2007 2 2 :100026788 (2007) 0220112206 1,2, 1 (11, 300072 ;21,050018) :. Copula (Generalized Pareto Distribution,GPD),. GPD Copula. Copula2GARCH2GPD. Clayton2GARCH2GPD., VaR Copula ;,Clayton Copula. : Copula ;; ; : F83019 : A Research on the Correlation of Portfolio Value at Risk in Financial Markets LI Xiu2min 1,2, SHI Dao2ji 1 (11College of Science, Tianjin University, Tianjin 300072, China ; 21College of Science, Hebei University of Science and Technology, Shijiazhuang 050018, China) Abstract : Research on the correlation between Shanghai and Shenzhen stock markets. By means of copula function and generalized Pareto distribution, we discussed the dependence structure of Shanghai and Shenzhen stock markets. The copula function can capture the correlation between random variables and GPD described the marginal distribution. So the Copula2GARCH2GPD model is established and used to study the financial market. The empirical results show that the dependence pattern of the two stock markets is Clayton2GARCH2GPD distribution. Moreover, the results presented through Monte Carlo Simulation told that the portfolio VaR under normal joint distribution is lower than under other Copulas. And the Clayton Copula gives secure conclusion at high level quantiles. Key words : Copula function ; generalized Pareto distribution ; dependence structure ; VaR 1,,.,.,,. 1952, (Markowitz).,.,, ARCH,GARCH. VaR,..,,.,, Pearson :2005212212 : (70573077) ;(06457225) :(1964 - ),,,;(1942 - ),,,,, :.
2 113.,,,,. Copula.,Copula,.,,,. Copula Sklar [1 ] 1959.. 1999 Nelsen [2 ],Embrechts et al Copula, [3 ],Patton Copula [4 ],Rob W.J. van den Goorbergh et al Copula [5 ]. Copula,VaR [6 ], Copula [7 ], [8 ] [9 ] Copula,., Copula,GARCH2GPD, Copula2GPD,. 2 Copula Sklar [2 ] :Copula, [ 0,1 ]. Copula (C) : U = F( x),v = G( y), X, Y : C( u, v) = P r ( U u,v v), (1) H( x, y) = C( F( x), G( y) ), (2) H( x, y) X, Y, F ( x), G( y) X, Y. Sklar, Copula. [ 2 ],Copula,Kendall,Spearman.,Copula,Copula,.,. Copula,, Pearson., Copula. Copula Copula. Copula Copula t Copula, [2 ] Copula., t Copula,Gumbel Copula,Clayton Copula Frank Copula Copula. t Copula Frank Copula,,. Gumbel copula,clayton Copula. Copula,, Copula. t Copula : 1 C( u, v) = 1 + κ x2-2 xy + y 2 21-2 (1-2 ) [ -, t - 1 ( u) ] [ -, t - 1 ( v) ] ( - 1 1),. Gumbel Copula : - +2 2 d xd y, (3) C( u, v) = exp - [ ( - ln u) + ( - ln v) ] 1, (4) (1 < + ). Clayton Copula : C( u, v) = [ u - + v - - 1 ] - 1, (5)
114 2007 2 (1 < + ). Frank Copula : C( u, v) = - ( - << +, 0). 1 ln 1 + ( e - u - 1) ( e - v - 1) e - - 1, (6) 1 Copula 3 GARCH2GPD Copula,., ;Copula ;Monte Carlo,Copula VaR,Copula.,, GARCH, : R t = + t = + t Z t, p 2 t = a 0 + a 2 i t - i + i = 1 Z t t - 1 N (0,1), a 0 > 0,0 a i, b j < 1, q j = 1 2 t - j, b j R t,,,z t t - 1 t - 1, t - 1 t - 1, t t t - 1. [10 ], { Z t },. Pareto [11 ], GARCH { Z t },GARCH2GPD, : ^F ( z) = 1 - {1 - F( u R ) } 1 + ^ R z - ur R - 1 R, z > u R F( ), u L z u R, (8) F( u L ) 1 - ^ L z - - 1 ul L L, z < u L GPD, u R u L GPD. () Pareto,.,,VaR. (7)
2 115,,., [11 ].,,. (8) [0,1 ] { u i },, Copula,Copula. Copula2GARCH2GPD 2, Copula. [0,1 ] [0,1 ] k k, i j E ( i, j), i, j = 1,2,, k,[ e i - 1, e i ] [ e j - 1, e j ],0 = e 0 < e i < e k = 1. e i. ( u t, v t ), e i - 1 u t < e i e j - 1 v t < e j, ( u t, v t ) E( i, j). k,,. M ij E( i, j), N ij Copula2GPD, 2 : 2 = k i = 1 k j = 1 ( M ij - N ij ) 2 N ij, (9) 2 ( k - 1) 2., ( 5), p, q, 2 ( k - 1) 2 - p - ( q - 1). k = 20. R. 4 411,,{ R t }R t = Log P t - Log P t - 1, SH SZ. 2000 1 4 2005 3 25, 1252,www. stockstar. com.,1. 1,SH SZ, Shapiro2Wilk (0105 ), p 0105.. 1 SH SZ W p SH - 1. 2e204 0. 0135 0. 7498 6. 2010 0. 9357 2. 2e216 SZ - 2. 4e204 0. 0141 0. 5341 5. 4031 0. 9416 2. 0e216 412, (7) GARCH(1,1) SH SZ,2,. GARCH(1,1), GPD,. GPD 3. 413 Copula 2 a 0 a 1 b 1 SH SZ 7. 75e206 (1. 48e206) 7. 024e206 (1. 374e206) 0. 1435 (0. 0150) 0. 1452 0. 01559 0. 8231 (0. 0177) 0. 8275 (0. 0166) GPD,(8), [ 0,1 ] { u i, v i },Box. test,.,{ u i, v i } Copula,Copula, 2. 4 Copula.
