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研究生：	曾毅豪 Tseng, Yi-Hao
論文名稱：	關聯結構在金融市場風險管理之研究 Copulas for Risk Management in Financial Market
指導教授：	黃銘欽 Huang, Min-Ching
學位類別：	碩士 Master
系所名稱：	管理學院 - 統計學系 Department of Statistics
論文出版年：	2007
畢業學年度：	95
語文別：	英文
論文頁數：	119
中文關鍵詞：	穩定分配、高狹峰、CML 、關聯結構、風險值、IFM 、Kendall’s tau
外文關鍵詞：	Kendall’s tau, CML, IFM, Leptokurtosis, Stable distribution, Copula, Value-at-Risk
相關次數：	點閱：180 下載：1
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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。

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.

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
                                    

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校內：2008-07-21公開
校外：2008-07-21公開

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