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系統識別號 U0026-0812200913455796
論文名稱(中文) 關聯結構在金融市場風險管理之研究
論文名稱(英文) Copulas for Risk Management in Financial Market
校院名稱 成功大學
系所名稱(中) 統計學系碩博士班
系所名稱(英) Department of Statistics
學年度 95
學期 2
出版年 96
研究生(中文) 曾毅豪
研究生(英文) Yi-Hao Tseng
電子信箱 kur_rose@hotmail.com
學號 r2694106
學位類別 碩士
語文別 英文
論文頁數 119頁
口試委員 口試委員-蘇永在
口試委員-梁雪富
指導教授-黃銘欽
中文關鍵字 穩定分配  高狹峰  CML  關聯結構  風險值  IFM  Kendall’s tau 
英文關鍵字 Kendall’s tau  CML  IFM  Leptokurtosis  Stable distribution  Copula  Value-at-Risk 
學科別分類
中文摘要 本論文研究關聯結構(Copulas)以及對金融資產報酬間之關聯性建模,並探討在投資組合報酬風險值之應用。文中以五種不同關聯結構描述歷史資料,再由關聯結構、常態模型模擬之風險值與歷史資料所得之風險值進行比較。我們將其理論方法應用於兩種複合型指數歷史資料,分別利用Kendall’s tau、IFM與CML方法估計關聯結構模型。模擬結果顯示,關聯結構能充份反應金融資料所具有高狹峰與厚尾的特性。在較高信賴水準下,由關聯結構模型模擬所得之風險值,其表現比常態模型佳。在信賴水準為90%與95%之間,以穩定分配為邊際之關聯結構模型有較佳的風險值估計值。若信賴水準高於97.5%,則以t分配為邊際之關聯結構模型較為適當,而以穩定分配為邊際者表現過於保守。最後,我們採用有母數與無母數方法選取合適之關聯結構模型。
英文摘要 This thesis studies the copula-based method to model relationship of financial asset returns and calculate the value-at-risk (VaR) of portfolio returns. Five candidate copulas are used to fit the empirical data, and the comparisons of the simulated VaR by copula-based and normal models to the empirical VaR are made. We apply the methodology to an empirical data set composed of two composite indexes, and estimate copulas using the nonparametric Kendall’s tau estimator, IFM and CML methods. The simulation results indicate that the copula-based model is capable of capturing the leptokurtosis inherent in financial data. The copula-based VaR gives more accurate approximation than the normal-based VaR for larger confidence level. Furthermore, copulas with stable margins produce reasonable 90% and 95% VaR estimates. Copulas with Student-t margins are more appropriate under a larger confidence level, e.g. 99% or 99.5%, where copulas with stable margins are too conservative. Finally, for selecting an appropriate copula we use parametric and nonparametric approach to implement the model selection.
論文目次 Contents

