2005 1 Frechet 2 31999-2003 1993-1998 4 Pearson GPD 500 200122 0216840-1002 han.gf@shfe.com.cn
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.
1 JP Morgan RiskMetrics Preto GPD Frechet Frechet Pearson Pearson Pearson Gencay & Selcuk20012003 Longin & Solnik2000Fernandez2003 Poon et. al. 2003
SHFE LME 2, Fisher-Tippett GPD 3 4 2 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)
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)
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 03/31/1993 11/14/2003
1 2 LME SHFELME SHFE SHFE LME LME 1996-1997 1994-1995 20-40 1997-1998 1 1. (E-6) LME 2623 15.945 0.0127-0.6774 10.8246-0.1136 0.0784 SHFE 2623 1.814 0.0095 0.0649 4.695-0.0454 0.0385 LME SHFE skewnessshfe
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 1 0.45, 0.3, 0.6 5 6 6 0.5, 0.8 R1(s)
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 1998-1999 SHFE LME Pearson 1998-1999 SHFE 1999 1999 12 1998 1999 1998 SHFE LME 1993-1998 1999-2003 1993-1998 3%
5% 1993-1998 SHFE SHFE LME SHFE LME SHFE 2 1993 2003 R1 R2 3% 5% T H 0 R2 = 1 3% 5% H 0 2: LME SHFE 1993-2003 1993-2003 3% 5% 3% 5% u 9.5836 6.0298 9.5836 6.541 79 132 80 132 R2 0.6615 0.6961 0.4328 0.4701 (0.187) (0.148) (0.160) (0.128) R1 0.2888 0.3036 0.2924 0.3293 (0.032) (0.026) (0.1) (0.028) 3 1993 1998 4 1999 2003 5% 3%
3: LME SHFE 1993-1998 1993-1998 3 % 5% 3% 5% u 7.2318 5.1847 8.0577 5.3468 44 74 45 73 R2 0.7923 0.5704 0.2141 0.384 (0.27) (0.182) (0.181) (0.162) R1 0.2201 0.2653 (0.033) (0.03) 4: LME SHFE 1999-2003 3% 5% 3% 5% u 13.0338 7.4483 11.3917 8.0882 36 59 37 59 R2 0.6868 0.8964 0.6868 0.5951 (0.281) (0.247) (0.277) (0.208) R1 0.399 0.3737 0.3584 0.4058 (0.066) (0.047) (0.058) (0.052) 3% 5%
1999-2003 3% 3% 5% 1993-1998 3% 5% 1999-2003 5 Pearson 5: Pearson Correlations 1993-2003 1993-1998 1999-2003 LME SHFE 0.3444 0.2352 0.5494 SHFE LME 0.0004-0.0166 0.0304 5 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
Pearson
References Coles, Stuart, Janet Heffernan, and Jonanthan Tawn,Dependence Measures for Extreme Value Analyses,Extremes, V3, 1999. Fernandez, Viviana, Extreme Value Theory: Value at Risk and Returns Dependence around the World, http://www.tis.cl/tis/16/g-09.00-12.00-p.pdf, 2003. Gencay, Ramazan, and Faruk Selcuk, Overnight Borrowing, Interest Rates and Extreme Value Theory, Working Papers #0103, Department of Economics, Bilkent University, 2001. Gencay, Ramazan, and Faruk Selcuk, Extreme Value Theory and Value-at-Risk: Relative Performance in Emerging Market, International Journal of Forecasting, 2003. Hill, B.M., A simple general approach to inference about the tail of a distribution, Annals of Statistics, V3, 1975. Ledford, A. W., and J.A. Tawn, Statistics for Near Independence in Multivariate Extreme Values, Biometrika, V83, 1996. Login, Francois, and Bruno Solnik, Extreme Correlation of International Equity Markets, The Journal of Finance, V56, 2000. Poon Ser-Huang, Michael Rokinger, and Jonathan Tawn, Modeling Extreme Dependence in International Stock Markets, Economic Seminar Paper, Cardiff Business School, 2003.