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镕娄
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2 Abstract We examine the tail characteristics of and tail dependence between return series of copper futures traded at the Shanghai Futures Exchange (SHFE) and the London Metal Exchange (LME). We find that both return series have fat tails comparing to normal distribution, and the tail fatness of LME s is more significant. While the right tails of the two series have strong finite dependence, they are asymptotically independent. However, the left tails are not only finitely dependent, but also asymptotically dependent in general. Therefore dependence of the two series is more significant during market downturn than market upswings. Further more, the degree of finite dependence of either side of tails is significantly higher than traditional Pearson correlations of the two series, implying the importance of tail dependence in risk analysis upon the portfolio investment. Key words: extreme value theory, futures market, GPD, tail dependence.

3 1 JP Morgan RiskMetrics Preto GPD Frechet Frechet Pearson Pearson Pearson Gencay & Selcuk Longin & Solnik2000Fernandez2003 Poon et. al. 2003

4 SHFE LME 2, Fisher-Tippett GPD R1= lim R1(s) = s lim P(T1>s T2>s) (1) s R1 T1 T2 T2 s T1 s s T1 T2 R1=1 T1 T2 R1 = 0 T1 T2 U V u 1 s u 1 P(U>u V>u) = P(U>u,V>u)/P(V>u) 2- lnc(u,u)/lnu Copula C(u,u) U V R1(u) = 2- lnc(u,u)/lnu, R1 lim R1(u) 1 u 1 Coles et.al. (1999)

5 R2 = lim s 2ln P( T1 > s) -1 (2) ln P( T1 > s, T 2 > s) T1 T2 U V P(U>u)= 1-u, u 1 s, R2(u) = 2ln(1-u)/lnC (u,u)-1 C (u,u) U V Survivor R2 lim R2(u) 2 u 1 R2>0 T1 T2 R2 = 0 T1 T2 R2<0 T1 T2 T1 T2 R2 Pearson R1R2 R1>0, R2=1 R1= 0-1<R2<1 R2 1 R1 Frechet R1 R2 Survivor Ledford and Tawn (1996), P(T1>s,T2>s) L(s)s -1/η, L(s) Y= min(t1,t2), P(Y>s) = P(T1>s,T2>s) L(s)s -1/η T1 T2 Y s L(s)η Y Hill (1975), η = (lny i i ln s) n s, L(s) = n s /ns 1/η n n s Coles et. al2002 R2 = 2η-1. (3)

6 R2 1 R1= sn s. (4) n s 3 s QQ-Plot, Mean Excess Function (MEF) Hill-PlotQQ-plot QQ QQ QQ s 2 Mean Excess FunctionMEFMEF s MEF GPD s MEF 3 Hill-Plot s s Hill MSE 3 SHFE SHFE SHFE LME 03/31/ /14/2003

7 1 2 LME SHFELME SHFE SHFE LME LME (E-6) LME SHFE LME SHFE skewnessshfe

8 LME kurtosis 3 LME SHFE 1 2 R1R2 3 4 SHFE LME SHFE LME 0 1 sr1r2 R1(s)R2(s) s 1 R1(s) R2(s) 95% 3 R1(s) s 1 4 R2(s) s , 0.3, , 0.8 R1(s)

9 Frechet Frechet MinS L S L SHFE LME QQ-Plot, MEFHill-Plot MSE 3% 5% 3% 5%1% 3% 5% LME SHFE SHFE LME 1998 LME SHFE LME Pearson SHFE SHFE LME %

10 5% SHFE SHFE LME SHFE LME SHFE R1 R2 3% 5% T H 0 R2 = 1 3% 5% H 0 2: LME SHFE % 5% 3% 5% u R (0.187) (0.148) (0.160) (0.128) R (0.032) (0.026) (0.1) (0.028) % 3%

11 3: LME SHFE % 5% 3% 5% u R (0.27) (0.182) (0.181) (0.162) R (0.033) (0.03) 4: LME SHFE % 5% 3% 5% u R (0.281) (0.247) (0.277) (0.208) R (0.066) (0.047) (0.058) (0.052) 3% 5%

12 % 3% 5% % 5% Pearson 5: Pearson Correlations LME SHFE SHFE LME SHFE LME Pearson LME SHFE Pearson SHFE LME 2 4 3% 5% Pearson Pearson LME SHFE Pearson VAR 4 LME SHFE 3 LME SHFE LME SHFE LME

13 Pearson

14 References Coles, Stuart, Janet Heffernan, and Jonanthan Tawn,Dependence Measures for Extreme Value Analyses,Extremes, V3, Fernandez, Viviana, Extreme Value Theory: Value at Risk and Returns Dependence around the World, Gencay, Ramazan, and Faruk Selcuk, Overnight Borrowing, Interest Rates and Extreme Value Theory, Working Papers #0103, Department of Economics, Bilkent University, Gencay, Ramazan, and Faruk Selcuk, Extreme Value Theory and Value-at-Risk: Relative Performance in Emerging Market, International Journal of Forecasting, Hill, B.M., A simple general approach to inference about the tail of a distribution, Annals of Statistics, V3, Ledford, A. W., and J.A. Tawn, Statistics for Near Independence in Multivariate Extreme Values, Biometrika, V83, Login, Francois, and Bruno Solnik, Extreme Correlation of International Equity Markets, The Journal of Finance, V56, Poon Ser-Huang, Michael Rokinger, and Jonathan Tawn, Modeling Extreme Dependence in International Stock Markets, Economic Seminar Paper, Cardiff Business School, 2003.

经济与管理耿庆峰 : 我国创业板市场与中小板市场动态相关性实证研究基于方法比较视角 87 Copula 模型均能较好地刻画金融市场间的动态关系, 但 Copula 模型效果要好于

经济与管理耿庆峰 : 我国创业板市场与中小板市场动态相关性实证研究基于方法比较视角 87 Copula 模型均能较好地刻画金融市场间的动态关系, 但 Copula 模型效果要好于第 19 卷第 6 期中南大学学报 ( 社会科学版 ) Vol.19 No.6 013 年 1 月 J. CENT. SOUTH UNIV. (SOCIAL SCIENCE) Dec. 013 我国创业板市场与中小板市场动态相关性实证研究基于方法比较视角耿庆峰 ( 闽江学院公共经济学与金融学