copula, 5 3 Copula Κ L = lim System s Engineering M ay., 2006 : (2006) ,,, copula Ξ A rch im edean copula (Joe,

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1 24 5 ( 149 ) V ol. 24, N o System s Engineering M ay., 2006 : (2006) copula Ξ, (, ) : (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 : 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 : : ( ) : (19812),,,, : ; (19562),,,, : portfolio

2 5, : copula ), 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= Σ, δ = (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, 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 : 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, δ Κ δ U = 2-2 1gδ (8) Κ δ L = 2-1gδ (9) (p i, qi), n= 1160, (x i, y i) = (log (p i+ 1gp i), log (qi+ 1gqi) ) 1,,,

3 ( ri, si) = (rank (x i), rank (y i) ), (ri, si) (x i, y i) (7) 1 1copula Σ δ Σ δ = 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 ) C layton copula C C l (u, v) = (u - + v - - 1) - 1g C C (u, v) = (1+ [ (u - 1-1) + (v - 1-1) - 1 ] - 1g ) - 1 δ = 2g3 (1- Σ δ ) , 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)

4 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 (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 C layton copula , 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) Α Gum bel2h copula, Α Κ U (Α), p , p , p , p , q0. 925, q0. 950, q0. 975, q , , , , , , , ; ,,,,,

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, [2 ]Juri A,W utrich M V. Copula convergence theorem s for tail events[j ]. InsuranceM athem atics and Econom ics, 2002, 30 (3): [3 ]Joe H. M ultivariate models and dependence concep ts[m ]. London: Chapm an & H all, [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: [5 ]D urrlem an V,N oleghbalia, Roncalli T. W h ich copula is the righ t one? [Z ] [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: [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, [9 ]H urlim ann W. F itting bivariate cum ulative returns w ith copulas[j ]. Computational Statistics & D ata A nalysis, 2004, 45: [10 ],. [J ]., 2003, 10: [11 ],,. copula [J ]. (), 2005, 33 (1): [12 ]. (copula) [J ]., 2002, 4: [13 ]. [J ]., 2002, 9: [14 ],. [M ]. :, 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 , 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