(自然科学),10(4):269~ 270
李夫明. 2005.金融市场风险VaR方法的研究.[硕士学位论文].上海:华东师范大学:1~2
刘凡,宋福铁.2006.基于VaR方法对开放式基金风险评估的实证研究.华东理工大学学报,2:34~39
刘红波,边宽江,程波等.2008.VaR数学模型及其计算方法.西北农业学报,17(4):334-338
刘宇飞.1999.VaR模型及其在金融监管中的应用.经济科学,1:39~50
马丽娜. 2009.投资组合中VaR及其实证分析[硕士学位论文].上海:华东师范大学:11~14
南京工学院数学教研组.1989.积分变换(第三版).北京:高等教育出版社:12~16
石芸. 2009.基于下界VaR和极值理论的风险管理模型及其实证分析.[硕士学位论文].合肥:中国科技大学:1~2
唐丽娜. 2009.股票投资风险管理的数学模型研究.[硕士学位论文].西安:西安建筑科技大学
王新宇,宋学锋.2006.拟合中国股票市场收益的统计分布.系统工程理论与实践.12:40~45
韦艳华,张世英等.2003.Copula理论在金融上的应用.西北农林科技大学学报(社会科学版),3(5):97~101
吴文俊.1988.现代数学新进展—刘徽数学讨论班报告集.合肥:安徽科学技术出版社:12~17
吴振翔,叶五一,缪柏其.2004.基于Copula的外汇投资组合风险分析.中国管理科学,12(4):1~5
小詹姆斯L,法雷尔,沃尔特,J雷哈特.投资组合管理理论及应用.第二版.齐寅峰等译. 2000.北京:机械工业出版社,16
邢琳琳.2009.基于极值理论和Copula函数的中国基金市场投资组合VaR研究.[硕士学位论文].重庆:重庆大学:4
徐永春,甘斌. 2008.VaR的不同计算方法研究.统计与决策.12:147~148.
曾霞,何平. 2008.基于g函数的多元Copula的构造.数理统计与管理.27(5):844~849
张国勇,杨宝臣. 2003.VaR计算方法综述.天津理工学院学报.19(4):74~76
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