Risk Analysis of Portfolio by Copula2GARCH

2006 3 3 :100026788 (2006) 0320045208 Copula2GARCH 1, 1, 2, 2 (11, 100080 ;21, 230026) : Copula GARCH, Copula2GARCH.,,. : ; ;Copula ; GARCH : F830159 ;O21113 : A Risk Analysis of Portfolio by Copula2GARCH WU Zhen2xiang 1, CHEN Min 1, YE Wu2yi 2, MIAO Bai2qi 2 (11Academy of Mathematics and Systems Science,CAS, Beijing 100008,China ; 21University of Science and Technology of China, Hefei 230026,China) Abstract : Copula can describe the dependency structure of multi2dimension random variable. In this paper, Copula and the forecast function of GARCH model are well combined, and a Copula2Garch model is built for risk analysis of portfolio investment. By this model, empirical portfolio risk analysis is made in Chinese stock market. At last, the mini2risk portfolio is given. Key words : portfolio ; risk analysis ; Copula ; GARCH 1 Copula.,Copula. Copula Schweizer Sklar (1983), Genest MacKay (1986) Joe. H. (1993),Copula. Embrechts etc. (1999) Copula,., Claudio Romano (2002) Copula ; Embrechts etc. (2003) Durrleman etc (2001) Copula ;Roberto De Matteis (2001) Copula, Archimedean Copula ;Ling Hu (2002) Copula ; Gabriel. F. etc (2003) Copula ;Davide. M. etc (2004) Copula (CDO BDS), t2copula ; Jean2David. F. (2004) Copula ; Hull,White (2004) Copula.,Copula. CSFB (Credit Suisse First Boston) Creditrisk + J P Morgan Creditmetrics Copula. 2002 Copula. (2002a,2002b) Copula ;(2004) VaR Copula ; (2003) Copula. (2004) Copula. :2005203229 : (70221001 ;70331001) :(1977 - ),( ),,,;,,.

46 2006 3,., GARCH Copula,. Jondeau. E. etc (2002) Copula2GARCH,,Jondeau. E. Copula, t Copula. (2004a,2004b) Copula2GARCH. Archimedean Copula, Copula Copula,2 3.,.,Copula. Copula2GARCH,,,. : t2 GARCH ; Copula2GARCH ; Copula2GARCH,,,,.. 2 t2garch,. GARCH,, GARCH,,, t2garch. GARCH(1,1) : r t a t = t + a t = t e, t t v 2 t = 0 + 1 a 2 t - 1 + 2 t - 1. (1), r t, t r t,arma ARIMA t, a t ; a t r t,, (1) t GARCH ( ). t i. i. d. v t2, v 0 1., (ARCH ), t, : r t a t = + a t = t t, t t v 2 v = 0 + 1 a 2 t - 1 + 2 t - 1 { r 1, r 2,, r T },(2), a t a T }, v 0 1, R T + 1 :. (2) P( R T+1 r T ) = P( a T+1 ( r - ) T ) = P( T+1 T+1 ( r - ) T ) { a 1, a 2,, = P T+1 r - r - = 0 + 1 a 2 t + 2 t v. (3) t 0 + 1 a 2 t + 2 t, t v ( ) v t2, T T. (3) R T + 1. ITSM2000, GARCH,(3) T.

3 Copula2GARCH 47 3 Copula2GARCH m..,,,,,. Copula, GARCH,. m,i { r 1 i, r 2 i,, r Ti }, i = 1,, m. t2garch, R i = R T + 1, i F i ( )., Copula : Copula t2 Copula m R = { R T + 1,1, R T + 1,2,, R T + 1, m }. r t = { r t1, r t2,, r tm }, t = 1,, T, Copula (Joe. H. (1993) Roberto D. M. (2001) ),: m Copula : C Normal ( u 1,, u m ) = ( - 1 ( u 1 ),, - 1 ( u m ) ). (4), Copula. m, - 1. r i F i ( r), { r t1, r t2,, r tm }, t = 1,, T,: (5) u t (4), C Normal : m t2 Copula : u t = ( u 1, t, u 2, t,, u m, t ) = ( F 1 ( r t1 ), F 2 ( r t2 ),, F m ( r tm ) ), t = ( - 1 ( u t1 ), - 1 ( u t2 ),, - 1 ( u tm ) ), t = 1,2,, T. (5) ( u t ) = ( t ), t m, ^= 1 T T t = 1 t t. (6) C t,( u 1,, u m ) = t n,( t - 1 ( u 1 ),, t - 1 ( u m ) ), (7) t2,t2, t2 Copula. r i F i ( r), { r t1, r t2,, r tm }, t = 1,, T, t2 Copula,: u t = ( u 1, t, u 2, t,, u m, t ) = ( F 1 ( r t1 ), F 2 ( r t2 ),, F m ( r tm ) ), t = ( t( - 1 u 1 t ), t( - 1 u 2 t ),, t( - 1 u mt ) ), t = 1,2,, T. (8) Copula, : 1) ^ 1 (6) ; 2) ^ k+1 = 1 T + m T t = 1 t t 1 + 1, k = 1,2,. (9) t ^ k t 3) step2,,t2 Copula. Copula C( u 1, u 2,, u m ),r : F( R 1, R 2,, R m ) = C( F 1 ( R 1 ), F 2 ( R 2 ),, F m ( R m ) ) (10) 4 A :,. 200211213120041112,,.,,, Monte2Carlo.,..,2003..

