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Research Article Multivariate Time-Varying - Copula GARCH Model and Its Application in the Financial Market Risk Measurement Qi-an Chen, 1 Dan Wang, 1 and Mingyong Pan 2 1 School of Economics and Business Administration, Chongqing University, Chongqing 400030, China 2 School of Mathematics and Statistics, Chongqing University, Chongqing 400030, China Correspondence should be addressed to Qi-an Chen; chenqi [email protected] and Mingyong Pan; [email protected] Received 23 March 2015; Accepted 11 May 2015 Academic Editor: Ruihua Liu Copyright © 2015 Qi-an Chen et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Taking full advantage of the strengths of - distribution, Copula function, and GARCH model in depicting the return distribution of financial asset, we construct the multivariate time-varying - Copula GARCH model which can comprehensively describe “asymmetric, leptokurtic, and heavy-tail” characteristics, the time-varying volatility characteristics, and the extreme-tail dependence characteristics of financial asset return. Based on the conditional maximum likelihood estimator and IFM method, we propose the estimation algorithm of model parameters. Using the quantile function and simulation method, we propose the calculation algorithm of VaR on the basis of this model. To apply this model on studying a real financial market risk, we select the SSCI (China), HSI (Hong Kong, China), TAIEX (Taiwan, China), and SP500 (USA) from January 3, 2000, to June 18, 2010, as the samples to estimate the model parameters and to measure the VaRs of various index risk portfolios under different confidence levels empirically. e results of the application example are in line with the actual situation and the risk diversification theory of portfolio. To a certain extent, these results also justify the feasibility and effectiveness of the multivariate time-varying - Copula GARCH model in depicting the return distribution of financial assets. 1. Introduction Financial market risk has always been one of the hottest topics in the field of financial investment, and many financial researchers put forward many different financial market risk measurement methods. Among them, Value-at-Risk (VaR) management technology is an assessment and measurement method of financial risk that has risen in recent years, playing an increasingly important role in the risk management and investment decision. It has been widely adopted by the major banks, nonbank financial intermediaries, corporations, and financial regulators in the world and has become the standard of risk measurement and risk management in financial industry. Accurate calculation of VaR is one of the keys to estimate the probability distribution of future return on financial assets. Usually, it is assumed that financial asset returns are independent of each other and obey the normal distribution in the calculation of VaR, but the movement of financial asset return in the financial market is extremely complex. e return of all kinds of financial assets usually does not satisfy the normal distribution hypothesis. However, it oſten shows “asymmetric, leptokurtic, and heavy-tail” characteristics [14]. At the same time, various financial asset returns do not satisfy the multivariate normal distribution hypothesis and present the extreme-tail dependence. On this occasion, a large error would be made by using normal distribution to fit financial asset return, and the estimation of the VaR may be overestimated or underestimated. To solve this problem, many scholars have proposed a lot of leptokurtic and heavy-tail distributions in recent years, such as the logistic distribution, Student’s -distribution, and the - distribution. e logistic distribution and Student’s - distribution can comprehensively describe the leptokurtic characteristics of financial asset return series, but they could not make a good explanation for the heavy-tail characteristics of financial asset return series [5]. e - distribution can comprehensively describe the asymmetric, leptokurtic, and heavy-tail characteristics of financial asset return series and it has a good fitting of the univariate unconditional return distribution of some financial assets; however, it could not Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 286014, 9 pages http://dx.doi.org/10.1155/2015/286014
Transcript
Page 1: Research Article Multivariate Time-Varying Copula GARCH ...downloads.hindawi.com/journals/mpe/2015/286014.pdf · model, they presented an empirical pricing study of China s market.

Research ArticleMultivariate Time-Varying 119866-119867 Copula GARCH Model andIts Application in the Financial Market Risk Measurement

Qi-an Chen1 Dan Wang1 and Mingyong Pan2

1School of Economics and Business Administration Chongqing University Chongqing 400030 China2School of Mathematics and Statistics Chongqing University Chongqing 400030 China

Correspondence should be addressed to Qi-an Chen chenqi an33163com and Mingyong Pan panmingyongaliyuncom

Received 23 March 2015 Accepted 11 May 2015

Academic Editor Ruihua Liu

Copyright copy 2015 Qi-an Chen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Taking full advantage of the strengths of 119866-119867 distribution Copula function and GARCH model in depicting the returndistribution of financial asset we construct the multivariate time-varying119866-119867Copula GARCHmodel which can comprehensivelydescribe ldquoasymmetric leptokurtic and heavy-tailrdquo characteristics the time-varying volatility characteristics and the extreme-taildependence characteristics of financial asset return Based on the conditional maximum likelihood estimator and IFM methodwe propose the estimation algorithm of model parameters Using the quantile function and simulation method we propose thecalculation algorithm of VaR on the basis of this model To apply this model on studying a real financial market risk we selectthe SSCI (China) HSI (Hong Kong China) TAIEX (Taiwan China) and SP500 (USA) from January 3 2000 to June 18 2010 asthe samples to estimate the model parameters and to measure the VaRs of various index risk portfolios under different confidencelevels empirically The results of the application example are in line with the actual situation and the risk diversification theory ofportfolio To a certain extent these results also justify the feasibility and effectiveness of the multivariate time-varying119866-119867 CopulaGARCHmodel in depicting the return distribution of financial assets

1 Introduction

Financial market risk has always been one of the hottesttopics in the field of financial investment and many financialresearchers put forward many different financial market riskmeasurement methods Among them Value-at-Risk (VaR)management technology is an assessment and measurementmethod of financial risk that has risen in recent years playingan increasingly important role in the risk management andinvestment decision It has been widely adopted by the majorbanks nonbank financial intermediaries corporations andfinancial regulators in the world and has become the standardof risk measurement and risk management in financialindustry Accurate calculation of VaR is one of the keysto estimate the probability distribution of future return onfinancial assets Usually it is assumed that financial assetreturns are independent of each other and obey the normaldistribution in the calculation of VaR but the movement offinancial asset return in the financial market is extremelycomplex The return of all kinds of financial assets usually

does not satisfy the normal distribution hypothesis Howeverit often shows ldquoasymmetric leptokurtic and heavy-tailrdquocharacteristics [1ndash4] At the same time various financial assetreturns do not satisfy the multivariate normal distributionhypothesis and present the extreme-tail dependence On thisoccasion a large error would be made by using normaldistribution to fit financial asset return and the estimationof the VaR may be overestimated or underestimated Tosolve this problem many scholars have proposed a lot ofleptokurtic and heavy-tail distributions in recent years suchas the logistic distribution Studentrsquos 119905-distribution and the119866-119867 distribution The logistic distribution and Studentrsquos 119905-distribution can comprehensively describe the leptokurticcharacteristics of financial asset return series but they couldnotmake a good explanation for the heavy-tail characteristicsof financial asset return series [5] The 119866-119867 distribution cancomprehensively describe the asymmetric leptokurtic andheavy-tail characteristics of financial asset return series andit has a good fitting of the univariate unconditional returndistribution of some financial assets however it could not

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 286014 9 pageshttpdxdoiorg1011552015286014

2 Mathematical Problems in Engineering

reflect the time-varying volatility characteristics of financialasset return and the extreme-tail dependence characteristicsof various financial assets return [6 7] Meanwhile Copulafunction can connect the joint distribution and the marginaldistribution of multiple random variables to construct flex-ible multivariate distribution functions which can be usedto measure the extreme-tail dependence of multiple financialassets return GARCH model can comprehensively describethe time-varying volatility characteristics of financial assetreturn Therefore building the multivariate time-varying 119866-119867 Copula GARCH Model by combining the 119866-119867 distribu-tion with Copula function and GARCH model can not onlycomprehensively describe the ldquoasymmetric leptokurtic andheavy-tailrdquo characteristics the time-varying volatility andextreme-tail dependence characteristics of the financial assetreturn and make measurement of VaR more accurately butalso enrich and expand the risk measurement theory andmethod of financial market theoretically and improve therisk control ability of investors corporations financial insti-tutions and policy authorities and reduce their unnecessarylosses in practice

