Portfolio Optimization via Pair Copula-GARCH-EVT-CVaR Model

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1 Avalable ole at Systems Egeerg Proceda 2 (2011) Portfolo Optmzato va Par Copula-GARCH-EVT-CVaR Model Lg Deg, Chaoqu Ma, Weyu Yag * Hua Uversty, Hua, Chagsha , PR Cha Abstract Ths paper uses CVaR as the rsk measure ad apples EVT to model the tals of the retur seres so as to estmate rsk of assets more accurately. Ths paper also apples par Copula to capture the ter-depedece structure betwee assets ad costructs par Copula-GARCH-EVT model; the, we combe t wth Mote Carlo Smulato ad Mea-CVaR model to optmze portfolo. Fally, a emprcal study of four Idexes from Chese Stock Market s performed ad the result suggests that par Copula ca better characterze the ter-depedece structure betwee assets ad the performace of par Copula-GARCH-EVT-CVaR model s better tha that of multvarate t Copula-GARCH-EVT-CVaR model portfolo optmzato Publshed by Elsever B.V. Ope access uder CC BY-NC-ND lcese. Keywords: par Copula; GARCH; EVT; CVaR; Portfolo Optmzato; Facal Egeerg 1. Itroducto Markowtz [1] poeers the costructo of the optmal portfolo takg the combato of rsk ad retur to accout ad proposes the Mea-Varace model, characterzg portfolo retur wth expected retur ad estmatg portfolo rsk wth varace, whch lad the corerstoe of the developmet of moder portfolo theory. Varace has o drecto, but most vestors oly regard the adverse devato betwee retur ad expected retur as rsk, thus subsequet researchers troduce other rsk measures such as sem-varace ad sem-devato. Baumol [2] frst proposes Value-at-Rsk (VaR), whch summarzes the worst loss over a target horzo at a gve cofdece level. Because VaR s easy to calculate ad agrees stcts of vestors, t has bee wdely accepted as the rsk measure by may facal sttutos ad facal regulators. However, whe the asset retur s o-ormal dstrbuto, VaR s ot sub-addtve ad covex, so t s ot the coheret measure of rsk proposed by Artzer et al. [3]. Takg the o-ormal features of asset returs to accout, order to overcome the shortcomgs of VaR, Uryasev ad Rockafellar [4] suggests a alteratve rsk measure: Codtoal Value at Rsk (CVaR). CVaR has close relatoshp wth VaR, herts most advatages of VaR ad satsfes sub-addtvty ad covexty, so t s a coheret measure of rsk. Ths paper uses CVaR as the rsk measure ad apples Mea-CVaR model for portfolo optmzato. Sce VaR ad CVaR are greatly affected by the tal dstrbuto of rsk factors, t s able to characterze tal dstrbutos ad estmate rsk of assets more accurately applyg extreme value theory (EVT) to research the tals of the retur seres. However, the retur seres are ot always depedetly ad detcally dstrbuted but E-mal address: deglg1212@163.com Publshed by Elsever B.V. do: /.sepro Ope access uder CC BY-NC-ND lcese.