Contents ………………………………………………………………Ⅰ
List of Tables …………………………………………………………Ⅲ
List of Figures …………………………………………………………Ⅵ
Chapter 1 Introduction ………………………………………………1
1.1 Motivation and Background ………………………………………………1
1.2 Objective …………………………………………………………………2
1.3 Organization ………………………………………………………………3
Chapter 2 Literature Review ………………………………………4
2.1 Asset Allocation Models …………………………………………………4
2.2 Copula Theory and Its Financial Applications ……………………………5
Chapter 3 Model Specification and Methodology …………………7
3.1 Distribution Functions ……………………………………………………7
3.1.1 Normal and Student-t distributions …………………………………7
3.1.2 Stable distribution ……………………………………………………9
3.2 Background on Copula …………………………………………………10
3.2.1 Definition of Copula ………………………………………………11
3.2.2 Fréchet-Hoeffding Bounds Inequality ……………………………12
3.2.3 Sklar’s Theorem ……………………………………………………15
3.2.4 Copula Densities ……………………………………………………16
3.2.5 Examples of Copulas ………………………………………………17
3.3 Dependence and Measures of Association ………………………………25
3.3.1 Linear Correlation Coefficient ……………………………………26
3.3.2 Rank Correlation ……………………………………………………27
3.3.3 Link Measures of Association to Correlation Coefficient …………29
3.3.4 Link Kendall’s tau to Archimedean Copulas ………………………30
3.3.5 Tail Dependence ……………………………………………………33
3.4 Value-at-Risk for a Portfolio of Assets……………………………………35
3.5 Parameter Estimation ……………………………………………………36
3.5.1 Maximum Likelihood Estimation …………………………………37
3.5.2 IFM Method ………………………………………………………37
3.5.3 CML Method ………………………………………………………38
3.6 Simulation Procedure ……………………………………………………39
3.6.1 Simulating from Elliptical Copulas ………………………………41
3.6.2 Simulating from Archimedean Copulas ……………………………42
3.6.3 Simulating VaR from a Copula with Given Margins ………………45
3.7 How to Select an Appropriate Copula? …………………………………46
Chapter 4 Empirical Results ………………………………………47
4.1 Data Collection …………………………………………………………47
4.2 Normal and Student-t Fitting for Empirical Daily Log-Returns …………49
4.3 Stable Fitting for Empirical Daily Log-Returns …………………………55
4.4 Comparison of Model-based VaR and Empirical VaR …………………59
4.5 Selecting an Appropriate Copula for a Given Data Set …………………66
Chapter 5 Conclusion ………………………………………………71
References ……………………………………………………………74
Appendix A ……………………………………………………………79
Appendix B ……………………………………………………………84
Appendix C ……………………………………………………………86
Appendix D ……………………………………………………………98


List of Tables

Table 3.1 Summary of three one-parameter families of Archimedean copulas …24
Table 3.2 Summary of measures of association for Archimedean copulas …31
Table 3.3 Coefficients of tail dependence of the Archimedean copulas ……34
Table 4.1 Descriptive statistics for each daily return series ………………48
Table 4.2 Correlation coefficient and measures of association ……………………52
Table 4.3 Estimation of the copula parameters using Kendall’s tau ………………52
Table 4.4 Estimated parameters of the stable distribution …………57
Table 4.5 Estimated copulas with Student-t margins using Kendall’s tau ………66
Table 4.6 Estimated copulas with stable margins using Kendall’s tau ………66
Table 4.7 Estimated copulas with empirical margins using Kendall’s tau ………67
Table 4.8 Estimated copulas with Student-t margins using IFM method ……67
Table 4.9 Estimated copulas with stable margins using IFM method ……………67
Table 4.10 Estimated copulas using CML method ……………………………68
Table 4.11 Parameter estimations using Kendall’s tau nonparametric estimator and distance for different type of copulas …………………………………69
Table 4.12 CML estimation of parameters and distance for different type of copulas …………………………70
Table C.1 Empirical VaR, Normal VaR, and simulated VaR from the estimated copulas using Kendall’s tau for equal investment in two assets with different confidence levels ………………………………………86
Table C.2 Biases of normal VaR and simulated VaR from the estimated copulas using Kendall’s tau for equal investment in two assets with different confidence levels …………………………………………………87
Table C.3 Empirical VaR, Normal VaR, and simulated VaR from the estimated copulas using Kendall’s tau for portfolio composed of 20% S&P 500, 80% DOW JONES with different confidence levels …………………88
Table C.4 Biases of normal VaR and simulated VaR from the estimated copulas using Kendall’s tau for portfolio composed of 20% S&P 500, 80% DOW JONES with different confidence levels ………………………89
Table C.5 Empirical VaR, Normal VaR, and simulated VaR from the estimated copulas using Kendall’s tau for portfolio composed of 70% S&P 500, 30% DOW JONES with different confidence levels …………………90
Table C.6 Biases of normal VaR and simulated VaR from the estimated copulas using Kendall’s tau for portfolio composed of 70% S&P 500, 30% DOW JONES with different confidence levels ………………………91
Table C.7 Empirical VaR, Normal VaR, and simulated VaR from the estimated copulas using IFM method for equal investment in two assets with different confidence levels …………………………………………92
Table C.8 Biases of normal VaR and simulated VaR from the estimated copulas using IFM method for equal investment in two assets with different confidence levels …………………………………………………93
Table C.9 Empirical VaR, Normal VaR, and simulated VaR from the estimated copulas using IFM method for portfolio composed of 20% S&P 500, 80% DOW JONES with different confidence levels ………………94
Table C.10 Biases of normal VaR and simulated VaR from the estimated copulas using IFM method for portfolio composed of 20% S&P 500, 80% DOW JONES with different confidence levels …………………………95
Table C.11 Empirical VaR, Normal VaR, and simulated VaR from the estimated copulas using IFM method for portfolio composed of 70% S&P 500, 30% DOW JONES with different confidence levels ……96
Table C.12 Biases of normal VaR and simulated VaR from the estimated copulas using IFM method for portfolio composed of 70% S&P 500, 30% DOW JONES with different confidence levels …………………………97