48 2006 3 2003111220041112,. 1 1,., Hong Y. M(1999) Q Chen M(2002) Cn Kn,, Ljung2Box. :.. 1 Q Hong Q 6. 1406 8. 1972 10. 1024 16. 9923 Chen Kn 2. 3547 2. 9274 2. 3712 1. 5637 Chen Cn 1. 8927 3. 0333 2. 5047 1. 2367 :Hong Q,,, ;Chen Kn Cn,,Chen M(2002). Hong Y. M(1999) Chen M(2002) Q Kn Cn, 0. 99,,, (2).

3 Copula2GARCH 49 : GARCH (2) GARCH(1,1),ITSM2000 2 GARCH(1,1) 0. 0024 0. 0011-0. 0006 0. 0015 0 0. 0003 0. 0000 0. 0004 0. 0000 1 0. 1190 0. 0918 0. 0680 0. 1080 0. 2594 0. 8825 0. 0011 0. 7098 v 3. 5948 3. 5651 7. 5543 3. 4117, (3), : F 1, F 2, F 3 F 4. r 2, t, 4, r t = { r 1, t,, r 3, t, r 4, t },,,,,. Copula (6) (9),Copula,, 4,,,(6) (9) Copula,. (6) (9),, u t = ( u 1, t, u 2, t, u 3, t, u 4, t ) = ( F 1 ( r 1, t ), F 2 ( r 2, t ), F 3 ( r 3, t ), F 4 ( r 4, t ) ),, t,,,.,,copula.,233, y t = { y 1, t, y 2, t, y 3, t, y 4, t }, t = 1,,233. u 4 ) ; a) Copula Step1. y t (6) ^,(4) Copula C ( u 1, u 2, u 3, Step2. t2garch step2 (1), F( x 1, x 2, x 3, x 4 ) ; Step3. : ( b 1, b 2, b 3,1 - b 1 - b 2 - b 3 ), b 1, b 2, b 3 0 1 - b 1 - b 2 - b 3 0. (0105 VaR 0101 VaR ) ; Step4. ^ Cholesky, ^= A A ; Step5. x = ( x 1, x 2, x 3, x 4 ), x i N (0,1),y = A x, z = ( F - 1 1 (( y 1 ) ), F - 1 2 (( y 2 ) ), F - 1 3 (( y 3 ) ), F - 1 4 (( y 4 ) ) ),, F i i,( ) ( ) ; Step6. step5 10000, ( z h1, z h2, z h3, z h4 ), h = 1,,10000, F, h = ( b 1, b 2, b 3,1 - b 1 - b 2 - b 3 ) ( z h1, z h2, z h3, z h4 ). { h },10000 { h } 0105 0101 VaR,

50 2006 3. b) 3 t2 Copula Step1. y t (9) ^,(7) 3 t2copula C( u 1, u 2, u 3, u 4 ) ; Step2. t2garch step2 (1), F( x 1, x 2, x 3, x 4 ) ; Step3. : ( b 1, b 2, b 3,1 - b 1 - b 2 - b 3 ), b 1, b 2, b 3 0 1 - b 1 - b 2 - b 3 0. (0105 VaR 0101 VaR ) ; Step4. ^ Cholesky, ^= A A ; Step5. x = ( x 1, x 2, x 3, x 4 ), x i N (0,1),y = A x, x 3 S, z = F - 1 1 t y 1 SΠ3, F - 1 2 t y 2 SΠ3, F - 1 3 t y 3 SΠ3, F - 1 4 t, F i i, t ( ) 3 t2 ( ) ; Step6. step5 10000, z k y 4 SΠ3 = ( z h1, z h2, z h3, z h4 ), h = 1,,10000, F, h = ( b 1, b 2, b 3,1 - b 1 - b 2 - b 3 ) ( z h1, z h2, z h3, z h4 ). { h },10000 { h } 0105 0101 VaR,. ( b 1, b 2, b 3 ),, VaR 0105 = risk1 ( b 1, b 2, b 3 ), VaR 0101 = rik2 ( b 1, b 2, b 3 ), (11) = risk ( b 1, b 2, b 3 ).,Copula t2copula : 3 Copula. Copula b 1 ( ) b 2 () b 3 ( ) 1 - b 1 - b 2 - b 3 ( ) VaR 0. 05 0. 0146768 0 0. 83 0. 14 0. 03 VaR 0. 05 0. 0259594 0. 03 0. 76 0. 15 0. 06 0. 0098527 0 0. 8 0. 18 0. 02 VaR 0. 05 0. 0088456 0. 01 0. 79 0. 09 0. 11 t2copula VaR 0. 05 0. 016567 0. 08 0. 87 0. 03 0. 02 0. 0165669 0 0. 83 0. 15 0. 02 3, Copula,,,, VaR,. 3, t2copula Copula,Copula

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