2 172 Lg Deg et al. / Systems Egeerg Proceda 2 (2011) leptokurtoss, fat tals ad volatlty clusterg, t s approprate to apply EVT to the retur seres drectly. Followg the approach of Frey ad McNel [5], we wll frst use GARCH model to ft the retur seres ad the apply EVT to the ovatos. Addtoally, order to accurately estmate CVaR of the portfolo, we eed to capture the o-lear terdepedece betwee tals of asset returs effectvely. Ths paper wll apply Copula to model the ter-depedece betwee ovatos of the retur seres, whch ca characterze varous depedece structures comprehesvely ad effectvely. Sklar [6] frst proposes Copula to measure the o-lear ter-depedece betwee varables. After that, Schwezer ad Wolff, Geest ad Mackay, Joe. H., Nelse ad so o further develop ad mprove Copula ad t has become a mportat approach to costruct multvarable ot dstrbuto ad capture depedece structure betwee varables. Embrechts et al. [7] frst troduce Copula to facal research; Cherub [8] systematcally summarzes the applcatos of Copula face; Jodeau ad Rockger [9] propose Copula-GARCH model ad apply t to extract the depedece structure betwee stock markets. I Cha, Zhag [10] frst troduces Copula; We ad Zhag [11] apply Copula-GARCH model to extract the depedece structure betwee Shagha Stock Market ad Shezhe Stock Market; Wu ad Che [12] apply Copula-GARCH model to aalyze portfolo rsk Chese Stock Market; We ad Yag propose to use a copula-based correlato measure to test the terdepedece amog stochastc varables ad fd that the ew correlato coeffcet Co s more sutable to descrbe the terdepedece amog stochastc varables tha the G correlato coeffcet [13]. However, most emprcal researches about Copula have focused o capturg bvarate ter-depedece structure. The reaso s that the complexty of multvarate Copula creases sharply alog wth ts dmesos ad t s cled to gore the fluece of ts dmesos ad the dffereces of tal depedece betwee varables. Berdford ad Cooke [14] troduce probablty structure model of multvarate dstrbuto va smple buldg block par Copula based o ve structure, whch decomposes multvarate desty to a seres of par Copula tmes margal desty. The paper s orgazed as follows. I secto 2, we troduce the par Copula decomposto of a geeral multvarate dstrbuto ad ts parameters estmato. I secto 3, combg GARCH model ad EVT we costruct par Copula-GARCH-EVT model. I secto 4, we costruct Mea-CVaR model ad combe par Copula-GARCH-EVT model to optmze portfolo. I sectos 5 we carry out emprcal aalyss. Secto 6 cocludes. 2. Par Copula decomposto of multvarate dstrbuto ad parameter estmato Accordg to Sklar theory, multvarate dstrbuto F wth margals F 1 (x 1 ), F 2 (x 2 ),, F (x ) ca be wrtte as follows wth some approprate -dmesoal Copula C F( x, x,, x ) C F( x ), F ( x ),, F ( x ) (1) If F s absolutely cotuous ad strctly creasg, ts desty ca be factorzed as f ( x, x, x ) c F( x ), F ( x ),, F ( x ) f ( x ) f ( x ) 1 2, (2) I formula (2), f (x ) s the probablty desty of F (x ), c 1 ( ) s -dmesoal Copula desty. We ca kow from formula (2) that multvarate desty cotas two parts: ter-depedece structure amog varables ad margal desty. Margal desty s easy to get but depedece structure s very complex. Par Copula decomposto provdes a approach to decompose multvarate ter-depedece structure, whch decomposes multvarate desty to a seres of par Copula tmes margal desty accordg to certa logcal structure [15]. Multvarate dstrbuto has a umber of possble par Copula costructos. For example, a fve-dmesoal desty has 240 possble dfferet costructos. Berdford ad Cooke [16] troduce the regular ve to descrbe par Copula decomposto. Kurowcka ad Cooke [17] troduce two specal cases of regular ves: the caocal ve ad the D-ve. Fgure 1 ad fgure 2 show the specfcatos correspodg to a four-dmesoal caocal ve ad a four-dmesoal D ve respectvely.

3 Lg Deg et al / Systems Egeerg Proceda 2 (2011) Caocal ve ad D ve are sutable for dfferet data sets. Whe a partcular varable acts as the key varable that govers teractos the data set, t s advatageous to ft a caocal ve ad locate the varable at the root of the caocal ve. However, the character lmts the freedom of par Copula decomposto ad ts use. Whe there s o plot varable, the data set s sutable for D ve. Fg. 1. (a) A caocal ve wth 4 varables; (b) A D-ve wth 4 varables Accordg to Bedford ad Cooke [14], the -dmesoal desty correspodg to a caocal ve ca be wrtte as 1 f( x, x,, x ) f( x ) c F( x x,, x ), F( x x,, x ) 1 2 k, 1,, k1 1 1 (3) the -dmesoal desty correspodg to a D ve ca be wrtte as 1 f( x, x,, x ) f( x ) c F( x x,, x ), F( x x,, x ) (4) 1 2 k, 1,, k1 1 1 There are a couple of margal codtoal dstrbutos F(x v) the par Copula costructo c xv () v from formula (3) ad (4). Accordg to Joe [18], for every, Cxv, ( ), ( ) v Fx v Fv v F( x v) Fv ( v ) (5) where v s a d-dmesoal vector, v s a arbtrarly chose compoet of v, v - deotes the v-vector excludg the compoet v. Parameters of the -dmesoal desty correspodg to a caocal ve ad D ve ca be estmated through maxmum lkelhood approach. The log-lkelhood for a caocal ve s gve by 1 T log c, 1,, 1 F( x, t x1, t,, x 1, t), F( x, t x1, t,, x 1, t) (6) 1 1 t1 the log-lkelhood for a D-ve s gve by 1 T log c, 1,, 1 F( x, t x 1, t,, x 1, t), F( x, t x 1, t,, x 1, t) (7) 1 1 t1