The 119866-119867 distribution Copula function and GARCHmodel as well as the financial risk measurement modelbased on them have been researched in the existing relevantliteratures A lot of innovative research results with referencevalue have been brought out Zhu and Pan [8] proposedthree kinds of 119866-119867 VaR methods based on the portfoliogains losses and extreme losses according to the statisticalcharacteristics of 119866-119867 distribution Their empirical resultsshowed that this method is superior to the commonly useddelta-normal method Kuester et al [9] believed that the 119866-119867 distribution can describe skewness and kurtosis of thefinancial asset return simultaneously and it plays a veryimportant role of VaR measurement of financial asset returnDegen et al [10] discussed the application of119866-119867distributionin operational riskmeasurement Jondeau andRockinger [11]Rodriguez [12] Fischer et al [13] and Sun et al [14] combinedtime series model with various Copulas functions by usingthe Sklar theorem to build a lot of highly flexible multivariatetime-varying models for risk measurement of portfolios Liuet al [15] proposed a GARCH-119872model with a time-varyingcoefficient of the risk premiumTheir study indicated that thecoefficient of the risk premium varies with the time and evenin a mature market the conditional skewness in the returndistribution is negatively correlated with the time-varyingcoefficient of the risk premium Wen et al [16] built a 119863-GARCH-119872 model by separating investorsrsquo return into gainsand losses on the basis of the characteristics of investorsrsquorisk preferenceThey found that investors become risk aversewhen they gain and risk-seeking when they lose whicheffectively explains the inconsistent risk-return relationshipAnd the degrees of investorsrsquo risk aversion and risk-seekingare both in direct proportion to the value of gains and lossesrespectively Wen et al [17] adopted aggregative indices of14 representative stocks around the world as samples andestablished a TVRA-GARCH-119872 model to investigate theinfluence of prior gains and losses on current risk attitudeThe empirical results indicated that the prior gains increasepeoplersquos current willingness to take risk asset at the whole

market level Huang et al [18] combined Studentrsquos 119905-marginaldistribution with Archimedean Copula functions to build theconditional Copula GARCH model They used this modelto estimate the VaR of portfolios Ghorbel and Trabelsi [19]built the conditional extremum Copula GARCH model byusing extreme value theory (EVT) and measured the risk offinancial asset according to this model Chollete et al [20]used multivariate regime-switching Copula function to buildinternational financial asset return model and accordinglyput forward the VaR calculation method Huggenberger andKlett [21] proposed a measurement model of multivari-ate risk asset return VaR based on 119866-119867 distribution andCopula function They used DAX30 (Germany) FTSE100(UK) and CAC40 (French) from January 2000 to May2010 as samples to test empirically Wang et al [22] appliedthe Gumbel Copula function in multivariate ArchimedeanCopula functions family to construct the joint distributionfunction which can describe the actual distribution and thecorrelation of various financial asset returns They also usedthe Monte Carlo simulation technology to analyze the port-folios VaR and its composition under different confidencelevels The result showed that using the multidimensionalGumbel Copula function to construct the risk measurementmodel of financial asset can make the assets chosen byinvestors more robust and it can also help investors todiversify and control the overall risk of the portfolios Daiand Wen [23] proposed a computationally tractable robustoptimization method for minimizing the CVaR of a portfoliounder a general affine data perturbation uncertainty setAnd they presented some numerical experiments with realmarket data to illustrate the behavior of robust optimiza-tion model Liu et al [24] proposed a pricing model forconvertible bonds based on the utility-indifference methodand got access to the empirical results by use of InformationTechnology Furthermore using the proposed theoreticalmodel they presented an empirical pricing study of Chinarsquosmarket They found that the theoretical prices are higherthan the actual market prices 024ndash458 and the utility-indifference prices are better than the Black-Scholes (B-S)prices

Based on the aforementioned analyses the VaR is stillthe mainstream measurement method of financial mar-ket risk In order to achieve the purpose of measuringVaR more precisely it has been the hot issue of existingresearch literatures to construct the distribution functionsas comprehensive as possible to describe the ldquoasymmetricleptokurtic and heavy-tailrdquo characteristics the time-varyingvolatility characteristics and the extreme-tail dependencecharacteristics of financial asset return through a variety ofmathematicalmethods However in the process of construct-ing the return distribution model and measuring VaR offinancial asset existing results only grasp some characteristicsof financial asset return distribution They are not able tocomprehensively describe the ldquoasymmetric leptokurtic andheavy-tailrdquo characteristics the time-varying volatility charac-teristics and the extreme-tail dependence characteristics ofthe financial asset returnThe rationality and accuracy of VaRcalculated based on the existing distribution models have alarge space for further improvement

Mathematical Problems in Engineering 3

In this paper we would take full advantage of thestrengths of119866-119867 distribution Copula function andGARCHmodel in depicting the return distribution of financial asset tobuild multivariate time-varying119866-119867Copula GARCHmodelwhich can simultaneously describe ldquoasymmetric leptokurticand heavy-tailrdquo characteristics the time-varying volatilitycharacteristics and the extreme-tail dependence character-istics of financial asset return and propose the estimationmethod of model parameters and the calculation algorithmof VaR Then this paper selects the SSCI (China) HSI(Hong Kong China) TAIEX (Taiwan China) and SP500(USA) from January 3 2000 to June 18 2010 as samples toestimate the parameters and calculate the VaRs of variousindex portfolios under different confidence levels

2 119866-119867 Distribution Copula Functionand Its Tail Dependence Index

21 119866-119867 Distribution

211 119866 Distribution Assuming that random variable 119885

obeys the standard normal distribution and 119892 is a realnumber then the random variable 119884119892 = 119866119892(119911) obeys 119866

distribution Consider

119866119892 (119911) =119890119892119911

minus 1119892

119911 sim 119873 (0 1) (1)

where 119892 controls the skewness of 119866 distribution When 119892 rarr

0 119866119892(119911) rarr 119911 and 119866 distribution tends to be symmetricWith the increase of the absolute value of 119892 the degree ofasymmetry increases Changing the sign of 119892 can changethe asymmetric direction of 119866 distribution but it does notchange its degree of asymmetry

212 119867 Distribution Assuming that random variable 119885

obeys the standard normal distribution and ℎ is a realnumber then the random variable 119884ℎ = 119867ℎ(119911) obeys 119867

distribution Consider

119867ℎ (119911) = 119890ℎ119911

22 119911 sim 119873 (0 1) (2)

119867 distribution stretches the tail of the standard normaldistribution ℎ controls the tail heaviness of 119867 distributionThe larger the ℎ is the heavier the tail is Because 119867ℎ(119911)

is an even function 119867 distribution is symmetric But theheaviness of its tail changes compared to the standard normaldistribution

213 119866-119867 Distribution The random variable 119884119892ℎ can beobtained by introducing both functions 119866119892(119911) and 119867ℎ(119911) torevise standard normal random variable 119885 Consider

119884119892ℎ = 119866119892 (119911)119867ℎ (119911) =119890119892119911

minus 1119892

119890ℎ119911

22 (3)

Then 119883119892ℎ can be obtained through linear transformation of119884119892ℎ Consider

119883119892ℎ = 119860+119861119890119892119911

minus 1119892

119890ℎ119911

22 119911 sim 119873 (0 1) (4)

The distribution of the random variable119883119892ℎ obeys the119866-119867 distribution 119860 119861 119892 and ℎ are real numbers 119892 describesthe asymmetry of 119866-119867 distribution and ℎ describes theheavy-tail characteristics of 119866-119867 distribution Obviously (3)is a special form of (4) The random variable 119884119892ℎ in (3)is the random variable of 119866-119867 distribution after centralstandardization

22 Copula Function and Its Tail Dependence Index Assum-ing that marginal distribution of random vector 119906119894 = 119865119894(119909119894)

(119894 = 1 2 119901) obeys uniformdistribution119880(0 1) accordingto the Sklar theorem the joint distribution function of 119875-dimensional random vectors 119865(1199091 119909119901) can be repre-sented as the following formula

119865 (1199091 119909119901) = 119862 (1198651 (1199091) 119865119901 (119909119901)) (5)

where 119862 is the Copula function of 119865 which is a hyper-cube [0 1]119901 multivariate density function defined on 119875-dimensional space R119901 If the marginal distribution is contin-uous there is a unique Copula function 119862 Then

119862 (1199061 119906119901) = 119865 (119865minus11 (1199061) 119865

minus1119901

(119906119901)) (6)

On the contrary given 119875-dimensional Copula func-tion 119862(1199061 119906119901) and its marginal distribution function1198651(1199091) 119865119901(119909119901) the density function of 119875-dimensionaljoint distribution function is

119891 (1199091 119909119901) = 119888 (1198651 (1199061) 119865119901 (119906119901))

119901

prod

119894=1119891119894 (119909119894) (7)

If 119891119894(119909119894) is the edge density 119888(1199061 119906119901) denotes Copuladensity derived from (6) Thus

119888 (1199061 119906119901) =

119891 (119865minus11 (1199061) 119865

minus1119901

(119906119901))

prod119901

119894=1119891119894 (119865minus1119894

(119906119894)) (8)

Since the joint distribution function of random vari-ables defines the correlation among its components Copulafunction determines the dependent structure among randomvariables uniquely The upper tail index 120582119906 and lower tailindex 120582119897 of tail dependence indicators can be defined asfollows

120582119906 = lim119902rarr 1

1 minus 2119902 + 119888 (119902 119902)

1 minus 119902

120582119897 = lim119902rarr 0

119888 (119902 119902)

119902

(9)

According to Nelsen [25] Gauss Copula function gen-erated by multivariate normal distribution function whosecorrelation matrix is R can be represented as follows

119862119866119906

R (1199061 119906119901)

= int

Φminus11 (1199061)

minusinfin

sdot sdot sdot int

Φminus1119901

(119906119901

)

minusinfin

1

radic(2120587)119901 |R|

expminusu1015840Rminus1u

2119889u

(10)

