混合连接函数模型及其在风险度量中的应用
摘要
本文在基金业迅猛发展的背景下提出基于GARCH-t及混合连接函数模型度量基金投资组合风险值VaR(Value-at-Risk)的方法。该方法的优点在于:摒弃了以往VaR测算中常用的正态性假设,简化了多维函数的计算,令模拟更贴近实际。准确地刻画出总体风险和个体风险的关系。充分考虑到金融数据厚尾分布的性质,灵活运用及扩展了连接函数模型,有效捕捉尾部信息。
本文首先介绍连接函数模型的定义及性质,给出椭圆族连接函数和阿基米德族连接函数的表达式及模拟散点图;充分利用阿基米德族连接函数尾部依赖性的特点,建立混合连接函数模型,运用解决复杂极大似然估计的EM算法进行参数估计,并采用AIC准则进行模型选择。而后介绍连接函数在风险度量中的应用,以GARCH模型拟合各资产收益率分布,以收益率残差分布为边际分布,用连接函数模型连接为收益率的联合分布,运用蒙特卡罗模拟拟合VaR。最后将GARCH-Copula模型应用于基金风险评价,针对2005年初发行的博时主题行业基金及景顺鼎益基金进行实证分析,结合基金投资组合时变性的特点分段计算VaR值,与t连接函数模拟结果进行比较,评价模型优劣。根据拟合结果,得出以下结论:运用混合连接函数模型拟合的VaR值与t连接函数及Gumbel连接函数模型所得结果基本相同,但混合连接函数可以更好地给出数据的尾部结构。
This paper introduces a GARCH-Mixed Copula model for VaR(Value-at-Risk) estimation with the background of high development in Chinese fund market. The advantage of this approach is significant. It abandons the assumption of multivariate normality in former research, simplifies the multivariate function’s computation and makes the simulation much closer to the reality. The relationship between the whole and the individual risk is also accurately calibrated. It flexibly applies and expands copula models to catch tail information while fully considering the financial data's fat tail traits.
The paper’s structure is as follows: the first three chapters introduce the definition and properties of copula model and its classifications as elliptical copulas and Archimedean copulas with expressions and simulation scatter plots. Then it takes advantage of the tail traits of the Archimedean copulas to construct mixed copula models. The EM algorithm which is always used in complicated MLE and Akaike information criterion(AIC) are applied in estimating the parameters and model selection. The next two chapters introduce the application of the copula models in risk measurement. GARCH is used to model the asset return series. The residual distributions are taken as the marginal distributions and the copula models are taken as union distributions. Finally I use Monte Carlo method to simulate VaR. At the end of my paper, I choose two stock funds which were both issued at the beginning of 2005 to apply my method and compute VaR in different times. Based on the results, I get the conclusion that the values of VaR are nearly the same in t and mixed copula models. But the mixed copula model can give the tail structure which is superior to t copula.
本文首先介绍连接函数模型的定义及性质,给出椭圆族连接函数和阿基米德族连接函数的表达式及模拟散点图;充分利用阿基米德族连接函数尾部依赖性的特点,建立混合连接函数模型,运用解决复杂极大似然估计的EM算法进行参数估计,并采用AIC准则进行模型选择。而后介绍连接函数在风险度量中的应用,以GARCH模型拟合各资产收益率分布,以收益率残差分布为边际分布,用连接函数模型连接为收益率的联合分布,运用蒙特卡罗模拟拟合VaR。最后将GARCH-Copula模型应用于基金风险评价,针对2005年初发行的博时主题行业基金及景顺鼎益基金进行实证分析,结合基金投资组合时变性的特点分段计算VaR值,与t连接函数模拟结果进行比较,评价模型优劣。根据拟合结果,得出以下结论:运用混合连接函数模型拟合的VaR值与t连接函数及Gumbel连接函数模型所得结果基本相同,但混合连接函数可以更好地给出数据的尾部结构。
This paper introduces a GARCH-Mixed Copula model for VaR(Value-at-Risk) estimation with the background of high development in Chinese fund market. The advantage of this approach is significant. It abandons the assumption of multivariate normality in former research, simplifies the multivariate function’s computation and makes the simulation much closer to the reality. The relationship between the whole and the individual risk is also accurately calibrated. It flexibly applies and expands copula models to catch tail information while fully considering the financial data's fat tail traits.
The paper’s structure is as follows: the first three chapters introduce the definition and properties of copula model and its classifications as elliptical copulas and Archimedean copulas with expressions and simulation scatter plots. Then it takes advantage of the tail traits of the Archimedean copulas to construct mixed copula models. The EM algorithm which is always used in complicated MLE and Akaike information criterion(AIC) are applied in estimating the parameters and model selection. The next two chapters introduce the application of the copula models in risk measurement. GARCH is used to model the asset return series. The residual distributions are taken as the marginal distributions and the copula models are taken as union distributions. Finally I use Monte Carlo method to simulate VaR. At the end of my paper, I choose two stock funds which were both issued at the beginning of 2005 to apply my method and compute VaR in different times. Based on the results, I get the conclusion that the values of VaR are nearly the same in t and mixed copula models. But the mixed copula model can give the tail structure which is superior to t copula.