List of Figures

Figure 3.1 Comparison of normal and Student- distributions. The normal is scaled to have variance 2, like Student- distribution ……………8
Figure 3.2 Left panel: Influence of on the resulting stable distribution . Right panel: Influence of on the resulting stable distribution …………………………………9
Figure 3.3 Graphs of product copula , the Fréchet-Hoeffding lower bound and the Fréchet-Hoeffding upper bound along with their contour diagrams ………………………………………13
Figure 3.4 Bounds for all copulas. The surface given by the bottom and back side of the pyramid is the lower bound, whereas the front side of the pyramid is the upper bound ………………………………………………………14
Figure 3.5 Left panel: Contours of the Fréchet-Hoeffding lower bound (dotted line), upper bound (solid line) and the product copula (dashed line). Right panel: The region which contains the level set ……………………15
Figure 3.6 Copula densities of Gaussian (left) and Student- copulas (right) with linear correlation coefficient 0.2 ……………………………………20
Figure 3.7 Contour plots of Gaussian (left) and Student- (right) copula densities with linear correlation coefficient 0.2 ………………………………20
Figure 3.8 Copula densities for three Archimedean families with parameters chosen such that Kendall’s are 0.3333 ……………………………………24
Figure 3.9 Contour plots of three copula densities of Archimedean families with parameters chosen such that Kendall’s are 0.3333 ………………24
Figure 3.10 Kendall’s tau and Spearman’s rho versus correlation coefficient ……30
Figure 3.11 Kendall’s tau versus parameters of Gumbel, Clayton and Frank copulas …………………………32
Figure 4.1 S&P 500 and DOW JONES indexes time series plots ………………47
Figure 4.2 S&P 500 and DOW JONES daily log-return time series plots ………48
Figure 4.3 Normal Q-Q plots for S&P 500 and DOW JONES ……………………49
Figure 4.4 Student-t Q-Q plots for S&P 500 and DOW JONES …………………50
Figure 4.5 Density histograms with normal (dashed line) and Student-t (solid line) fitted densities for S&P 500 and DOW JONES ……………………50
Figure 4.6 Log-likelihood function of the Student-t distribution versus degrees of freedom for S&P 500 and DOW JONES …………………………51
Figure 4.7 Fitted distributions of S&P 500 and DOW JONES daily log-returns …51
Figure 4.8 Left panel: S&P 500 versus DOW JONES daily log-returns. Right panel: The pseudo-observations with uniform margins corresponding to empirical asset returns ………………………………………………52
Figure 4.9 Contour plots of the estimated bivariate normal distribution, Gaussian and Student- copula with Student-t margins ………………………53
Figure 4.10 Contour plots of the Archimedean copulas with Student-t margins and parameters chosen such that the Kendall’s for all three distributions are 0.7837 ……………………………………………………………53
Figure 4.11 Top left panel: Simulated bivariate normal data. Middle panel: The empirical data. The other panels: Samples from bivariate distributions via copula with Student-t margins and Kendall’s tau 0.7837 …………54