116 2007 2 3 GPD u R u L ^ R ^ L ^ R ^ L SH 1. 55-1. 66 0. 1460 0. 0194 0. 6236 0. 5266 SZ 1. 53-1. 67 0. 1923-0. 0502 0. 5037 0. 6236 4 Copula Copula t Copula Gumbel2C Clayton2C Frank2C = 2. 11 = 0. 86 5. 01 4. 87 18. 52 2 ( n) 67. 26 (48) 68. 32 (39) 73. 15 (69) 83. 49 (42) 4,0105,Clayton Copula,t Copula 2,, Gumbel Copula Frank Copula. 414 Monte Carlo Copula, Monte Carlo Copula,. ( 5000, GARCH2GPD. Copula ( u n, v n ), n, qgpd GPD { Z i, n }, i = 1,2, x n = 1 + 1, n Z 1, n, y n = 2 + 2, n Z 2, n ( i, i, n ),Copula ( x n, y n ).,015, VaR. 5, VaR. 5,95 %, VaR 5 Copula VaR Copula,,Copula VaR. t Copula VaR, Clayton Copula, 99 % VaR, Gumbel Copula Frank Copula., 90 % 95 % 97 % 99 % Normal 1. 2040 1. 5979 1. 8623 2. 3666 t Copula 0. 9078 3. 7867 4. 5067 5. 3333 Gumbel2C 0. 8028 1. 6054 2. 3406 4. 2457 Clayton2C 0. 7817 2. 0959 4. 3432 5. 4723 Frank2C 0. 8473 2. 1306 4. 0646 4. 8568,t Copula,, Clayton Copula. 5 Copula VaR.,, Copula,.,., Clayton Copula, VaR. : [ 1 ] Sklar A. Functions de repartition an dimensions et leurs marges [J ]. Publication de IUniversit de Pairs,1959,(8) :229-231. [ 2 ] Nelsen R B. An Introduction to Copulas [M]. Springe,NewYork,1999,18 125-137 90-95. [ 3 ] Embrechts P,Lindskog F,McNeil A. Modeling dependence with copulas and application to risk management [ C ]ΠΠHandbook of Heavy Tailed Distributions in Finance,ed. S. Rachev, Elsevier,2003,Chapter 8 : 329-384. [ 4 ] Patton A J. Modeling time2varing exchange rate dependence using the conditional copula [ R ]. San Diego : Department of
2 117 Economics,University of California,2003,(5) :97-101. [ 5 ] Rob W J. van den Goorbergh, Christian Genest,Bas J M. Werker Bavariate option pricing using dynamic copula models [J ]. Insurance : Mathematics and Economics,2005,37 : 101-114. [ 6 ],. VaR [J ].,2004 : 22(9) :42-45. Shi Daoji,Wang Aili. The bounds of Value2at2Risk for functions of dependent risks[j ]. Systems Engineering,2004 : 22(9) :42-45. [ 7 ],. Copula [J ].,2004,24(4) :49-55. Shi Daoji, Yao Qingzhu. A method of improving copula fited to datd[j ]. Systems Engineering - Theory & Practice,2004,24(4) :49-55. [ 8 ]. (Copula) [J ].,2002, (4) :41-44. Zhang Yaoting. Copula function technique and financial risk analysis[j ]. Statistical Research,2002,(4) :41-44. [ 9 ],,. [J ].,2004,19(4) :355-362. Wei Yanhua,Zhang Shiying, Guo Yan. Research on degree and patterns of dependence in financial markets[j ]. Journal of Systems Engineering,2004,19(4) :355-362. [10 ]. VaR [J ].,2002, (4) :34-38. Fen Jianqiang. Rearch on the VaR of return ratio of Shanghai and Shenzheng Stock market [J ]. Statistical Research,2002,(4) :34-38. [11 ] Stuart Coles. An Introduction to Statistical Modeling of Extreme Values [M]. Springer,2001. (104 ) [2 ]. [J ].,2005,14(5) :36-40. Xie Fanrong. A heuristic numerical algorithm for solving the problem of curriculum scheduling [ J ]. Operations Research and Management Science,2005,14(5) :36-40. [3 ] Ronen D. Ship scheduling : The last decade[j ]. European J. Operation Research,1993,71(3) :325-333. [4 ] Chrisliansen M, Fagerholt K, Ronen D. Ship routing and scheduling : Status and perspectives [J ]. Transportation Science,2004, 38 (1) :1-18. [5 ],. DFS [J ].,2002,28(8) :224-226. Liu Yunfeng, Qi Huan. Application of DFS algorithm in the arranging of Three2Gorges permanent lock chamber [J ]. Computer Engineering,2002,28(8) :224-226. [6 ],. [J ].,2002, (1) : 1-3. Liu Yunfeng, Qi Huan. The two2dimension optimization arranging heuristic algorithm and its application in the Yangtse gorges permanent ship lock decision system [J ]. Computer and Modernization,2002,(1) :1-3. [7 ],. [J ].,2002,17(2) :163-166. Lai Wei, Qi Huan. The MADM of three gorges ship gates running [J ]. Control and Decision,2002,17(2) :163-166. [8 ] Maturana, Francisco P,Tichy, Pavel, Slechta, Petr,et al. Distributed multi2agent architecture for automation systems [J ]. Expert Systems with Applications, 2004, 26(1) :49-56.