Research on the Correlation of Portfolio Value at Risk in Financial Markets

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1 : (2007) ,2, 1 (11, ;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 , China ; 21College of Science, Hebei University of Science and Technology, Shijiazhuang , 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 : : ( ) ;( ) :( ),,,;( ),,,,, :.

2 2 113.,,,,. Copula.,Copula,.,,,. Copula Sklar [1 ] 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 (1-2 ) [ -, t - 1 ( u) ] [ -, t - 1 ( v) ] ( - 1 1),. Gumbel Copula : d xd y, (3) C( u, v) = exp - [ ( - ln u) + ( - ln v) ] 1, (4) (1 < + ). Clayton Copula : C( u, v) = [ u - + v ] - 1, (5)

3 (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 ul L L, z < u L GPD, u R u L GPD. () Pareto,.,,VaR. (7)

4 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 ,,{ R t }R t = Log P t - Log P t - 1, SH SZ , 1252,www. stockstar. com.,1. 1,SH SZ, Shapiro2Wilk (0105 ), p SH SZ W p SH e e216 SZ e e , (7) GARCH(1,1) SH SZ,2,. GARCH(1,1), GPD,. GPD Copula 2 a 0 a 1 b 1 SH SZ 7. 75e206 (1. 48e206) e206 (1. 374e206) ( ) ( ) ( ) GPD,(8), [ 0,1 ] { u i, v i },Box. test,.,{ u i, v i } Copula,Copula, 2. 4 Copula.

5 GPD u R u L ^ R ^ L ^ R ^ L SH SZ Copula Copula t Copula Gumbel2C Clayton2C Frank2C = = ( n) (48) (39) (69) (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 t Copula Gumbel2C Clayton2C Frank2C ,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) : [ 2 ] Nelsen R B. An Introduction to Copulas [M]. Springe,NewYork,1999, [ 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 : [ 4 ] Patton A J. Modeling time2varing exchange rate dependence using the conditional copula [ R ]. San Diego : Department of

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