4 174 Lg Deg et al. / Systems Egeerg Proceda 2 (2011) Estmatg the parameters of formula (6) ad (7), we should frst get values of the parameters from each tree the ve, take them as the tal values of parameters ad maxmze the log-lkelhood to get the fal values of parameters[19]. 3. Costructo of par Copula-GARCH-EVT model EVT s a effectve approach to estmate the extreme case of market rsk. It oly focuses o the dstrbuto of extreme values ad t ca descrbe the tal quatles of a dstrbuto accurately wth the overall dstrbuto ukow. EVT maly cludes Block Maxma model ad Peaks over Threshold model. We have to set wdow parameters artfcally ad ca ot mport other explaatory varables whe usg BMM, so t lacks flexblty practcal applcato. Whle POT s bult o the hypothess that the dstrbuto of returs over thresholds follows the geeralzed Pareto dstrbuto (GPD), whch oly model returs data over some hgh eough threshold. POT focuses o the approxmate descrpto to the tals, ot the overall dstrbuto, so t overcomes the shortcomgs of other approaches to ft fat tal dstrbuto [20]. Therefore POT based o GPD s wdely used actual research. The geeralzed Pareto dstrbuto ca be wrtte as G, x 1/ 1 (1 ), 0, ( x) x 1 exp( ), 0, (8) where β, ξ are the scale ad shape parameters respectvely. Whe ξ 0, x 0 ad 0 x -β/ξ whe ξ<0. Facal tme seres have typcal o-ormal characters, such as leptokurtoss, fat tals, volatlty clusterg ad leverage effect, t s able to descrbe the tme seres effectvely usg GJR-GARCH (1, 1) ad so ths paper apples GJR-GARCH (1, 1) to ft each asset retur seres. Assumg that a portfolo cossts of assets, the retur seres for asset s {y,t, t=1,2, }. The GJR-GARCH (1, 1) ca be wrtte as y t, t, t, t, zt, zt, ~. d. w [ 0] t, t, 1 t, 1 t, 1 t, 1 (9) where μ, w, α, β, γ are parameters, y,t s the retur seres of asset, γ s the leverage coeffcet ad z,t s the ovato process, whch s depedetly ad detcally dstrbuted. Typcally, z,t follows a fat-taled dstrbuto. I the emprcal studes, we assume z,t follows Studet t dstrbuto. The, we use GPD to model the ovato z the lower ad upper tals ad the emprcal dstrbuto the remag part. The the margal dstrbuto of each ovato s gve by L L 1/ T L u L u z L 1 L z u T L R F( z) ( z) u z u (10) R R 1/ T R u R z u R 1 1 z R u T L R where u, u are the lower ad upper threshold respectvely, φ(z ) s the emprcal dstrbuto o the terval L R [ u, u ], T s the umber of z ad Tu L s the umber of ovato whose value s less tha u L ad T R s the u