4 Mathematical Problems in Engineering

where u = (1199061 119906119901) and Φminus1 is the inverse function

of single normal distribution Because the Gauss Copulafunction does not have the characteristics of tail dependencewe often use the119879-Copula function whose degree of freedomis 120578 and correlation matrix is R to measure tail dependencestructure of risk asset in empirical analysis that is

119862119905

120578R (1199061 119906119901)

= int

119905minus1120578

(1199061)

minusinfin

sdot sdot sdot int

119905minus1120578

(119906119901

)

minusinfin

Γ ((120578 + 119901) 2) (1 + u1015840Rminus1u2)minus(120578+119901)2

Γ (1205782)radic(120587120578)119901|R|

119889u(11)

where 119905minus1120578

is the inverse function of simple standard Studentrsquos119905-distribution whose degree of freedom is 120578 When 120578 rarr infin119879-Copula function degenerates to Gauss Copula function Itstail index 120582119906 = 120582119897 = 0 that is the tail is independent The tailindex of 119879-Copula function is

120582119906 = 120582119897 = 2119905120578+1 (minus

radic(120578 + 1) (1 minus 120588)

radic1 + 120588) (12)

where 119905120578+1 is simple standard Studentrsquos 119905-distribution whosedegree of freedom is 120578 + 1 Considering that the innovationimpacts on the price of risk asset in varying degrees at differ-ent times 120578 and 120588 should have time-varying characteristicsFor this reason tail index also has the same characteristics

3 Multivariate Time-Varying 119866-119867 CopulaGARCH Model

Let r119905 = (1199031119905 119903119901119905) denote return time series of 119901 riskassets The prior information set before time 119905 is

I119905minus1 = r119905minus1 h119905minus1 r119905minus2 h119905minus2 =

119901

prod

119894=1I119894119905minus1 (13)

where I119894119905minus1 = 119903119894119905minus1 ℎ119894119905minus1 119903119894119905minus2 ℎ119894119905minus2 ℎ119894119905 is conditionalvolatility of 119903119894119905 about single asset prior information set I119894119905minus1Let 119862(sdot | I119905minus1) denote 119875-dimensional conditional Copulafunction and 119865119894(119903119894119905 | I119894119905minus1) be the conditional distributionof the 119894th component According to Sklar theorem theconditional joint distribution of 119901 risk assets return is

119865 (r119905 | I119905minus1)

= 119862 (1198651 (1199031119905 | I1119905minus1) 119865119901 (119903119901119905 | I119901119905minus1) | I119905minus1)

(14)

Numerous empirical studies show that the risk assetreturn series obey GARCH (1 1) model Based on thisassuming that 119903119894119905 satisfies the GARCH (1 1) model we canget the following 119866-119867 Copula GARCH (1 1) model whichdescribes the time-varying dependence structure of 119901 risk

assets return after filtering the time-varying characteristics ofsingle series

119903119894119905 = 120583119894 + 120576119894119905 119894 = 1 2 119901

120576119894119905 = radicℎ119894119905119911119894119905

ℎ119894119905 = 120603119894 +1205721198941205762119894119905minus1 +120573119894ℎ119894119905minus1

119865 (z119905 | I119905minus1)

= 119862 (1198651 (1199111119905 | I1119905minus1) 119865119901 (119911119901119905 | I119901119905minus1) | I119905minus1)

(15)

where the parameters satisfy the conditions 120603119894 120572119894 120573119894 gt 0and 120572119894 + 120573119894 lt 1 These parameters can ensure the stabilityof conditional volatility series The innovation series z119905obey 119866-119867 distribution whose parameter is (119892 ℎ) in (4) Butin order to simplify the analysis we only consider 119866-119867distribution after central standardization given by (3) and itsdensity function is written as 119891119884

119894

(119910119894) The Copula function119862(sdot | I119905minus1) is given by (11) and its density 119888(sdot) can berepresented as the following time-varying119879-Copula functionwhose degree of freedom is 120578

119888119905

120578120588119905

(1199061119905 119906119901119905)

=

119891119905

120578120588119905

(119891minus1V1 (1199061119905) 119891

minus1V119901

(119906119901119905))

prod119901

119894=1119891120578 (119891minus1V119894

(119906119894119905))

(16)

where 119891119905

120578120588119905

denotes the multivariate Studentrsquos 119905-distributionwhose degree of freedom is 120578 and time-varying correlationmatrix is 120588

119905= (120588119894119895119905)119901times119901 and

120588119894119894119905 = 1

119891V119894

(119906119894119905) =Γ ((V119894 + 1) 2)Γ (V1198942)radicV119894120587

(1+1199062119894119905

V119894)

minus(V119894

+1)2

(17)

The joint density function of 119901 risk assets return is

119891 (y | u h119905) = 119888119905

120578120588119905

(1199091119905 119909119901119905)

119901

prod

119894=1119891119884119894

(119910119894)

= Γ (120578 + 119901

2) Γ (

120578

2)

119901minus1(1+

x1015840119905120588119905x119905

120578)

minus(120578+119901)2

sdot (1003816100381610038161003816120588119905

1003816100381610038161003816)minus12119901

prod

119894=1(1 +

1199092119894119905

120578)

(120578+1)2 119901

prod

119894=1119891119884119894

(119910119894)

(18)

Mathematical Problems in Engineering 5

where x119905 = (1199091119905 119909119901119905) and 119909119894119905 = 119905minus1120578

(119905V119894

(120576119894119905)) Then thelikelihood function of overall samples is

119897 (120579 | y) =

119879

prod

119905=1Γ (

120578 + 119901

2) Γ (

120578

2)

119901minus1

sdot (1+x1015840119905120588119905x119905

120578)

minus(120578+119901)2

(1003816100381610038161003816120588119905

1003816100381610038161003816)minus12

Γ (120578 + 12

)

minus119901

sdot

119901

prod

119894=1(1+

1199092119894119905

120578)

(120578+1)2 119901

prod

119894=1119891119884119894

(119910119894)

(19)

where 120579 = (120583119894 120603119894 120572119894 120573119894 V119894)119901

119894=1 119886 119887120588119905 120578 and x119905 = (1199091119905 119909119901119905)The value of correlationmatrix 120588

119905is similar to the time-

varying correlation matrix of multivariate Copula GARCHmodel proposed by Jondeau and Rockinger [11] That is 120588

119905

satisfies the following evolution equation

120588119905= (1minus 119886minus 119887)120588+ 119886Ψ119905minus1 + 119887120588

119905minus1 (20)

where 0 le 119886 119887 le 1 119886 + 119887 le 1 120588 is a positive definite matrixwhose main diagonal elements are 1 and other elements arestatic correlation coefficientsΨ119905minus1 is a 119901times119901matrix in whichevery element

120595119894119895119905minus1 =sum119898

119897=1 119909119894119905minus119897119909119895119905minus119897

radicsum119898

119897=1 1199092119894119905minus119897

radicsum119898

119897=1 1199092119895119905minus119897

119894 119895 = 1 2 119901 (21)

denotes the correlation coefficients of 119901 risk asset returns(119898 ge 119901 + 2) 119909119905 = (1199091119905 119909119901119905) = (119905

minus1V1 (119891V1(1199111119905))

119905minus1V119901

(119891V119901

(119911119901119905))) Each element 120588119894119895119905 of 120588119905 satisfies minus1 le 120588119894119895119905 le 1

4 Parameter Estimation Algorithm ofthe Multivariate Time-Varying 119866-119867 CopulaGARCH Model

On the basis of Huggenberger and Klett [21] this section willuse dynamic correlation matrix 120588

119905instead of static correla-

tion matrix in the multidimensional discrete-time stochasticprocess to estimate the parameters of multivariate time-varying119866-119867CopulaGARCHmodel established in Section 3Assuming thatΘ denotes the parameter space defined by themodel and (r1 r2 r119879) denotes the log return samples of119875-dimensional risk asset which is generated by multivariateconditional density function119891120588

119905

|I119905minus1

(r119905 | I119905minus1 1205790) where 1205790 isin

Θ I119905minus1 is 120590 algebra of time 119905 minus 1 and before the maximumlikelihood estimation of parameter vector 120579 can be calculatedby the following equation

= argmax120579isinΘ

119879

sum

119905=1log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 120579) (22)

where 119891120588119905

|I119905minus1

(r119905 | I119905minus1 120579) can be obtained by calculating thederivative of (5) Let 119888120579 denote Copula density functionThus

119891120588119905

|I119905minus1

(1199031119905 119903119901119905 | I119905minus1 120579)

= 119888120579 (1198651119905 (1199031119905 120579) 119865119901119905 (119903119901119905 120579))

sdot

119901

prod

119894=1119891119894119905 (119903119905119894 120579)

(23)