Figure 4.12 The empirical p.d.f. of S&P 500 and DOW JONES daily log-returns, the normal fit and Student-t fit ……………………………………………56
Figure 4.13 The empirical p.d.f. of S&P 500 and DOW JONES daily log-returns with the normal fit and Stable fit ……………………………………56
Figure 4.14 The empirical c.d.f. of daily log-returns with the Stable fitted distribution ………57
Figure 4.15 Contour plots of the estimated bivariate normal distribution, Gaussian and Student- copula with stable margins …………………………57
Figure 4.16 Contour plots of the Archimedean copulas with stable margins and parameters chosen such that the Kendall’s for all three distributions are 0.7837 ……………………………………………………………58
Figure 4.17 Top left panel: Simulated bivariate normal data. Middle panel: The empirical data. The Other panels: Samples from bivariate distributions via copula with stable margins and Kendall’s tau 0.7837 ……………58
Figure 4.18 VaR versus confidence level. (Empirical VaR, simulated Normal VaR, and simulated VaR from the estimated copulas using Kendall’s tau for portfolio composed of 50% S&P 500, 50% DOW JONES) …………60
Figure 4.19 Biases versus confidence level. (Biases are computed by subtracting the Empirical VaR from simulated VaR, from normal and estimated copulas using Kendall’s tau for portfolio composed of 50% S&P 500, 50% DOW JONES) ………………………………………………………61
Figure 4.20 Biases versus confidence level. (Biases are computed by subtracting the Empirical VaR from simulated VaR, from normal and estimated copulas using Kendall’s tau for portfolio composed of 20% S&P 500, 80% DOW JONES) ………………………………………………………61

Figure 4.21 Biases versus confidence level. (Biases are computed by subtracting the Empirical VaR from simulated VaR, from normal and estimated copulas using Kendall’s tau for portfolio composed of 70% S&P 500, 30% DOW JONES) ………………………………………………………62
Figure 4.22 Biases of daily VaR from normal, copulas with Student-t margins (CTM), and copulas with stable margins (CSM) using Kendall’s tau for equal investment in two assets …………………..……………………63
Figure 4.23 Biases of daily VaR from normal, copulas with Student-t margins (CTM), and copulas with stable margins (CSM) using Kendall’s tau for portfolio composed of 20% S&P 500, 80% DOW JONES …………64
Figure 4.24 Biases of daily VaR from normal, copulas with Student-t margins (CTM), and copulas with stable margins (CSM) using Kendall’s tau for portfolio composed of 70% S&P 500, 30% DOW JONES …………65
Figure 4.25 Empirical copula of S&P 500 versus DOW JONES constructed by 100 sub-samples in the period from Sep. 1, 2006 to Feb. 2, 2007 ………68
Figure 4.26 Left panel: Contour plot of empirical copula constructed by 100 sub-samples in the period from Sep. 1, 2006 to Feb. 2, 2007. Right panel: Contour plot of the empirical copula using in total 2532 daily log-returns …69
Figure 4.27 Log-CDFs of Stable (solid line), Student-t (dotted line), Normal (dashed line) fits and the empirical data (circles) ……………………………72
Figure D.1 Biases of simulated VaR from normal and estimated copulas using IFM method for portfolio composed of 50% S&P 500, 50% DOW JONES ………………………………………………………………98


Figure D.2 Biases of simulated VaR from normal and estimated copulas using IFM method for portfolio composed of 20% S&P 500, 80% DOW JONES ………………………………………………………………98
Figure D.3 Biases of simulated VaR from normal and estimated copulas using IFM method for portfolio composed of 70% S&P 500, 30% DOW JONES ………………………………………………………………99
Figure D.4 Biases of daily VaR from normal, copulas with Student-t margins (CTM), and copulas with stable margins (CSM) using IFM method for equal investment in two assets ……………………………………………100
Figure D.5 Biases of daily VaR from normal, copulas with Student-t margins (CTM), and copulas with stable margins (CSM) using IFM method for portfolio composed of 20% S&P 500, 80% DOW JONES ……………………101
Figure D.6 Biases of daily VaR from normal, copulas with Student-t margins (CTM), and copulas with stable margins (CSM) using IFM method for portfolio composed of 70% S&P 500, 30% DOW JONES ……………………102
Figure D.7 99% and 95% VaR estimates for S&P 500 daily log-returns ………103
Figure D.8 99% and 95% VaR estimates for DOW JONES daily log-returns …103
參考文獻 References