5 Lg Deg et al / Systems Egeerg Proceda 2 (2011) R umber of ovato whose value s greater tha u. I the emprcal studes, we choose the exceedaces to be 10 percet of the ovato for lower ad upper tals respectvely. Fally, we apply par Copula to extract the depedece structure betwee the ovatos of each asset retur, whch eeds to select caocal ve or D ve to descrbe the logcal structure of the decomposto frstly. Accordg to the character of caocal ve ad D ve, we frst ft a bvarate Studet Copula to each par of ovatos ad obta the estmated degrees of freedom for each par; a small umber of degrees of freedom dcates strog depedece. The we ca select a approprate ve for the par Copula decomposto [19]. Theoretcally, we ca choose the best bvarate Copula for each Copula, but ths wll make the decomposto so complcated ad the result may ot be the best, such as Aas ad Czado [19]. Because Studet Copula s both lower ad upper tal depedet, we choose Studet Copula for all pars of the decomposto. 4. Mea-CVaR model CVaR s the expected losses that exceed the VaR at some cofdece level, whch ca be wrtte as [21]: CVaR ( y) E[ y y VaR ( y) ] (11) where y s the returs of a portfolo, β s the cofdece level, VaR β (y) s the VaR at the β cofdece level ad CVaR β (y) s the expected losses of the portfolo at the β cofdece level, whch reflects the umber of the potetal losses whe the losses exceed the threshold VaR β (y). So CVaR β (y) VaR β (y). I rsk maagemet, f we ca cotrol CVaR, the we ca also cotrol VaR at the same tme [22]. Assumg that there are assets a portfolo, X=(x 1,,x ) T s the posto for each asset, x 0(=1,,) ad 1, correspodg asset retur s Y=(y 1,,y ) T x, so the expected retur of the portfolo s x y, the 1 1 T expected loss s x 1 y. The loss fucto of the portfolo s f ( X, Y) xy 1 X Y, assumg Y has desty P(Y). Rockafellar ad Uryasev [23] combe CVaR ad VaR through a specal fucto F β (X, α) ad trasform the problem of mmzg CVaR to the problem of mmzg a cotuously dfferetable ad covex fucto. 1 F ( X, ) [ f( X, ) ] P( Y ) dy m 1 Y (12) yr VaR ( X ) arg m F ( X, ) (13) R CVaR ( X ) m F ( X, ) F ( X, VaR ( X )) (14) R where [U] + =max(u,0), F β (X,α) s covex ad cotuously dfferetable wth respect to α. I practce, the desty of Y s usually ukow, we ca geerate a collecto of Y uder q dfferet stuatos by Mote Carlo smulatos, the the correspodg approxmato to F β (X, α) s ~ 1 q T k F ( X, ) [ X Y ] (15) k 1 q(1 ) The we costruct Mea-CVaR model usg CVaR to take the place of varace Mea-Varace model:

6 176 Lg Deg et al. / Systems Egeerg Proceda 2 (2011) m q(1 ) q k u k 1 T k k X Y u 0 k u 0 1 T q k st.. X Y k 1 q x 1 1 x 0 (16) where ρ s the vestors expected retur ad Y k ca be geerated through Mote Carlo smulatos combg par Copula-GARCH-EVT model. I secto 3, we use par Copula to capture the depedece structure betwee ovatos, so we have to smulate the ovatos frst ad the smulate the returs of the portfolo usg GJR- GARCH (1, 1). The progress to smulate a retur sample correspodg to a caocal ve s as follows: Step 1. Sample w 1, w 2,, w depedet uform o [0, 1], assumg z 1 = w 1; Step 2. Accordg to formula (5), we ca get F (z 2 z 1 ), assumg w 2 = F (z 2 z 1 ); C3,2 1 F( z3 z1 ), F( z2 z1 ) Step 3. Accordg to formula (5), we ca get F (z 3 z 1 ), the F( z3 z1, z2), assumg F( z2 z1) w 3 = F (z 3 z 1, z 2 ); Step 4. Through the same way, we ca get w 4 =F (z 4 z 1, z 2, z 3 ),, w =F(z z 1, z 2,, z -1 ); Step 5. Accordg to the margal dstrbuto of z, we ca get z =F -1 (w z 1, z 2,, z -1 ), the we ca get a smulato of the ovato {z 1, z 2,, z }; Step 6. Smulate the returs of the portfolo usg GJR-GARCH (1, 1); Step 7. Repeat the above steps ad smulate q tmes, we ca get the smulated retur seres of the portfolo {y 1,t,, y,t, t=1,,q}. 5. Emprcal studes ad aalyss 5.1. Data source ad statstcs I the emprcal studes, we choose four dexes from Chese Stock Market: Shagha Composte Idex (H), Shezhe Compoet Idex (Z), SME Composte Idex (M) ad Shagha Fud Idex (F). The data employed s from December 1, 2008 to May 31, 2011 ad the data source s from CSMAR soluto. We employ the daly log returs defed as R t =l p t - l p t-1.the sample cossts of 1334 observatos of log returs total. All the umercal computatos are ru MATLAB The prelmary descrptve statstcs of the sample data are preseted Table 1. Table 1. The prelmary descrptve statstcs of the sample data Shagha Composte Idex Shezhe Compoet Idex SME Composte Idex Shagha Fud Idex Mea Maxmum Mmum Std.Dev Skewess Kurtoss

