MPRA Munich Personal RePEc Archive Exreme Risk Value and Dependence Srucure of he China Securiies Index 300 Terence Tai Leung Chong and Yue Ding and Tianxiao Pang The Chinese Universiy of Hong Kong, The Chinese Universiy of Hong Kong, Zhejiang Universiy 6 March 017 Online a hps://mpra.ub.uni-muenchen.de/80556/ MPRA Paper No. 80556, posed Augus 017 14:51 UTC
Exreme Risk Value and Dependence Srucure of he China Securiies Index 300 Terence Tai-Leung Chong 1 and Yue Ding Deparmen of Economics The Chinese Universiy of Hong Kong Tianxiao Pang School of Mahemaical Sciences Zhejiang Universiy, China 6/3/17 Absrac: A ime-varying copulas condiional value a risk (CVaR) model is esimaed o analyze he exreme risk value and dependence srucure of he China Securiies Index 300 (CSI 300) and index fuures porfolios. The goodness-of-fi es as well as he in-sample and ou-of-sample ess show ha ime-varying copulas ouperform consan copulas. Specifically, he Suden s, normal, Placke, and he roaed Gumbel copulas ouperform he roaed Clayon copulas. Keywords: CVaR model; Time-varying copulas. 1 We would like o hank Min Chen, Yingshi Chen and Margare Loo for heir able research assisance. Any remaining errors are ours. Corresponding Auhor: Terence Tai-Leung Chong, Deparmen of Economics, The Chinese Universiy of Hong Kong, Shain, N.T., Hong Kong. E-mail: chong064@cuhk.edu.hk. Webpage: hp://www.cuhk.edu.hk/eco/saff/lchong/lchong3.hm. 1
1. Inroducion Time-varying copulas have been widely applied in recen years. For example, Chollee e al. (009) develop a regime-swiching copula (RSC). Creal e al. (011) propose he generalized auoregressive score (GAS) ime-varying copulas. Time-varying copula models also include condiional normal, condiional Gumbel, and condiional symmerized Joe-Clayon copulas. In his paper, a ime-varying copulas-condiional value a risk (CVaR) model is applied o analyze he exreme risk value and dependence srucure of he China Securiies Index 300 (CSI 300) and index fuures porfolios. An AR-GARCH (1, 1)- model is esimaed. Nine consans and wo ime-varying copula models are compared; he goodness-of-fi es as well as he in-sample and ou-of-sample ess show ha ime-varying copulas ouperform consan copula models.. Mehodology and Daa Time-varying copulas allow he copula parameers o change over ime. Consider a bivariae ime series, wih H(x 1, x ) being he join disribuion. Le F 1 (x 1 ) and F (x ) be he marginal disribuions. There exiss a copula C wih he following join disribuion funcion: x, F x ) H ( x1, x ) С( F1 1. (1) The copula funcion C(x ) is shown as (1) if eiher one of F 1 (x 1 ) and F (x ) is coninuous. This paper considers wo ime-varying copulas: he ellipical copula (he Suden s GAS) and he Archimedean copula (he roaed Gumbel GAS). The inference funcion for margins (IFM) mehod of Joe (1997), which considers he esimaion error from marginal disribuions, is used for parameer esimaion. The daily reurn of he CSI 300 and he consecuive price of index fuures (IFLX0) of he curren monh are used o characerize specific feaures of he sock markes and index fuures markes of China. The sample is drawn during he period from April 16, 010 (he sar dae of he IFLX0) o Sepember 7, 01, which covers 585 rading days. Table 1 shows he summary saisics, where he daily reurn series of boh indices clearly indicae fa-ail disribuions. The kurosis of IFLX0 is 5.57, which is slighly higher han he 4.456 of CSI 300, indicaing ha he index fuures are more volaile. Table 1: Summary saisics IFLX0 CSI 300 Summary Saisics Mean -0.001-0.001 Sd dev 0.015 0.014 Skewness 0.019-0.170 Kurosis 5.57 4.456 Correl (lin/rnk) 0.944 0.98