[1] Aas, K. (2004). Modelling the Dependence Structure of Financial Assets: A Survey of Four Copulas. Research report SAMBA/22/04, Norwegian Computer Center.
[2] Blattberg, R. C. and Gonedes, N. J. (1974). A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices. Journal of Business, 47, 244-280.
[3] Bouyé, E., Durrleman, V., Bikeghbali, A., Riboulet, G. and Rconcalli, T. (2000). Copulas for Finance – A Reading Guide and Some Application. Working paper, Groupe de Recherche Opérationelle, Crédit Lyonnais, Paris, France.
[4] Cherubini, U., Luciano, E. and Vecchiato, W. (2004). Copula Methods in Finance. Wieley, Chichester.
[5] Demarta, S. and McNeil, A. J. (2004). The t Copula and Related Copulas. Technical report, ETH Zurich.
[6] Durrleman, V., Nikeghbali, A. and Rconcalli, T. (2000). Which Copula is the Right One? Working paper, Goupe de Recherche Opérationelle, Crédit Lyonnais.
[7] Embrechts, P., Lindskog, F. and McNeil, A. J. (2003). Modelling Dependence with Copulas and Applications to Risk Management. In: Rachev, S.T. (Ed.), Handbook of Heavy Tailed Distribution in Finance, pp. 329-384. Elsevier.
[8] Embrechts, P., McNeil, A. J. and Straumann, D. (1999). Correlation: Pitfalls and Alternatives. Risk, 12, 69-71.



[9] Embrechts, P., McNeil, A. J. and Straumann, D. (2000). Correlation and dependency in risk management: Properties and pitfalls. In: Dempster, M. (Ed.), Risk Management: Value at Risk and Beyond, pp. 176-223. Cambridge University Press, UK.
[10] Escarela, G. and Carriére, J. F. (2003). Fitting Competing Risks with an Assumed Copula. Statistical Method in Medical Research, 12, 333-349.
[11] Fama, E. F. (1965). The Behavior of Stock Market Prices. Journal of Business, 38, 34-105.
[12] Fermanian, J.D. (2005). Goodness-of-Fit Tests for Copulas. Journal of Multivariate Analysis, 95, 119-152.
[13] Frank, M. J. (1979). On the Simultaneous Associativity of F(x, y) and x + y − F(x, y). Aequationes Mathematicae, 19, 194-226.
[14] Frees, E. W., Carriére, J. and Valdez, E. A. (1996). Annuity Valuation with Dependent Mortality. The Journal of Risk and Insurance, 63, 229-261.
[15] Frees, E. W. and Valdez, E. A. (1998). Understanding Relationships Using Copulas. North American Actuarial Journal, 2, 1-25.
[16] Frees, E. W. and Wang, P. (2005). Credibility Using Copulas. North American Actuarial Journal, 9, 31-48.
[17] Genest, C. and Favre, A. C. (2007). Everything You Always Wanted to Know About Copula Modeling But Were Afraid to Ask. Journal of Hydrologic Engineering, 12, in press.
[18] Genest, C., Ghoudi, K. and Rivest, L.-P. (1995). A Semiparametric Estimation Procedure of Dependence Parameters in Multivariate Families of Distributions. Biometrika, 82, 543-552.
[19] Genest, C. and MacKay, J. (1986b). The Joy of Copulas: Bivariate Distribution with Uniform Marginals. The American Statistician, 40, 280-283.
[20] Genest, C. and Rivest, L.-P. (1993). Statistical Inference Procedures for Bivariate Archimedean Copulas. Journal of the American Statistical Association, 88, 1034-1043.
[21] Jansons, V., Kozlovskis, K. and Lace, N. (2005). Portfolio Modeling Using the Theory of Copula in Latvian and American Equity Market. Faculty of Engineering Economics, Riga Technical University.
[22] Joe, H. (1997). Multivariate Models and Dependence Concepts, volume 73 of Monographs on Statistics and Applied Probability. Chapman & Hall, London.
[23] Joe, H. and Xu, J. (1996). The Estimation Method of Inference Functions for Margins for Multivariate Models. Technical Report 166, Department of Statistics, University of British Columbia.
[24] Kole, E., Koedijk, K. and Verbeek, M. (2006). Selecting Copulas for Risk Management. Journal of Banking & Finance, forthcoming.
[25] Kon, S. (1984). Models of Stock Returns – A Comparison. Journal of Finance, 39, 147-165.
[26] Lindskog, F., McNeil, A. and Schmock, U. (2001). A Note on Kendall’s tau for Elliptical Distributions. ETH preprint.
[27] Mandelbort, B. B. (1963). The Variation of Certain Speculative Prices. Journal of Business, 36, 394-419.
[28] Marshall, A. W. and Olkin, I. (1988). Families of Multivariate Distributions. Journal of the American Statistical Association, 83, 834-841.
[29] Mashal, R. and Zeevi, A. (2002). Beyond correlation: Extreme Co-movements Between Financial Assets. Technical report, Columbia University.
[30] McNeil, A. J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press.

