Risk Analysis of Portfolio by Copula2GARCH

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1 : (2006) Copula2GARCH 1, 1, 2, 2 (11, ;21, ) : Copula GARCH, Copula2GARCH.,,. : ; ;Copula ; GARCH : F ;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 ,China ; 21University of Science and Technology of China, Hefei ,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 Copula. (2002a,2002b) Copula ;(2004) VaR Copula ; (2003) Copula. (2004) Copula. : : ( ; ) :( ),( ),,,;,,.

2 ,., 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 = a 2 t 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 = a 2 t 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 - = a 2 t + 2 t v. (3) t a 2 t + 2 t, t v ( ) v t2, T T. (3) R T + 1. ITSM2000, GARCH,(3) T.

3 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 :, ,,.,,, Monte2Carlo.,..,2003..

4 ,. 1 1,., Hong Y. M(1999) Q Chen M(2002) Cn Kn,, Ljung2Box. :.. 1 Q Hong Q Chen Kn Chen Cn :Hong Q,,, ;Chen Kn Cn,,Chen M(2002). Hong Y. M(1999) Chen M(2002) Q Kn Cn, 0. 99,,, (2).

5 3 Copula2GARCH 49 : GARCH (2) GARCH(1,1),ITSM GARCH(1,1) v , (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 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 (( y 1 ) ), F (( y 2 ) ), F (( y 3 ) ), F (( y 4 ) ) ),, F i i,( ) ( ) ; Step6. step , ( 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 } VaR,

6 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 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 t y 1 SΠ3, F t y 2 SΠ3, F t y 3 SΠ3, F t, F i i, t ( ) 3 t2 ( ) ; Step6. step , 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 } 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 VaR VaR t2copula VaR , Copula,,,, VaR,. 3, t2copula Copula,Copula

7 3 Copula2GARCH 51, t2copula,,,var ;,,.,,, step6 h.,, h,.,,,,,. 5 t2garch,copula,,, VaR,,. Copula2GARCH, t2garch,copula,t2copula [Jondeau. E. etc (2002) ],,., : 1) t2copula,,,,; 2) t2garch, ; 3),EGARCH,,. 4) Archimedean Copula,,. : [ 1 ] Schweizer B,Sklar A. Probabilistic Metric Spaces[M]. North2HollandΠElsevier,1983,NewYork. [ 2 ] Genest C,Mackay J. The Joy of Copulas :Bivariate distributions with uniform marginals[j ]. American Statistician,1986,40, [ 3 ] Joe H. Parametric families of multivariate distributions with given marginals[j ]. Journal of Multivariate Analysis,1993,46, [ 4 ] Embrechts P,Mcneil A J,Straumann D. Correlation and dependence in risk management :Properties and pitfalls[a ]. Dempster M. Risk Management :Value at Risk and Beyond. Cambridge University Press,1999, [ 5 ] Claudio Romano. Calibrating and Simulating Copula Functions :an Application to the Italian Stock Market[ R] [ 6 ] Embrechts P, Hoeing A,Juri A. Using copulae to bound the value2at2risk for functions of dependent risks [J ]. Finance and Stochastics,2003,7, [ 7 ] Durrleman V A,Nikeghbali G. Riboulet and T. Roncalli. Copulas :An open field for risk management[ R] [ 8 ] Roberto De Matteis. Fitting Copulas to Data[D]. IMU,Zurich,2001. [ 9 ] Ling Hu. Essays in econometrics with applications in macroeconomic and financial modeling[d]. Yale University,2002. [10 ] Gabriel F,Markus J,Alexander S. Elliptical copulas :Applicability and limitations[j ]. Statistics and Probability Letters,2003,63 : t., 2.

8 [11 ] Davide M,Walter V. Copula Sensitivity in Collateralized Debt Obligations and Basket Default Swaps Pricing and Risk Monitoring [ R] [12 ] Jean2David F. Goodness2of2. t tests for Copulas[J ]. Journal of Multivariate Analysis. [13 ] Hull J,White A. Valuation of a CDO and an nth to Default CDS without Monte Carlo Simulation[ R] [14 ] Credit Suisse of First Boston. Creditrisk + [ R]. Credit Risk Management,Framework,1997. [15 ] Morgan J P. Credit metrics[ R]. Credit Risk Management Framework,1997. [16 ]. (Copula) [J ].,2002,4 : Zhang Y T. Copula and financial risk analysis[j ]. Statistic Research,2002,4 : [17 ]. [J ].,2002,9 : Zhang Y T. Which correlated index should we choose[j ]. Statistic Research,2002,9 : [18 ]. Copula [J ].,2004,4 : Zhang M H. Quantitative research of multi2asset VaR by copula[j ]. Quantitative&Technical Economics,2004,4 : [19 ],,. Copula [J ]. 2003,3 (5) : Wei Y H,Zhang S Y,Meng L F. The application of Copula in finance[j ]. Journal of Northwest A&F University,2003,3 (5) : [20 ],,. [J ].,2004,12 (4) :1-5. Wu Z X,Ye W Y,Miao B Q. Risk analysis of foreign exchange markets by Copula[J ]. Management Science of China,2004,12 (4) : 1-5. [21 ] Jondeau Eric,Rockinger. Micheal,Conditional Dependency of Financial Series :The Copula2GARCH Model,working paper,2002. [22 ],. Copula2GARCH [J ].,2004,22 (4) :7-12. Wei Y H, Zhang S Y. The correlation analysis of financial market - the model and application of copula2garch [ J ]. System Engineer,2004,22 (4) :7-12. [23 ],,. [J ].,2004,19 (4) : Wei Y H,Zhang S Y,Guo Y. The study of correlation in financial market[j ].Journal of System Engineer,2004,19 (4) : [24 ] Hong Y M. A new test for ARCH effects and its finite2sample performance[j ].Journal of Business & Economic Statistics,1999,17 : [25 ] Chen M,Chen G. A nonparametric test of conditional autoregressive heteroscedasticity for threshold autoregressive models[j ]. The Canadian Journal of Statistics,2002,30. [26 ],. [A].,2003. Zhang J, Yang Z H. Copula and Its Application in Calculating the VaR of Portfolio[A]. the No. 11 Annual Meeting of Field Statistics of China,2003.

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