引文
[1] Trevor Hastie,Robert Tibshirani,Jerome Friedman. The Elements of Statistical Learning,Data Mining Inference and Prediction,Canada,Springer,2001
[2] Enzo Giacomini,Wolfgang K. Hardle and Vladimir Spokoiny,Inhomogeneous Dependency Modeling with Time Varying Copulae. CASE,Weierstrass Institute for Applied Analysis and Stochastics,2007.
[3] Jeff A. Bilmes. A Gentle Tutorial of the EM Algorithm and its Application to Parameter Estimation for Gaussian Mixture and Hidden Markov Models. International Computer Science Institute,1998.
[4] Till Groβmaβ,Copulae and Tail Dependence. Diploma Thesis,School of Business and Economics,Humboldt-University Berlin,2007.
[5] Alexander J.McNeil,Rudiger Frey,Paul Embrechts,Quantitative Risk Management Concepts Techniques and Tools,USA,Princeton University Press,2005.
[6] Xiaohong CHEN and Yanqin FAN , Pseudo-likelihood Ratio Tests for Semi-parametric Multivariate Copula Model Selection. The Canadian Journal of Statistics,2005,Vol.33,No 2.
[7] Xiaohong CHEN , Yanqin FAN , Estimation and Model Selection of Semi-parametric Copula-based Multivariate Dynamic Models under Copula Misspecification. Journal of Econometrics,2006,135.
[8] Jun Yan,Enjoy the Joy of Copulas: With a Package Copula. Journal of Statistical Software, October 2007,Volume 21,Issue 4.
[9] 张尧庭,连接函数 (Copula) 技术与金融风险分析,统计研究,2002,第四期。
[10] 肖春来,宋然,VaR 理论及其应用研究,数理统计与管理,2003,第 2期。
[11] 韦艳华,张世英,金融市场的相关性分析——Copula-GARCH 模型及其应用,系统工程,2004,第 4 期。
[12] 刘国光,许世刚,基于 Copula 方法的 A 股,B 股投资组合风险值实证分析,淮海工学院学报,2004,第 4 期。
[13] 刘大伟,杜子平,Copula 方法在投资组合选择与 VaR 计量中的应用,统计与决策,2006,第 3 期。
[14] 王红莲,连接函数 (Copula) 理论及其在金融中的应用,上海财经大学硕士学位论文,2006。
[15] 刘汉伟,基于连接函数 Copula 理论的 VaR 算法及应用,暨南大学硕士学位论文,2007。
[2] Enzo Giacomini,Wolfgang K. Hardle and Vladimir Spokoiny,Inhomogeneous Dependency Modeling with Time Varying Copulae. CASE,Weierstrass Institute for Applied Analysis and Stochastics,2007.
[3] Jeff A. Bilmes. A Gentle Tutorial of the EM Algorithm and its Application to Parameter Estimation for Gaussian Mixture and Hidden Markov Models. International Computer Science Institute,1998.
[4] Till Groβmaβ,Copulae and Tail Dependence. Diploma Thesis,School of Business and Economics,Humboldt-University Berlin,2007.
[5] Alexander J.McNeil,Rudiger Frey,Paul Embrechts,Quantitative Risk Management Concepts Techniques and Tools,USA,Princeton University Press,2005.
[6] Xiaohong CHEN and Yanqin FAN , Pseudo-likelihood Ratio Tests for Semi-parametric Multivariate Copula Model Selection. The Canadian Journal of Statistics,2005,Vol.33,No 2.
[7] Xiaohong CHEN , Yanqin FAN , Estimation and Model Selection of Semi-parametric Copula-based Multivariate Dynamic Models under Copula Misspecification. Journal of Econometrics,2006,135.
[8] Jun Yan,Enjoy the Joy of Copulas: With a Package Copula. Journal of Statistical Software, October 2007,Volume 21,Issue 4.
[9] 张尧庭,连接函数 (Copula) 技术与金融风险分析,统计研究,2002,第四期。
[10] 肖春来,宋然,VaR 理论及其应用研究,数理统计与管理,2003,第 2期。
[11] 韦艳华,张世英,金融市场的相关性分析——Copula-GARCH 模型及其应用,系统工程,2004,第 4 期。
[12] 刘国光,许世刚,基于 Copula 方法的 A 股,B 股投资组合风险值实证分析,淮海工学院学报,2004,第 4 期。
[13] 刘大伟,杜子平,Copula 方法在投资组合选择与 VaR 计量中的应用,统计与决策,2006,第 3 期。
[14] 王红莲,连接函数 (Copula) 理论及其在金融中的应用,上海财经大学硕士学位论文,2006。
[15] 刘汉伟,基于连接函数 Copula 理论的 VaR 算法及应用,暨南大学硕士学位论文,2007。