The probability density function and distribution func-tion can be obtained in the process ofmodel built in Section 3Using the IFM method proposed by Joe [26] we can convert(22) into an optimization problem Therefore we need todivide the parameter vector 120579 into two subparameter vectors120579119888 and 120579119903 that is 120579 = (120579119888 120579119903) where 120579119903 = (1205791199031

120579119903119901

)120579119903119894

is the parameter vector of 119894th marginal distribution and120579119888 is the parameter vector of Copula function Because IFMmethod is a two-step likelihood estimation method themodel parameters should be estimated through the followingtwo steps

Step 1 Solving the maximum likelihood estimator of theparameter vector of each risk asset return

119903119894

= argmax120579119903

119894

119879

sum

119905=1log119891119894119905 (119903119894119905 | 120579119903

119894

) 119894 = 1 2 119901 (24)

Thismeans that we need to estimate parameters vector 119903119894

of 119901 distributions continuously

Step 2 Taking each 119903119894

into the likelihood equation (22) wecan obtain the parameter vector 120579119888 of Copula function and itsmaximum likelihood estimator 119888 Consider

119888

= argmax120579119888

119879

sum

119905=1log 119888120579

119888

(1198651119905 (10038161003816100381610038161199031119905

1003816100381610038161003816 1199031) 119865119901119905 (

10038161003816100381610038161003816119903119901119905

10038161003816100381610038161003816119903119901

))

(25)

In themaximum likelihood estimationweneed to use thederivative function of the density function of 119866-119867 marginaldistribution with respect to the component of parametervector Because the density function of 119866-119867 marginal distri-bution is very complex this paper uses the implicit functiondifferentiation rule to take its derivative The estimator 2119904of parameter vector 120579 obtained by the above-mentionedtwo-step method obeys normal distribution consistentlyand asymptotically under the standard regularity conditionsproposed inHuggenberger andKlett [21] Joe [26] andPatton[27] that is

radic119879(2119904119879 minus 1205790)119889

997888rarr119879rarrinfin

119873(0Ωminus1ΣΩ) (26)

6 Mathematical Problems in Engineering

where

Ω = minus119864(1205972

1205971205791205971205791015840log119891120588

119905

|I119905minus1

(120588119905| I119905minus1 1205790))

Σ = 119864(120597

120597120579(log119891120588

119905

|I119905minus1

(120588119905| I119905minus1 1205790))

sdot120597

120597120579(log119891120588

119905

|I119905minus1

(120588119905| I119905minus1 1205790))

1015840

)

(27)

Because the matrixes Σ and Ω can be estimated by theestimated parameter vector consistently

Ω119879 = minus119879minus1119879

sum

119905=1

1205972

1205971205791205971205791015840log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 2119904119879)

Σ119879 = 119879minus1119879

sum

119905=1

120597

120597120579(log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 2119904119879))

sdot120597

120597120579(log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 2119904119879))1015840

(28)

Thus (26) can be used to calculate the standard deviationof the estimator 2119904119879

5 VaR Algorithm Based on the MultivariateTime-Varying 119866-119867 Copula GARCH Model

After estimating the parameters of multivariate time-varying119866-119867 Copula GARCH model VaR of the risk portfolio canbe measured VaR of risk portfolio indicates the expectedmaximum losses of risk portfolio held by investors within agiven confidence level and a certain period of time Assumingthat r119905 = (1199031119905 119903119901119905) (119905 = 1 2 119879) are the return samplesof 119901 risk assets which satisfy themultivariate time-varying119866-119867 Copula GARCH (1 1)model in Section 3 andsum

119901

119894=1 120582119894119903119894119905 isthe portfolio of 119901 risk assets in which the weight of the riskasset 119894 is 120582119894 (119894 = 1 2 119901) that can be less than 0 because ofpermitting short-purchasing and short-selling the risk assetsand meet sum

119901

119894=1 120582119894 = 1 the VaR of risk portfolio underconfidence level 119902 at time 119905 should satisfy Pr(sum119901

119894=1 120582119894119903119894119905 le

VaR119905) = 119902 The confidence level 119902 can reflect the differentrisk preferences of investors or financial institutions to acertain extent Choosing a larger confidence level means thatinvestors or financial institutions have greater risk aversionand they hope to get a forecast result with larger probability

Although the conditional distributions of 1199031119879+1 1199032119879+1 119903119901119879+1 can be calculated through the known marginaldistributions it is very difficult to calculate quantile fromtime-varying Copula density function and it is adverse tomeasure and calculate the VaR of risk portfolio Thereforethis paper measures the dynamic risk of portfolio and its esti-mation value approximately through simulating119866-119867CopulaGARCH model Based on the parameters 120579(119899) of the samplethe return series of risk assets [119903

(119899119898)

11+119879 119903(119899119898)

1199011+119879] 119898 =

1 119872 and the one-step measurement and estimationvalues of VaR of their portfolios can be obtained through esti-mating (ℎ(119899)1119879+1 ℎ

(119899+1)119901119879+1) according to the volatility equation

Table 1 Moment estimation results of the daily log return of SSCIHSI TAIEX and SP500

Types of stock index Mean Std Skewness KurtosisSSCI 25078119890 minus 004 00181 minus02144 74467HSI 74505119890 minus 005 00177 minus02765 116866TAIEX 39511119890 minus 005 00140 minus09091 172823SP500 minus97173119890 minus 005 00148 minus03701 120177

of119866-119867 Copula GARCHmodel and calculating 120588(119899)

119879+1 by usingCopula dynamic evolution equation and then repeating thefollowing algorithm for 119872 times (119872 ge 3119901)

Step 1 It is simulating 119872 groups of random vectors[119906(119899119898)

1119879+1 119906(119899119898)

119901119879+1] according to the multivariate 119879-Copuladensity function whose degree of freedom is 120578(119899) and correla-tion matrix is 120588(119899)

119879+1

Step 2 Calculating 119903(119899119898)

119894119879+1 = 120583(119899)

119894+ 119911(119899119898)

119894119879+1radicℎ(119899)

119894119879+1 119894 = 1 119901

Step 3 Firstly one calculates the return rate of risk portfoliothat is equal to sum

119901

119894=1 120582119894119903(119899119898)

119894119879+1 119898 = 1 2 119872 Secondly oneevaluates its 119902-quantile VaR(119899)

119879+1 Thirdly one measures theVaR of the risk portfolio by VaR119879+1 = (1119873)sum

119873

119899=1 VaR(119899)

119879+1

6 Application of the Multivariate Time-Varying 119866-119867 Copula GARCH Model

61 Date Sample and Moment Estimation USA and Chinaas the most developed capitalism country and the largestdeveloping country in the world respectively rank top two ofthe world economy Their stock markets should have strongrepresentation in the world At the same time due to thehistorical reasons there exist several regions with differentpolitical systems such as Mainland China Hong KongTaiwan and Macau in Greater China Macau is similar toHong Kong on the whole For the above-mentioned reasonsthis paper selects the SSCI (China) HSI (HongKong China)TAIEX (Taiwan China) and SP500 (USA) from January3 2000 to June 18 2010 as data samples to estimate theVaR of various index portfolios under different confidencelevels by using the multivariate time-varying 119866-119867 CopulaGARCHmodelThe data comes fromYahoo Financewebsitehttpfinanceyahoocom

The moment estimation results of the daily log return ofSSCI HSI TAIEX and SP500 are shown in Table 1

Table 1 shows that the skewness of daily log returns ofSSCI HSI TAIEX and SP500 is less than 0 and their kurtosisis much larger than that of standard normal distributionwhich is equal to 3 These results demonstrate that thedaily log returns of these indices have the right skew andleptokurtic characteristics Therefore it is appropriate to fitthe daily log return of SSCI HSI TAIEX and SP500 byapplying 119866-119867 distribution which has leptokurtic heavy-tailcharacteristics and it is reasonable to apply the multivariate

Mathematical Problems in Engineering 7

Table 2 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on SSCI index risk asset

Model parameters 1205831

1205961

1205721

1205731

Estimate 000025lowastlowastlowast 00551lowastlowastlowast 00617lowastlowastlowast 08583lowastlowastlowast

119879-statistic 48653 46851 65764 44009Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 3 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on HSI index risk asset

Model parameters 1205832

1205962

1205722

1205732

Estimate 0000075lowastlowastlowast

00342lowastlowastlowast

00586lowastlowastlowast

08987lowastlowastlowast

119879-statistic 46539 48518 61796 66079Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 4 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on TAIEX index risk asset

Model parameters 1205833

1205963

1205723

1205733

Estimate 000040lowastlowastlowast 00343lowastlowastlowast

00517lowastlowastlowast

09054lowastlowastlowast

119879-statistic minus64538 43523 59817 65935Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 5 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on SP500 index risk asset