3. Esimaion of he Marginal Disribuion Using he Schwarz crierion, he AR ()-GJR (1, 1)- model and AR (0) - GJR (1, 1) - model are seleced o fi he marginal disribuions of he IFLX0 and CSI 300 Index, respecively. The models are R R R 0 1 1 () ( ) 1 1 1I 1 where iid Skewed (, ), (1,1 ) is he skewness degree, and (, ] is he degree of fa ail, R is he daily reurn, and is he new informaion. The key elemen among differen GARCH models depends on he form of he condiional covariance. In he GJR-GARCH condiional variance equaion, 0, 0 I( 1) 1, 0. 0 0 demonsraes posiive informaion or posiive impac, while indicaes bad news or negaive impac ha affecs condiional variance. The impac of good news on condiional variance is, whereas he impac of bad news is. Therefore, invesors are more responsive o bad news if 0, and more sensiive o good news if 0. Table : Marginal Disribuion Parameer Esimaion IFLX0 CSI 300 Condiional Mean 0 0.000-0.001 1-0.054-0.015 - Condiional Variance 0.000 0.000 0.09 0.014 0.019 0.009 0.946 0.965 Skew Densiy 4.33 6.08 0.058 0.06 GoF Tess Cross-equaion Effec p-value 0.0860 0.5687 KS p-value 0.57 0.053 3
Table shows he esimaion resuls of he marginal disribuion parameer. Boh series have 0 (0.946 for IFLX0 and 0.965 for CSI 300), which means invesors are more responsive o bad news. A cross-equaion effec is shown in Table, which suggess ha here is no crossequaion effec for boh series. The Kolmogorov-Smirnov es shows ha he AR-GJR (1, 1)- disribuion is well-specified for he marginal disribuions of boh series. 4. Esimaion of he Time-Varying Copula Parameer Muli-sage maximum likelihood (MSML) based on he esimaion of marginal disribuions is applied o esimae ime-varying parameers of copulas. Several consan copulas are also esimaed for comparison: Table 3 shows ha Suden s, normal, Placke, and roaed Gumbel copulas ouperform he roaed Clayon copula in erms of he log-likelihood values. Table 3: Consan Copula Model Parameer Esimaes Parameric Es. Param Normal 0.949 64.4 Clayon 4.6731 53.9 Roaed Clayon 4.5163 50.3 Placke 86.338 637.5 Frank 9.0000 506.6 Gumbel 4.350 69.4 Roaed Gumbel 4.391 633.3 L U Sym Joe Clayon(, ) 0.7343, 0.889 Inf -1 Suden s (, ) 0.9000, 0.476 647.4 Tail dependence is esimaed using he roaed Gumbel (lower ail) and Suden s models (upper and lower ail). Resuls for he roaed Clayon copula are omied because i only feaures he upper ail. Figure 1 illusraes he resuls. The Gumbel copula is higher han he Suden s copula during he sample period, which indicaes greaer dependence beween he wo indices in a bear marke as compared o a bull marke. Tail dependence is low during he marke downurn from July 010 o February 01. This resul indicaes ha invesmen in boh markes can beer diversify marke risk in a crisis, while a shor hedge may no, as he wo indices are likely o fall ogeher during a marke crash. 4
Figure 1: Condiional ail dependence 0.9 RoGumbel lower ail Sud upper and lower ail Tail dependence from ime-varying copula models 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 Apr10 Jan1 Table 4 presens he sandard errors of boosrap esimaes. Time-varying copulas ouperform consan copulas because he log-likelihood of he Suden s GAS copula is he highes among all six esimaed copulas. For he in-sample es, he pairwise comparison es of Rivers and Vuong (00) is applied. The Giacomini and Whie (006) es is applied in he ou-of-sample es. The ou-of-sample forecasing models are considered based on a fixed window esimaion using daa from April 16, 010 o November 3, 011, which is wo-hirds of he sample period. The esimaed model is evaluaed based on he remaining daa from November 4, 011 o Sepember 7, 01, and he ou-of-sample log-likelihood values of he models are compared. Tables 5 and 6 show he resuls of he model selecion es. A -value greaer han suggess ha he lef copula ouperforms he op one. The resuls show ha ime-varying copulas ouperform consan models. 5
Normal Table 4: Sandard Errors of Esimaed Copulas Consan Naive MSML Boo Sim ˆ 0.949 s.e. 0.003 0.0075 0.0071 0.0056 64.4 ˆ 4.6731 Clayon Roaed Gumbel Suden s s.e. 0.111 0.566 0.4087 0.3807 53.9 ˆ 4.391 s.e. 0.158 0.3401 0.50 0.11 633.3 ˆ 0.9000 s.e. 0.0000 0.0404 0.0000 0.0074 ˆ 1 0.476 s.e. 0.0000 0.785 0.016 0.0517 647.4 Roaed Gumbel GAS Suden s GAS Time-varying Naive MSML Boo ˆ 0.875 s.e. 0.0834 6.356 0.869 ˆ 0.51 s.e. 0.0389.5404 0.0789 ˆ 0.7655 s.e. 0.0636 46.774 0.330 649.3 ˆ 1.011 s.e. 0.04 0.0000 0.6468 ˆ 0.406 s.e. 0.057 0.0000 0.0878 ˆ 0.7078 s.e. 0.47 66.948 0.1888 ˆ 1 0.355 s.e. 0.0857 64.440 0.0603 676.5 Noe: NAÏVE means naïve sandard errors, where he esimaion error from he earlier sages of esimaion (AR, GARCH and marginal disribuions) is ignored. 6
