24 5 ( 149 ) V ol. 24, N o. 5 2006 5 System s Engineering M ay., 2006 : 100124098 (2006) 0520088205 copula Ξ, (, 230052) : (copula), A rch im edean copula, Gum bel2hougard copula,, : Copula; ; : F830 : A 1,,, copula,,,,,,, copula,, copula,,,,,, copula, 2 copula, 3 copula, 4 copula, 5 2copula,, x, y (A. Juri, 2002)copula Copula : 1-25 + C (t, t) Κ U = lim t1-1 - t C (t, t) Κ L = lim t0 + t 3Copula Copulas, ( E llip tic copulas) A rch im edean (A rch im edean copulas), copula, copula, A rch im edean copula : ; copula copula, copula log A rch im edean copula (Joe, Ξ : 2006203225 : (10201029) : (19812),,,, : ; (19562),,,, : portfolio
5, : copula 89 1997), A rch im edean copula C layton copula Gum bel2hougard copula ( Gum bel2h copula) copula A rch im edean copula, Σ Sk lar,, copula,, IFM (inference function for m argins), (V. D urrlem an, 2000), copula,, copula (N elsen R B, 1999), A rch im edean copula,, Σ copula 3. 1copula (1) Gum bel2h copula : C Gu (u, v) = exp - ( (- lnu) + (- lnv) ) 1g, 1 Gum bel2h copula,, = 1, ;, Gum bel2h copula copula,, Κ U = 2-2 1g, Σ:, =, Σ= - 1 1 1- Σ, δ = (1) 1 1 - Σ δ (2) Κ U (Α) = P (Y > p ΑgX > qα) = 1-2Α+ C (Α, Α) 1 - Α Α Κ U (Α) (2) C layton copula C layton copula : C C l (u, v) = (u - + v - - 1) - 1g, - 1 0 C layton copula,, (3) (4) = 0, Σ :, = Σ= 2Σ 1- Σ,, δ = + 2 2Σ δ 1 - Σ δ (5), C layton copula Κ L (Α) = P (Y p ΑgX qα) = C (Α, Α) Α Α Κ L (Α) (3) A rch im edean copula (6) Gum bel2h copula, C layton copula, A rch im edean copula, C (u, v) = (1 + [ (u - 1-1) + (v - 1-1) - 1 ] - 1g ) - 1, Σ δ δ : 3. 2-1 δ = 2g3 (1 - Σ δ ), Kendall Σ, Σ, Σ= P [ (X 1 - X 2) (Y 1 - Y 2) > 0 ] - P [ (X 1 - X 2) (Y 1 - Y 2) < 0 ], (X 1- X 2) (Y 1- Y 2) { (x i, y i) }1in ( (x i, y i) i ) Σ (F rees, 1998): Σ δ = n 2-1 1i< jn sgn ( (x i - x j) (y i - y j) ) (7), δ, copula, 4 4. 1 δ Κ δ U = 2-2 1gδ (8) Κ δ L = 2-1gδ (9) 2000 1 4 2004 12 31 5 (p i, qi), n= 1160, (x i, y i) = (log (p i+ 1gp i), log (qi+ 1gqi) ) 1,,,
90 2006 ( ri, si) = (rank (x i), rank (y i) ), (ri, si) (x i, y i) (7) 1 1copula Σ δ Σ δ = 0. 66438 copula 1 C (u, v) δ (Σ δ ) δ 1 Gum bel2h copula C Gu (u, v) = exp - ( (- lnu) + (- lnv) ) 1g δ = 1g(1- Σ δ ) 1. 0712 2 C layton copula C C l (u, v) = (u - + v - - 1) - 1g 1. 1423 3 C C (u, v) = (1+ [ (u - 1-1) + (v - 1-1) - 1 ] - 1g ) - 1 δ = 2g3 (1- Σ δ ) 0. 7141, Gum bel2h copula C layton copula δ ; copula δ (1, + ),, copula, copula 4. 2copula copula? copula, Kendall Σ, K C copula (Roberto, 2001), K2S K C (t)
5, : copula 91, K C (t) = t - Υ(t) Υ (t + ) Υ( t) copula (V. D urrlem an, 2000), K C (F (X ), G (Y ) ) (t) X, Y, δ, K C (F (X ), G (Y ) ) (t) Gum bel2h copula, C layton copula, K Gu C (F (X ), G (Y ) ) (t) = t - t ln t (10) K C l C (F (X ), G (Y ) ) (t) = t + t- - 1 t - - 1 (11) copula : { (x i, y i) }1in F (x i), G (y i) (F G ); t Gu i = C Gu (x i), G (y i) ); (F (x i), G (y i) ) t C l i = C C l (F ti (10) (11) K Gu C (F (X ), G (Y ) ) (ti) K C l C (F (X ), G (Y ) ) (ti); K CU (0, 1), Q 2Q Q 2Q 2 2Gum bel2h copula Clayton copula K C Q 2Q K2S, 2 2K2S Copula Z P Gum bel copula 0. 