Model parameters 1205834

1205964

1205724

1205734

Estimate minus00001lowastlowastlowast

00251lowastlowastlowast

00861lowastlowastlowast

08738lowastlowastlowast

119879-statistic minus33873 52684 76324 63786Note lowastlowastlowastin the table denotes that the parameter is significant at 1 level

time-varying 119866-119867 Copula GARCH model to measure theirVaR

62 Parameter Estimates of the Multivariate Time-Varying119866-119867 Copula GARCH Model Based on the parameter esti-mation algorithm proposed in Section 4 the parameters ofthe multivariate time-varying 119866-119867 Copula GARCH modelwith SSCI HSI TAIEX and SP500 can be estimated Theparameter estimation results are shown in Tables 2 3 4 5and 6

From Tables 2 to 6 the following results can be obtained

(1) Consider 1205721 + 1205731 = 092 1205722 + 1205732 = 09573 1205723 +

1205733 = 09571 and 1205724 + 1205734 = 09599 This shows thatthe volatility persistence of Shanghai stock market isthe strongest Taiwan and Hong Kong stock marketrank second and third and the volatility persistenceof USA stock market is minimum It indicates thatthe investorsrsquo expectation of risk compensation inthe emerging markets represented by Chinarsquos stockmarket is stronger than that in the mature marketsrepresented by the USArsquos stock market and the pricediscovery efficiency of innovation in the emergingmarkets represented by Chinarsquos stock market is lowerthan that in the mature markets represented by USArsquosstockmarket In addition the sumof the coefficients120572

and120573 is very close to 1 which indicates that the impactand shock of innovation on the index volatility of eachstock market has a long memory

(2) The degree of freedom 120578 = 1457 and the correlationcoefficients 120588119894119895 of 119879-Copula in Table 6 show thatthere exists the strongest correlation between HongKong stock market and Taiwan stock market andthe correlation between Shanghai stock market andHong Kong stock market is also relatively largeThe above-mentioned facts indicate that the extremeevents probably result in the phenomena that HongKong stock market and Taiwan stock market are upand down synchronously and there exist comovingbehaviors between Shanghai stock market and HongKong stock market

(3) The time-varying coefficient 119887 = 0987 indicatesthat the time-varying correlation coefficient of 119866-119867 Copula GARCH model has a long memory thatis the impact of historical values of each otherrsquoscorrelation coefficient among SSCI HSI TAIEX andSP500 on the expected correlation is relatively large

63 VaR Measurement Based on the Multivariate Time-Varying 119866-119867 Copula GARCH Model Based on the multi-variate time-varying119866-119867 Copula GARCHmodel with SSCIHSI TAIEX and SP500 whose parameters have been esti-mated the VaRs of various index portfolios under differentconfidence levels can be measured The measurement resultsare shown in Table 7

From Table 7 the following results can be obtained

(1) The inequalities VaR (SSCI) lt VaR (HSI) lt VaR(SP500) lt VaR (TAIEX) can be satisfied for anyconfidence level It shows that the risk of extremelosses in Shanghai stock market is higher than that inHong Kong stock market Taiwan stock market andUSA stock market This measurement result is in linewith the actual situation that thematurity of Shanghaistockmarket is far lower than that ofHongKong stockmarket Taiwan stock market and USA stock market

(2) For any confidence level the extreme losses risk ofthe investors who equally allocate their total assetsamong SSCI HSI TAIEX and SP500 is lower thanthat of the investors who put their total assets into oneindex asset The extreme losses risk of the investorsincreases with the concentration of risk asset in theindex portfolios This measurement result is consis-tent with the risk diversification theory of portfolio

7 Conclusion

Considering the ldquoasymmetric leptokurtic and heavy-tailrdquocharacteristics the time-varying volatility characteristicsand extreme-tail dependence characteristics of financialasset return this paper combined the 119866-119867 distributionCopula function and GARCH model to construct a mul-tivariate time-varying 119866-119867 Copula GARCH model whichcan comprehensively describe the ldquoasymmetric leptokurtic

8 Mathematical Problems in Engineering

Table 6 Estimates of correlation coefficients time-varying parameters and degree of freedom of four-variate time-varying G-H CopulaGARCH (1 1) model based on SSCI HSI TAIEX and SP500

Model parameters 12058812

12058813

12058814

12058823

12058824

12058834

a b 120578

Estimate 0256lowastlowastlowast

0139lowastlowastlowast

0021lowastlowastlowast

0528lowastlowastlowast

0193lowastlowastlowast

0128lowastlowast

0008lowastlowastlowast

0987lowastlowastlowast

1457lowastlowastlowast

119879-statistic 1351 1102 3431 5763 9564 2529 5179 3561 6871Note lowastlowastlowast and lowastlowast in the table denote that the parameter is significant at 1 and 5 level respectively

Table 7 VaR estimation results based on the four-variate time-varying G-H Copula GARCHmodel with SSCI HSI TAIEX and SP500

Ratio of index portfolio 1 5 10 15 Ratio of index portfolio 1 5 10 15025 025 025 025 minus00303 minus00165 minus00116 minus00086 025 025 025 025 minus00303 minus00165 minus00116 minus00086050 000 025 025 minus00329 minus00177 minus00128 minus00093 025 050 000 025 minus00361 minus00189 minus00132 minus00100

075 000 000 025 minus00418 minus00226 minus00157 minus00116 025 075 000 000 minus00426 minus00236 minus00165 minus00123

100 000 000 000 minus00536 minus00285 minus00199 minus00150 000 100 000 000 minus00491 minus00269 minus00190 minus00138

025 025 025 025 minus00303 minus00165 minus00116 minus00086 025 025 025 025 minus00303 minus00165 minus00116 minus00086025 025 050 000 minus0332 minus00176 minus00119 minus00093 000 025 050 050 minus00359 minus00181 minus00128 minus00094

000 025 075 000 minus00403 minus00196 minus00127 minus00097 000 000 025 075 minus00402 minus00206 minus00146 minus00108

000 000 100 000 minus00420 minus00211 minus00139 minus00101 000 000 000 100 minus00425 minus00230 minus00157 minus00119

Note the ratio of index portfolio is ranked by the sequence of SSCI HSI TAIEX SP500 in Table 7

and heavy-tailrdquo characteristics the time-varying volatilitycharacteristics and extreme-tail dependence characteristicsof financial asset return It proposed the parameter estimationalgorithm of the multivariate time-varying 119866-119867 CopulaGARCH model by using condition maximum likelihoodmethod and IFM two-step method An algorithm was con-structed to calculate VaR by using the quantile functionand the simulation method based on 119866-119867 Copula GARCHmodel In addition this paper selected the daily log returnof SSCI (China) HSI (Hong Kong China) TAIEX (TaiwanChina) and SP500 (USA) from January 3 2000 to June18 2010 as samples to estimate the parameters of themultivariate time-varying 119866-119867 Copula GARCH model andit also estimated theVaR for various index risk asset portfoliosunder different confidence levelsThe research results showedthat the multivariate time-varying 119866-119867 Copula GARCHmodel constructed in this paper could reasonably estimateand measure the extreme losses of risk portfolios in financialmarket and the measurement results were in line with theactual situation of stock market and the risk diversificationtheory of portfolio The achievement of this paper provideda practical and effective method for measuring the extremelosses of financial market

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees for all help-ful comments and suggestions This work is supported byNatural Science Foundation Project of CQ CSTC (Grant noCSTC 2011BB2088) and Key Project of National NaturalScience Foundation (Grant no 71232004)

References

[1] S J Kon ldquoModels of stock returnsmdasha comparisonrdquoThe Journalof Finance vol 39 no 1 pp 147ndash165 1984

[2] J B Gray and D W French ldquoEmpirical comparisons of dis-tributional models for stock index returnsrdquo Journal of BusinessFinance amp Accounting vol 17 no 3 pp 451ndash459 1990

[3] E Jondeau andMRockinger ldquoTesting for differences in the tailsof stock-market returnsrdquo Journal of Empirical Finance vol 10no 5 pp 559ndash581 2003

[4] C Jiang S Li and S Liang ldquoEmpirical investigation ofdistribution feature of return in china stockmarketrdquoApplicationof Statistics and Management vol 26 no 4 pp 710ndash717 2007

[5] D Dong and W Jin ldquoA subjective model of the distribution ofreturns and empirical analysisrdquo Chinese Journal of ManagementScience vol 15 no 1 pp 112ndash120 2007

[6] J Martinez and B Iglewicz ldquoSome properties of the tukey g andh family of distributionsrdquoCommunications in StatisticsmdashTheoryand Methods vol 13 no 3 pp 353ndash369 1984

[7] T C Mills ldquoModelling skewness and kurtosis in the Londonstock exchange FT-SE index return distributionsrdquo The Statisti-cian vol 44 no 3 pp 323ndash332 1995

[8] H Zhu and Z Pan ldquoA study of portfolios VaRmethod based ong-h distributionrdquo Chinese Journal of Management Science vol13 no 4 pp 7ndash12 2005

[9] K Kuester S Mittnik andM S Paolella ldquoValue-at-risk predic-tion a comparison of alternative strategiesrdquo Journal of FinancialEconometrics vol 4 no 1 pp 53ndash89 2006