[31] Nelsen, R. B. (2006). An Introduction to Copulas, volume 139 of Lecture Notes in Statistics, 2nd ed. Springer Verlag. Berlin Heidelberg New York.
[32] Nolan, J. P. (1997). Numerical Calculation of Stable Densities and Distribution Functions. Commun. Statist. -Stochastic Models, 13, 759-774.
[33] Nolan, J. P. (2001a). Maximum Likelihood Estimation of Stable Parameters. In Barndorff-Nielsen, O. E., Mikosch, T. and Resnick, S. I. (Eds.), Lévy Processes: Theory and Applications, pp. 379-400. Boston: BirkhÄauser.
[34] Nolan, J. P. (2007b). Stable Distributions - Models for Heavy Tailed Data. Boston: BirkhÄauser. In progress, Chapter 1 online at academic2.american.edu/~jpnolan.
[35] Ojeda, D. (2001). Comparison of Stable Estimates, Ph.D Thesis, Department of Mathematics and Statistics, American University.
[36] Rachev, S. T., Menn, C. and Fabozzi, F. J. (2005). Fat-tailed and Skewed Asset Return Distributions: Implications for Risk Management, Portfolio Selection, and Option Pricing. John Wiley & Sons.
[37] Romano, C. (2002). Calibrating and Simulating Copula Functions: An Application to the Italian Stock Market. Working paper n. 12, CIDEM.
[38] Schmidt, T. (2006). Coping with Copulas. In: Rank, J. (Ed.), Copulas – From Theory to Application in Finance, pp. 329-384.
[39] Schoutens, W. (2003). Lévy Processes in Finance. Wiley Series in Probability and Statistics. John Wiley & Sons, England.
[40] Schweuser, B. and Wolff, E. (1981). On nonparametric measures of dependence for random variables. The Annals of Statistics, 9, 879-885.
[41] Shih, J. H. and Louis, T. A. (1995). Inferences on the Association Parameter in Copula Models for Bivariate Survival Data. Biometrics, 51, 1384-1399.

[42] Sklar, A. (1959). Fonctions de répartitions à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris, 8, 229-231.
[43] Wang, W. and Wells, M. T. (2000). Model Selection and Semiparametric Inference for Bivariate Failure-Time Data. Journal of the American Statistical Association, 95, 62-72.
[44] Yan, J. (2006). Enjoy the Joy of Copulas. Department of Statistics and Actuarial Science, University of Iowa.
[45] Yan, J. (2006b). Multivariate Modeling with Copulas and Engineering Applications. In: Pham, H. (Ed.), Handbook in Engineering Statistics. Springer, pp.973-990.
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