7 Lg Deg et al / Systems Egeerg Proceda 2 (2011) Jarque-bera (0.0010) (0.0010) (0.0010) (0.0010) As show Table 1, the kurtoss of each dex s larger tha 3 ad the skewess s less tha 0, whch dcates that all seres have fat tals ad leptokurtoss. From the JB statstcs we kow that they do ot follow ormal dstrbuto. So t s effectve to ft the seres usg GJR-GARCH (1, 1) model GARCH-EVT applcato We use GARCH-EVT Model to ft the margal dstrbuto of each retur seres, estmatos of the parameters are preseted Table 2. Table 2. Estmated parameters for GARCH-EVT model Shagha Composte Idex Shezhe Compoet Idex SME Composte Idex Shagha Fud Idex θ μ w E E E E-06 α β γ v u L ξ L β L u R ξ R β R Par Copula decomposto ad parameter estmato We use Studet Copula to obta the estmated degrees of freedom for each couple of ovatos, whch s gve Table 3, ad the we ca determe the decomposto structure of par Copula. Table 3. Estmate degrees of freedom for each couple of ovatos Shezhe Compoet Idex SME Composte Idex Shagha Fud Idex Shagha Composte Idex Shezhe Compoet Idex Shagha Fud Idex As show Table 3, the degrees of freedom betwee Shagha Fud Idex ad other three dexes are smaller, so t s advatageous to select the caocal ve to descrbe the par Copula decomposto, as show fgure 3.

8 178 Lg Deg et al. / Systems Egeerg Proceda 2 (2011) Fg.2. Caocal ve structure for the sample data Havg chose the decomposto structure for par Copula, we estmate the parameters. The result s gve Table 4. Table 4. Estmated parameters for the par Copula decomposto Parameter Ital values Fal values ρ FH ρ FZ ρ FM ρ HZ F ρ HM F ρ ZM HF ν FH ν FZ v FM ν HZ F ν HM F ν ZM HF log-lkelh E E Portfolo optmzato We geerate a set of 10,000 samples usg the smulato procedure descrbed secto 4. The at the cofdece level of β=95%, we apply the Mea-CVaR model to optmze the portfolo ad obta the CVR effcet froter of the portfolo uder dfferet expected returs, as show fgure 4.

9 Lg Deg et al / Systems Egeerg Proceda 2 (2011) x CVaR VaR Expected returs VaR ad CVaR Fg.4. The effcet froters of VaR ad CVaR uder Mea-CVaR model Fgure 4 shows that CVaR s always larger tha VaR uder Mea-CVaR model ad VaR s ot covex. At the same tme, we ca obta the optmal portfolos usg Mea-Varace model ad calculate the CVaR effcet froter, as show fgure 5. 5 x 10-4 Mea-CVaR model Mea-Varace model Expected returs CVaR Fg.5. The effcet froters of CVaR uder Mea-CVaR ad Mea-Varace models From fgure 5, we ca kow that the CVaR uder Mea-CVaR model s always less tha that uder Mea- Varace model whe the expected returs are the same, but the dfferece s very small, whch s because the effcet froter of Mea-CVaR model coverges to the effcet froter of Mea-Varace model whe the cofdece level approaches 1 [24]. We ca also kow that the mmum CVaR portfolo s ot the mmum varace portfolo. Fally, we geerate a set of 10,000 samples usg Studet Copula-GARCH-EVT model ad Mote Carlo smulato. The at the cofdece level of β=95%, we apply the Mea-CVaR model to optmze the portfolo ad

10 180 Lg Deg et al. / Systems Egeerg Proceda 2 (2011) obta the CVR effcet froter of the portfolo uder dfferet expected returs, as show fgure x par Copula t Copula Expected returs CVaR Fg.6. The effcet froters of CVaR uder par Copula ad Studet Copula From fgure 6, we ca kow that whe the expected retur s equal, CVaR uder par Copula-GARCH-EVT model s larger tha that of Studet Copula-GARCH-EVT model. Ths dcates that t s better to apply par Copula to capture the depedece structure amog assets, so as to estmate CVaR of portfolo more accurately, but CVaR wth Studet Copula uderestmates the rsk of portfolo. 6. Cocluso Ths paper estmates the rsk of portfolo usg CVaR ad apples Mea-CVaR model to optmze portfolo. I order to estmate the rsk of portfolo more accurately, we apply EVT to model the tals of the ovato of each asset retur; the, we capture the depedece structure betwee ovatos of asset returs by par Copula. Par Copula decomposto model cosders both the fluece of portfolo dmesos ad the dffereces of tal depedece betwee assets. Fally, the results of the emprcal studes dcate that t s more effcet to optmze portfolo usg Mea-CVaR model tha Mea-Varace model; the optmal portfolo s better va par Copula- GARCH-EVT model tha that va Studet Copula-GARCH-EVT model. 7. Copyrght All authors must sg the Trasfer of Copyrght agreemet before the artcle ca be publshed. Ths trasfer agreemet eables Elsever to protect the copyrghted materal for the authors, but does ot relqush the authors' propretary rghts. The copyrght trasfer covers the exclusve rghts to reproduce ad dstrbute the artcle, cludg reprts, photographc reproductos, mcroflm or ay other reproductos of smlar ature ad traslatos. Authors are resposble for obtag from the copyrght holder permsso to reproduce ay fgures for whch copyrght exsts. Ackowledgemets Ths work s supported by the Fud of Natoal Outstadg Youg Scholar Cha ( ).