[10] M Degen P Embrechts and D Lambrigger ldquoThe quantitativemodeling of operational risk between g-and-h and EVTrdquo AstinBulletin vol 37 no 2 pp 265ndash291 2007

[11] E Jondeau and M Rockinger ldquoThe Copula-GARCH modelof conditional dependencies an international stock marketapplicationrdquo Journal of International Money and Finance vol25 no 5 pp 827ndash853 2006

[12] J C Rodriguez ldquoMeasuring financial contagion a Copulaapproachrdquo Journal of Empirical Finance vol 14 no 3 pp 401ndash423 2007

Mathematical Problems in Engineering 9

[13] M Fischer C Kock S Schluter and F Weigert ldquoAn empiricalanalysis of multivariate copula modelsrdquo Quantitative Financevol 9 no 7 pp 839ndash854 2009

[14] W Sun S Rachev F J Fabozzi and S S Kalev ldquoA newapproach to modeling co-movement of international equitymarkets evidence of unconditional copula-based simulation oftail dependencerdquo Empirical Economics vol 36 no 1 pp 201ndash229 2009

[15] Z Liu T Zhang and F Wen ldquoTime-varying risk attitude andconditional skewnessrdquo Abstract and Applied Analysis vol 2014Article ID 174848 11 pages 2014

[16] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 3 no 48 pp235ndash254 2014

[17] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[18] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics and Economics vol 45 no 3 pp 315ndash324 2009

[19] A Ghorbel and A Trabelsi ldquoMeasure of financial risk usingconditional extreme value copulas with EVT marginsrdquo Journalof Risk vol 11 no 4 pp 51ndash85 2009

[20] L Chollete A Heinen and A Valdesogo ldquoModeling interna-tional financial returns with a multivariate regime-switchingcopulardquo Journal of Financial Econometrics vol 7 no 4 pp 437ndash480 2009

[21] M Huggenberger and T Klett ldquoA g-and-h copula approach torisk measurement in multivariate financial modelsrdquo WorkingPaper 2010 httpssrncomabstract=1677431

[22] J Wang W Bao and J Hu ldquoPortfolios VaR analyses basedon the multi-dimensional gumbel copula functionrdquo Journal ofApplied Statistics and Management vol 29 no 1 pp 137ndash1432010

[23] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis and Applications vol 16 no 1 pp93ndash103 2014

[24] J Liu M Tao C Ma and F Wen ldquoUtility indifference pricingof convertible bondsrdquo International Journal of InformationTechnology and Decision Making vol 13 no 2 pp 429ndash4442014

[25] R B Nelsen An Introduction to Copulas Springer Science ampBusiness Media New York NY USA 2007

[26] H JoeMultivariate Models and Dependence Concepts vol 73 ofMonographs on Statistics and Applied Probability Chapman ampHall London UK 1997

[27] A J Patton ldquoEstimation of multivariate models for time seriesof possibly different lengthsrdquo Journal of Applied Econometricsvol 21 no 2 pp 147ndash173 2006

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Research Article Multivariate Time-Varying Copula GARCH ...downloads.hindawi.com/journals/mpe/2015/286014.pdf · model, they presented an empirical pricing study of China s market.

2 Mathematical Problems in Engineering

reflect the time-varying volatility characteristics of financialasset return and the extreme-tail dependence characteristicsof various financial assets return [6 7] Meanwhile Copulafunction can connect the joint distribution and the marginaldistribution of multiple random variables to construct flex-ible multivariate distribution functions which can be usedto measure the extreme-tail dependence of multiple financialassets return GARCH model can comprehensively describethe time-varying volatility characteristics of financial assetreturn Therefore building the multivariate time-varying 119866-119867 Copula GARCH Model by combining the 119866-119867 distribu-tion with Copula function and GARCH model can not onlycomprehensively describe the ldquoasymmetric leptokurtic andheavy-tailrdquo characteristics the time-varying volatility andextreme-tail dependence characteristics of the financial assetreturn and make measurement of VaR more accurately butalso enrich and expand the risk measurement theory andmethod of financial market theoretically and improve therisk control ability of investors corporations financial insti-tutions and policy authorities and reduce their unnecessarylosses in practice

The 119866-119867 distribution Copula function and GARCHmodel as well as the financial risk measurement modelbased on them have been researched in the existing relevantliteratures A lot of innovative research results with referencevalue have been brought out Zhu and Pan [8] proposedthree kinds of 119866-119867 VaR methods based on the portfoliogains losses and extreme losses according to the statisticalcharacteristics of 119866-119867 distribution Their empirical resultsshowed that this method is superior to the commonly useddelta-normal method Kuester et al [9] believed that the 119866-119867 distribution can describe skewness and kurtosis of thefinancial asset return simultaneously and it plays a veryimportant role of VaR measurement of financial asset returnDegen et al [10] discussed the application of119866-119867distributionin operational riskmeasurement Jondeau andRockinger [11]Rodriguez [12] Fischer et al [13] and Sun et al [14] combinedtime series model with various Copulas functions by usingthe Sklar theorem to build a lot of highly flexible multivariatetime-varying models for risk measurement of portfolios Liuet al [15] proposed a GARCH-119872model with a time-varyingcoefficient of the risk premiumTheir study indicated that thecoefficient of the risk premium varies with the time and evenin a mature market the conditional skewness in the returndistribution is negatively correlated with the time-varyingcoefficient of the risk premium Wen et al [16] built a 119863-GARCH-119872 model by separating investorsrsquo return into gainsand losses on the basis of the characteristics of investorsrsquorisk preferenceThey found that investors become risk aversewhen they gain and risk-seeking when they lose whicheffectively explains the inconsistent risk-return relationshipAnd the degrees of investorsrsquo risk aversion and risk-seekingare both in direct proportion to the value of gains and lossesrespectively Wen et al [17] adopted aggregative indices of14 representative stocks around the world as samples andestablished a TVRA-GARCH-119872 model to investigate theinfluence of prior gains and losses on current risk attitudeThe empirical results indicated that the prior gains increasepeoplersquos current willingness to take risk asset at the whole

market level Huang et al [18] combined Studentrsquos 119905-marginaldistribution with Archimedean Copula functions to build theconditional Copula GARCH model They used this modelto estimate the VaR of portfolios Ghorbel and Trabelsi [19]built the conditional extremum Copula GARCH model byusing extreme value theory (EVT) and measured the risk offinancial asset according to this model Chollete et al [20]used multivariate regime-switching Copula function to buildinternational financial asset return model and accordinglyput forward the VaR calculation method Huggenberger andKlett [21] proposed a measurement model of multivari-ate risk asset return VaR based on 119866-119867 distribution andCopula function They used DAX30 (Germany) FTSE100(UK) and CAC40 (French) from January 2000 to May2010 as samples to test empirically Wang et al [22] appliedthe Gumbel Copula function in multivariate ArchimedeanCopula functions family to construct the joint distributionfunction which can describe the actual distribution and thecorrelation of various financial asset returns They also usedthe Monte Carlo simulation technology to analyze the port-folios VaR and its composition under different confidencelevels The result showed that using the multidimensionalGumbel Copula function to construct the risk measurementmodel of financial asset can make the assets chosen byinvestors more robust and it can also help investors todiversify and control the overall risk of the portfolios Daiand Wen [23] proposed a computationally tractable robustoptimization method for minimizing the CVaR of a portfoliounder a general affine data perturbation uncertainty setAnd they presented some numerical experiments with realmarket data to illustrate the behavior of robust optimiza-tion model Liu et al [24] proposed a pricing model forconvertible bonds based on the utility-indifference methodand got access to the empirical results by use of InformationTechnology Furthermore using the proposed theoreticalmodel they presented an empirical pricing study of Chinarsquosmarket They found that the theoretical prices are higherthan the actual market prices 024ndash458 and the utility-indifference prices are better than the Black-Scholes (B-S)prices

Based on the aforementioned analyses the VaR is stillthe mainstream measurement method of financial mar-ket risk In order to achieve the purpose of measuringVaR more precisely it has been the hot issue of existingresearch literatures to construct the distribution functionsas comprehensive as possible to describe the ldquoasymmetricleptokurtic and heavy-tailrdquo characteristics the time-varyingvolatility characteristics and the extreme-tail dependencecharacteristics of financial asset return through a variety ofmathematicalmethods However in the process of construct-ing the return distribution model and measuring VaR offinancial asset existing results only grasp some characteristicsof financial asset return distribution They are not able tocomprehensively describe the ldquoasymmetric leptokurtic andheavy-tailrdquo characteristics the time-varying volatility charac-teristics and the extreme-tail dependence characteristics ofthe financial asset returnThe rationality and accuracy of VaRcalculated based on the existing distribution models have alarge space for further improvement