864 0. 445 C layton copula 2. 899 0. 000 2, Gum bel2h copula K C (F (X ), G (Y ) ) (t), C layton copula K Gu C (F (X ), G (Y ) ) (t), ; 2 Gum bel2h copula K Gu C (F (X ), G (Y ) ) (t) C layton copula,, Gum bel2h copula, Gum bel2h copula, C layton copula A rch im edean copula C layton copula copula,, Q 2Q K2S, Gum bel2h copula 4. 3, (3) Α Α Gum bel2h copula Κ U (Α), 3 3Α p Α, qα Κ U (Α) Α p Α qα H S I log2returns SZ I log2returns ΚU (Α) 0. 925 0. 02011 0. 01723 0. 1553 0. 950 0. 02399 0. 02119 0. 1336 0. 975 0. 02871 0. 02804 0. 1116 0. 995 0. 04331 0. 05647 0. 0940 Α1 0. 0901 Gum bel2h copula, Α Κ U (Α), p 0. 925, p 0. 950, p 0. 975, p 0. 995, q0. 925, q0. 950, q0. 975, q0. 995 0. 1553, 0. 1336, 0. 1116, 0. 0940, 0. 075, 0. 050, 0. 025, 0. 005; 0. 0901,,,,,
92 2006 5 copula, A rch im edean copula Gum bel2h copula C layton copula copula K C, Gum bel2h copula,,,,,, A rch im edean copula copula, copula : [1 ]N elsen R B. A n introduction to copulas [M ]. N ew York: Sp ringer2v erlag, 1999. [2 ]Juri A,W utrich M V. Copula convergence theorem s for tail events[j ]. InsuranceM athem atics and Econom ics, 2002, 30 (3): 405420. [3 ]Joe H. M ultivariate models and dependence concep ts[m ]. London: Chapm an & H all, 1997. [4 ]Beatriz V az dem elo M endes, R afaelm artins de Souza. M easuring financial risk sw ith copulas[j ]. International R eview of F inancial A nalysis, 2004, 13: 2745. [5 ]D urrlem an V,N oleghbalia, Roncalli T. W h ich copula is the righ t one? [Z ]. 2000. [6 ]F reesw E,V aldez E A. U nderstanding relationsh ip s using copulas[j ]. N orth Am. A ctuarial Journal, 1998, 2: 125. [7 ]F rahm G, Junker M, Schm idt R. E stim ating the tail2dependence coefficient: p roperties and p itfalls [J ]. Insurance: M athem atics and Econom ics, 2005, 37: 80100. [8 ]D e M atteis R,M cn eil A. F itting copulas to data[d ]. Institute of M athem atics of the U niversity of Zurich, 2001. [9 ]H urlim ann W. F itting bivariate cum ulative returns w ith copulas[j ]. Computational Statistics & D ata A nalysis, 2004, 45: 355372. [10 ],. [J ]., 2003, 10: 4549. [11 ],,. copula [J ]. (), 2005, 33 (1): 114116. [12 ]. (copula) [J ]., 2002, 4: 4851. [13 ]. [J ]., 2002, 9: 4144. [14 ],. [M ]. :, 2004. Ta il D ependence Analysis of SZI & HSI Based on Copula M ethod L I Yue, CH EN G X i2jun (School of M anagem ent,u niversity of Science and Technology of Ch ina, H efei 230052, Ch ina) Abstract: T he correlations betw een ShangZheng Index and H angseng Index are studied in th is paper by using tail dependence of copula functionṡ The results show that among som e of A rch im edean copulas w h ich include asymm etric tail dependence p roperties, Gum bel2hougard copula is selected to analyse the tail dependence on ShangZheng Index and H angseng Index, the research results indicate that the two indexescorrelation can be characterized th rough upper tail dependence, and the tail quantified dependence could forecast the change in stock m arket in the future. Key words: Copula; T ail D ependence; Em p irical D istribution Function