11 Lg Deg et al / Systems Egeerg Proceda 2 (2011) Refereces 1. Markowtz H., Portfolo Selecto. Joural of Face. 25(1952) Baumol W.J., A expected ga-cofdece lmt crtero for portfolo selecto. Maagemet Scece. 10(1963) Artzer P., Delbae F., Eber J.M, Heath D., Coheret measures of rsk. Mathematcal Face. 9(1999) Uryasev S., Codtoal Value-at-Rsk (CVaR): Algorthms ad Applcatos. Facal Egeerg News. 14(2000) McNel A.J., Frey R., Estmato of tal-related rsk measures for heteroscedastc facal tme seres: a extreme value approach. Joural of Emprcal Face. 7(2000) Sklar A., Foctos de repartto ad dmesos et leurs marges. Publcatos de l Isttut Statstque de l Uverste de Pars. 8(1959) Embrechts P., McNel A.J., Strauma D., Correlato: ptfalls ad alteratves. RISK. 5(1999) Cherub U., Lucao E., Vecchato W., (2004) Copula Methods Face. West Sussex: Wley. 9. Jodeau E., Rockger M., The Copula-GARCH model of codtoal depedeces: A teratoal stock market applcato. Joural of Iteratoal Moey ad Face. 25(2006) X. Zhag, Copula ad facal rsk aalyss. Statstc Research. 4(2002) Y. We, S. Zhag, Y. Guo, Research o degree ad patters of depedece facal markets. Joural of System Egeerg. 19(2004) Z. Wu, M. Che, W. Ye, Rsk Aalyss of Portfolo by Copula-GARCH. Systems Egeerg Theory ad Practce. 26(2006) F. WEN, Z. LIU. A Copula-based Correlato Measure ad Its Applcato Chese Stock Market. Iteratoal Joural of Iformato Techology & Decso Makg, 8(2009) Bedford T., Cooke R.M., Probablty desty decomposto for codtoally depedet radom varables modeled by ves. Aals of Mathematcs ad Artfcal Itellgece. 32(2001) E. Huag, X. Cheg, Aalyss of portfolo VaR by par Copula-GARCH. Joural of the Graduate school of the Chese Academy of Scece. 27(2010) Bedford T., Cooke R.M., Ves a ew graphcal model for depedet radom varables. Aals of Statstcs. 30(2002) Kurowcka D., Cooke R.M., Dstrbutos free cotuous bayesa belef ets. Fourth Iteratoal Coferece o Mathematcal Methods Relablty Methodology ad Practce. Sata Fe, New Mexco, Joe H., Famly of m-varate dstrbutos wth gve margs ad m(m-1)/2 bvarate depedece parameters. Dstrbutos wth fxed margals ad related topcs. 28(1996) Aas K., Czado C., Frgess A., Bakke H., Par-Copula costructos of multple depedece. Isurace: Mathematcs ad Ecoomcs. 44(2009) X. Lu, G. Qu, Research of Lqudty Adusted VaR ad ES Cha Stock Market Based o Copula-EVT model. Joural of Appled Statstcs ad Maagemet. 29(2010): Rockafellar R.T., Uryasev S., Codtoal Value-at-rsk for geeral loss dstrbutos. Joural of Bakg & Face. 26(2002) L. He, F. We, C. Ma, Portfolo Optmzato Model of Coheret Value-at-Rsk. Joural of Hua Uversty (Natural Sceces). 32(2005) Rockafellar R.T., Uryasev S., Optmzato of codtoal Value-at-rsk. Joural of Rsk. 2(2002) Z. Lu, A Comparso of the Portfolo Effcet Froter uder Varous Mea-Rsk Crteros. Mathematcs Ecoomcs. 23(2006)