Table 5: In-Sample Model Comparison Roaed Normal Clayon Gumbel Suden s Normal - - - - Clayon -4.9706 - - - Roaed Gumbel -0.6674 10.105 - - Suden s 1.7096 7.1378 1.4469 - Log L 64.4 53.9 633.3 647.4 Rank 4 3 1 Table 6: Ou-of-Sample Model Comparison Roaed Normal Clayon Gumbel Sud RGum-GAS Sud -GAS Normal - - - - - - Clayon -1.0 - - - - - Roaed Gumbel 1.18 3.60 - - - - Sud 0.98 3.03-0.10 - - - RGum-GAS 1.56 4.01 1.88 1.08 - - Sud -GAS.05 3.81.3.93 1.49 - Log L 195.8 179.9 06.6 06.3 11.8 19.5 Rank 5 6 3 4 1 5. VaR and CVaR under Copulas and Opimal Porfolio Selecion We conduc he following Mone Carlo simulaion o calculae he ime-varying VaR and CVaR values: ( u j, x, u j, y ) ~ C(.) Sep 1: Generae a pair of uniformly disribued random numbers. ˆ 1 ˆ 1 ˆ ˆ ˆ j ( j, x, j, y ) ( F1 ( u j, x ), F ( u j, y )) Sep : Calculae, where F 1 (x) and F (x) are he skewed -disribuions. ˆ ˆ ( R, ) ( ˆ ˆ ˆ, ˆ ˆ ˆ x, Ry, x, j, x x, y, j, y y, ) Sep 3: Calculae by using he resul from he GARCH (1, 1) model. R w Rˆ 1 x, w Rˆ y, Sep 4: Simulae he porfolio reurn. The above process is repeaed 5000 imes o calculae he opimal weigh on index fuures deermined by opimal porfolio selecion. I is based on he esimaed mean-variance under CVaR consrains. The confidence level of he opimal VaR and CVaR is 99%. 7
Figure : Time-varying opimal weigh according o Suden s copula Opimal weigh according o ime-varying copula models, q=0.01 1.5 1 0.5 0-0.5-1 Apr10 RoGumbel Sud Jan1 Noe: q =0.01 means significan a he 1% level. Figure shows ha ime-varying opimal weighs on index fuures are mosly posiive. A larger weigh on index fuures han on he CSI 300 lends o beer risk hedging feaures of index fuures. Figure 3 shows ha he VaR and CVaR are less han -6% from July 010 o February 01. This is consisen wih he increase of risks during a crisis. CVaR is lower han VaR in mos of our sample period because CVaR considers ail risk beyond he VaR and hus is a beer risk measure. 8
Figure 3: Value-a-risk and condiional value-a-risk -0.0 Value-a-Risk from ime-varying copula models,q=0.01-0.03-0.04-0.05-0.06-0.07-0.08-0.09 Apr10 RoGumbel Sud Jan1-0.0 Condiional Value-a-Risk from ime-varying copula models, q=0.01-0.03-0.04-0.05-0.06-0.07-0.08-0.09-0.1 Apr10 RoGumbel Sud Jan1 Noe: q =0.01 means significan a he 1% level. Figure 4 shows Paon s (01) concluding conversions of hree copulas a he rank correlaion of 1. CVaR values are clearly lower han he VaR values beween he rank correlaions of 0. and 0.6. The CVaR model during a crisis considers he fricions of informaion ransmission beween hese wo markes. Meanwhile, he correlaion beween he wo markes drops o less han 0.6. Thus, CVaR is lower han VaR when he rank correlaion is lower han 0.6. In conras, he correlaion in Paon (01) is beween 0.3 and 0.7, which is more balanced around 0.5. 9
Figure 4: Value-a-risk and condiional value-a-risk and rank correlaion -1.8 - -. -.4 Porfolio Value-a-Risk, q=0.01 -.6-4 -.8 0 0.5 1 0 0.5 1 Rank correlaion Rank correlaion Porfolio Condiional Value-a-Risk, q=0.01porfolio Condiional Value-a-Risk, q=0.001 -. -.4 -.6 -.8-3 -3. 0 0.5 1 Rank correlaion -.5-3 -3.5 -.5-3 -3.5-4 Porfolio Value-a-Risk, q=0.001 Normal Sud rogumbel 0 0.5 1 Rank correlaion Noe: q =0.01 means significan a he 1% level and q=0.001 means significan a he 0.1% level. 6. Conclusion In his paper, a ime-varying copulas-cvar model is employed o analyze he exreme risk value and dependence srucure beween CSI 300 Index and index fuures. Nine consan and wo ime-varying copula models are esed. I is found ha he Suden s, normal, Placke, and roaed Gumbel copulas ouperform he roaed Clayon copulas, and ha he ime-varying copulas ouperform all consan copulas. The value of he Gumbel copula is higher han ha of he Suden s copula during he sample period, which indicaes higher dependence beween he wo indices in a bear marke han in a bull marke. Tail dependence is low during he marke downurn from July 010 o February 01, which indicaes ha invesmen in boh markes can beer diversify marke risk during crises. The endency oward lower ail correlaion reflecs he underdevelopmen of invesmen insrumens in China and he relaively high risk aversion of Chinese invesors. 10
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