Mathematical Problems in Engineering 3

In this paper we would take full advantage of thestrengths of119866-119867 distribution Copula function andGARCHmodel in depicting the return distribution of financial asset tobuild multivariate time-varying119866-119867Copula GARCHmodelwhich can simultaneously describe ldquoasymmetric leptokurticand heavy-tailrdquo characteristics the time-varying volatilitycharacteristics and the extreme-tail dependence character-istics of financial asset return and propose the estimationmethod of model parameters and the calculation algorithmof VaR Then this paper selects the SSCI (China) HSI(Hong Kong China) TAIEX (Taiwan China) and SP500(USA) from January 3 2000 to June 18 2010 as samples toestimate the parameters and calculate the VaRs of variousindex portfolios under different confidence levels

2 119866-119867 Distribution Copula Functionand Its Tail Dependence Index

21 119866-119867 Distribution

211 119866 Distribution Assuming that random variable 119885

obeys the standard normal distribution and 119892 is a realnumber then the random variable 119884119892 = 119866119892(119911) obeys 119866

distribution Consider

119866119892 (119911) =119890119892119911

minus 1119892

119911 sim 119873 (0 1) (1)

where 119892 controls the skewness of 119866 distribution When 119892 rarr

0 119866119892(119911) rarr 119911 and 119866 distribution tends to be symmetricWith the increase of the absolute value of 119892 the degree ofasymmetry increases Changing the sign of 119892 can changethe asymmetric direction of 119866 distribution but it does notchange its degree of asymmetry

212 119867 Distribution Assuming that random variable 119885

obeys the standard normal distribution and ℎ is a realnumber then the random variable 119884ℎ = 119867ℎ(119911) obeys 119867

distribution Consider

119867ℎ (119911) = 119890ℎ119911

22 119911 sim 119873 (0 1) (2)

119867 distribution stretches the tail of the standard normaldistribution ℎ controls the tail heaviness of 119867 distributionThe larger the ℎ is the heavier the tail is Because 119867ℎ(119911)

is an even function 119867 distribution is symmetric But theheaviness of its tail changes compared to the standard normaldistribution

213 119866-119867 Distribution The random variable 119884119892ℎ can beobtained by introducing both functions 119866119892(119911) and 119867ℎ(119911) torevise standard normal random variable 119885 Consider

119884119892ℎ = 119866119892 (119911)119867ℎ (119911) =119890119892119911

minus 1119892

119890ℎ119911

22 (3)

Then 119883119892ℎ can be obtained through linear transformation of119884119892ℎ Consider

119883119892ℎ = 119860+119861119890119892119911

minus 1119892

119890ℎ119911

22 119911 sim 119873 (0 1) (4)

The distribution of the random variable119883119892ℎ obeys the119866-119867 distribution 119860 119861 119892 and ℎ are real numbers 119892 describesthe asymmetry of 119866-119867 distribution and ℎ describes theheavy-tail characteristics of 119866-119867 distribution Obviously (3)is a special form of (4) The random variable 119884119892ℎ in (3)is the random variable of 119866-119867 distribution after centralstandardization

22 Copula Function and Its Tail Dependence Index Assum-ing that marginal distribution of random vector 119906119894 = 119865119894(119909119894)

(119894 = 1 2 119901) obeys uniformdistribution119880(0 1) accordingto the Sklar theorem the joint distribution function of 119875-dimensional random vectors 119865(1199091 119909119901) can be repre-sented as the following formula

119865 (1199091 119909119901) = 119862 (1198651 (1199091) 119865119901 (119909119901)) (5)

where 119862 is the Copula function of 119865 which is a hyper-cube [0 1]119901 multivariate density function defined on 119875-dimensional space R119901 If the marginal distribution is contin-uous there is a unique Copula function 119862 Then

119862 (1199061 119906119901) = 119865 (119865minus11 (1199061) 119865

minus1119901

(119906119901)) (6)

On the contrary given 119875-dimensional Copula func-tion 119862(1199061 119906119901) and its marginal distribution function1198651(1199091) 119865119901(119909119901) the density function of 119875-dimensionaljoint distribution function is

119891 (1199091 119909119901) = 119888 (1198651 (1199061) 119865119901 (119906119901))

119901

prod

119894=1119891119894 (119909119894) (7)

If 119891119894(119909119894) is the edge density 119888(1199061 119906119901) denotes Copuladensity derived from (6) Thus

119888 (1199061 119906119901) =

119891 (119865minus11 (1199061) 119865

minus1119901

(119906119901))

prod119901

119894=1119891119894 (119865minus1119894

(119906119894)) (8)

Since the joint distribution function of random vari-ables defines the correlation among its components Copulafunction determines the dependent structure among randomvariables uniquely The upper tail index 120582119906 and lower tailindex 120582119897 of tail dependence indicators can be defined asfollows

120582119906 = lim119902rarr 1

1 minus 2119902 + 119888 (119902 119902)

1 minus 119902

120582119897 = lim119902rarr 0

119888 (119902 119902)

119902

(9)

According to Nelsen [25] Gauss Copula function gen-erated by multivariate normal distribution function whosecorrelation matrix is R can be represented as follows

119862119866119906

R (1199061 119906119901)

= int

Φminus11 (1199061)

minusinfin

sdot sdot sdot int

Φminus1119901

(119906119901

)

minusinfin

1

radic(2120587)119901 |R|

expminusu1015840Rminus1u

2119889u

(10)

4 Mathematical Problems in Engineering

where u = (1199061 119906119901) and Φminus1 is the inverse function

of single normal distribution Because the Gauss Copulafunction does not have the characteristics of tail dependencewe often use the119879-Copula function whose degree of freedomis 120578 and correlation matrix is R to measure tail dependencestructure of risk asset in empirical analysis that is

119862119905

120578R (1199061 119906119901)

= int

119905minus1120578

(1199061)

minusinfin

sdot sdot sdot int

119905minus1120578

(119906119901

)

minusinfin

Γ ((120578 + 119901) 2) (1 + u1015840Rminus1u2)minus(120578+119901)2

Γ (1205782)radic(120587120578)119901|R|

119889u(11)

where 119905minus1120578

is the inverse function of simple standard Studentrsquos119905-distribution whose degree of freedom is 120578 When 120578 rarr infin119879-Copula function degenerates to Gauss Copula function Itstail index 120582119906 = 120582119897 = 0 that is the tail is independent The tailindex of 119879-Copula function is

120582119906 = 120582119897 = 2119905120578+1 (minus

radic(120578 + 1) (1 minus 120588)

radic1 + 120588) (12)

where 119905120578+1 is simple standard Studentrsquos 119905-distribution whosedegree of freedom is 120578 + 1 Considering that the innovationimpacts on the price of risk asset in varying degrees at differ-ent times 120578 and 120588 should have time-varying characteristicsFor this reason tail index also has the same characteristics

3 Multivariate Time-Varying 119866-119867 CopulaGARCH Model

Let r119905 = (1199031119905 119903119901119905) denote return time series of 119901 riskassets The prior information set before time 119905 is

I119905minus1 = r119905minus1 h119905minus1 r119905minus2 h119905minus2 =

119901

prod

119894=1I119894119905minus1 (13)

where I119894119905minus1 = 119903119894119905minus1 ℎ119894119905minus1 119903119894119905minus2 ℎ119894119905minus2 ℎ119894119905 is conditionalvolatility of 119903119894119905 about single asset prior information set I119894119905minus1Let 119862(sdot | I119905minus1) denote 119875-dimensional conditional Copulafunction and 119865119894(119903119894119905 | I119894119905minus1) be the conditional distributionof the 119894th component According to Sklar theorem theconditional joint distribution of 119901 risk assets return is

119865 (r119905 | I119905minus1)

= 119862 (1198651 (1199031119905 | I1119905minus1) 119865119901 (119903119901119905 | I119901119905minus1) | I119905minus1)

(14)

Numerous empirical studies show that the risk assetreturn series obey GARCH (1 1) model Based on thisassuming that 119903119894119905 satisfies the GARCH (1 1) model we canget the following 119866-119867 Copula GARCH (1 1) model whichdescribes the time-varying dependence structure of 119901 risk

assets return after filtering the time-varying characteristics ofsingle series

119903119894119905 = 120583119894 + 120576119894119905 119894 = 1 2 119901

120576119894119905 = radicℎ119894119905119911119894119905

ℎ119894119905 = 120603119894 +1205721198941205762119894119905minus1 +120573119894ℎ119894119905minus1

119865 (z119905 | I119905minus1)

= 119862 (1198651 (1199111119905 | I1119905minus1) 119865119901 (119911119901119905 | I119901119905minus1) | I119905minus1)

(15)

where the parameters satisfy the conditions 120603119894 120572119894 120573119894 gt 0and 120572119894 + 120573119894 lt 1 These parameters can ensure the stabilityof conditional volatility series The innovation series z119905obey 119866-119867 distribution whose parameter is (119892 ℎ) in (4) Butin order to simplify the analysis we only consider 119866-119867distribution after central standardization given by (3) and itsdensity function is written as 119891119884

119894

(119910119894) The Copula function119862(sdot | I119905minus1) is given by (11) and its density 119888(sdot) can berepresented as the following time-varying119879-Copula functionwhose degree of freedom is 120578

119888119905

120578120588119905

(1199061119905 119906119901119905)

=

119891119905

120578120588119905

(119891minus1V1 (1199061119905) 119891

minus1V119901

(119906119901119905))

prod119901

119894=1119891120578 (119891minus1V119894

(119906119894119905))

(16)

where 119891119905

120578120588119905

denotes the multivariate Studentrsquos 119905-distributionwhose degree of freedom is 120578 and time-varying correlationmatrix is 120588

119905= (120588119894119895119905)119901times119901 and

120588119894119894119905 = 1

119891V119894

(119906119894119905) =Γ ((V119894 + 1) 2)Γ (V1198942)radicV119894120587

(1+1199062119894119905

V119894)

minus(V119894

+1)2

(17)

The joint density function of 119901 risk assets return is

119891 (y | u h119905) = 119888119905

120578120588119905

(1199091119905 119909119901119905)

119901

prod

119894=1119891119884119894

(119910119894)

= Γ (120578 + 119901

2) Γ (

120578

2)

119901minus1(1+

x1015840119905120588119905x119905

120578)

minus(120578+119901)2

sdot (1003816100381610038161003816120588119905

1003816100381610038161003816)minus12119901

prod

119894=1(1 +

1199092119894119905

120578)

(120578+1)2 119901

prod

119894=1119891119884119894

(119910119894)

(18)

Mathematical Problems in Engineering 5

where x119905 = (1199091119905 119909119901119905) and 119909119894119905 = 119905minus1120578

(119905V119894

(120576119894119905)) Then thelikelihood function of overall samples is

119897 (120579 | y) =

119879

prod

119905=1Γ (

120578 + 119901

2) Γ (

120578

2)

119901minus1

sdot (1+x1015840119905120588119905x119905

120578)

minus(120578+119901)2

(1003816100381610038161003816120588119905

1003816100381610038161003816)minus12

Γ (120578 + 12

)

minus119901

sdot

119901

prod

119894=1(1+

1199092119894119905

120578)

(120578+1)2 119901

prod

119894=1119891119884119894

(119910119894)

(19)

where 120579 = (120583119894 120603119894 120572119894 120573119894 V119894)119901

119894=1 119886 119887120588119905 120578 and x119905 = (1199091119905 119909119901119905)The value of correlationmatrix 120588

119905is similar to the time-

varying correlation matrix of multivariate Copula GARCHmodel proposed by Jondeau and Rockinger [11] That is 120588

119905

satisfies the following evolution equation

120588119905= (1minus 119886minus 119887)120588+ 119886Ψ119905minus1 + 119887120588

119905minus1 (20)

where 0 le 119886 119887 le 1 119886 + 119887 le 1 120588 is a positive definite matrixwhose main diagonal elements are 1 and other elements arestatic correlation coefficientsΨ119905minus1 is a 119901times119901matrix in whichevery element

120595119894119895119905minus1 =sum119898

119897=1 119909119894119905minus119897119909119895119905minus119897

radicsum119898

119897=1 1199092119894119905minus119897

radicsum119898

119897=1 1199092119895119905minus119897

119894 119895 = 1 2 119901 (21)

denotes the correlation coefficients of 119901 risk asset returns(119898 ge 119901 + 2) 119909119905 = (1199091119905 119909119901119905) = (119905

minus1V1 (119891V1(1199111119905))

119905minus1V119901

(119891V119901

(119911119901119905))) Each element 120588119894119895119905 of 120588119905 satisfies minus1 le 120588119894119895119905 le 1

4 Parameter Estimation Algorithm ofthe Multivariate Time-Varying 119866-119867 CopulaGARCH Model

On the basis of Huggenberger and Klett [21] this section willuse dynamic correlation matrix 120588

119905instead of static correla-

tion matrix in the multidimensional discrete-time stochasticprocess to estimate the parameters of multivariate time-varying119866-119867CopulaGARCHmodel established in Section 3Assuming thatΘ denotes the parameter space defined by themodel and (r1 r2 r119879) denotes the log return samples of119875-dimensional risk asset which is generated by multivariateconditional density function119891120588

119905

|I119905minus1

(r119905 | I119905minus1 1205790) where 1205790 isin

Θ I119905minus1 is 120590 algebra of time 119905 minus 1 and before the maximumlikelihood estimation of parameter vector 120579 can be calculatedby the following equation

= argmax120579isinΘ

119879

sum

119905=1log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 120579) (22)

where 119891120588119905

|I119905minus1

(r119905 | I119905minus1 120579) can be obtained by calculating thederivative of (5) Let 119888120579 denote Copula density functionThus

119891120588119905

|I119905minus1

(1199031119905 119903119901119905 | I119905minus1 120579)

= 119888120579 (1198651119905 (1199031119905 120579) 119865119901119905 (119903119901119905 120579))

sdot

119901

prod

119894=1119891119894119905 (119903119905119894 120579)

(23)

The probability density function and distribution func-tion can be obtained in the process ofmodel built in Section 3Using the IFM method proposed by Joe [26] we can convert(22) into an optimization problem Therefore we need todivide the parameter vector 120579 into two subparameter vectors120579119888 and 120579119903 that is 120579 = (120579119888 120579119903) where 120579119903 = (1205791199031

120579119903119901

)120579119903119894

is the parameter vector of 119894th marginal distribution and120579119888 is the parameter vector of Copula function Because IFMmethod is a two-step likelihood estimation method themodel parameters should be estimated through the followingtwo steps

Step 1 Solving the maximum likelihood estimator of theparameter vector of each risk asset return

119903119894

= argmax120579119903

119894

119879

sum

119905=1log119891119894119905 (119903119894119905 | 120579119903

119894

) 119894 = 1 2 119901 (24)

Thismeans that we need to estimate parameters vector 119903119894

of 119901 distributions continuously

Step 2 Taking each 119903119894

into the likelihood equation (22) wecan obtain the parameter vector 120579119888 of Copula function and itsmaximum likelihood estimator 119888 Consider

119888

= argmax120579119888

119879

sum

119905=1log 119888120579

119888

(1198651119905 (10038161003816100381610038161199031119905

1003816100381610038161003816 1199031) 119865119901119905 (

10038161003816100381610038161003816119903119901119905

10038161003816100381610038161003816119903119901

))

(25)

In themaximum likelihood estimationweneed to use thederivative function of the density function of 119866-119867 marginaldistribution with respect to the component of parametervector Because the density function of 119866-119867 marginal distri-bution is very complex this paper uses the implicit functiondifferentiation rule to take its derivative The estimator 2119904of parameter vector 120579 obtained by the above-mentionedtwo-step method obeys normal distribution consistentlyand asymptotically under the standard regularity conditionsproposed inHuggenberger andKlett [21] Joe [26] andPatton[27] that is

radic119879(2119904119879 minus 1205790)119889

997888rarr119879rarrinfin

119873(0Ωminus1ΣΩ) (26)

6 Mathematical Problems in Engineering

where

Ω = minus119864(1205972

1205971205791205971205791015840log119891120588

119905

|I119905minus1

(120588119905| I119905minus1 1205790))

Σ = 119864(120597

120597120579(log119891120588

119905

|I119905minus1

(120588119905| I119905minus1 1205790))

sdot120597

120597120579(log119891120588

119905

|I119905minus1

(120588119905| I119905minus1 1205790))

1015840

)

(27)

Because the matrixes Σ and Ω can be estimated by theestimated parameter vector consistently

Ω119879 = minus119879minus1119879

sum

119905=1

1205972

1205971205791205971205791015840log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 2119904119879)

Σ119879 = 119879minus1119879

sum

119905=1

120597

120597120579(log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 2119904119879))

sdot120597

120597120579(log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 2119904119879))1015840

(28)

Thus (26) can be used to calculate the standard deviationof the estimator 2119904119879

5 VaR Algorithm Based on the MultivariateTime-Varying 119866-119867 Copula GARCH Model

After estimating the parameters of multivariate time-varying119866-119867 Copula GARCH model VaR of the risk portfolio canbe measured VaR of risk portfolio indicates the expectedmaximum losses of risk portfolio held by investors within agiven confidence level and a certain period of time Assumingthat r119905 = (1199031119905 119903119901119905) (119905 = 1 2 119879) are the return samplesof 119901 risk assets which satisfy themultivariate time-varying119866-119867 Copula GARCH (1 1)model in Section 3 andsum

119901

119894=1 120582119894119903119894119905 isthe portfolio of 119901 risk assets in which the weight of the riskasset 119894 is 120582119894 (119894 = 1 2 119901) that can be less than 0 because ofpermitting short-purchasing and short-selling the risk assetsand meet sum

119901

119894=1 120582119894 = 1 the VaR of risk portfolio underconfidence level 119902 at time 119905 should satisfy Pr(sum119901

119894=1 120582119894119903119894119905 le

VaR119905) = 119902 The confidence level 119902 can reflect the differentrisk preferences of investors or financial institutions to acertain extent Choosing a larger confidence level means thatinvestors or financial institutions have greater risk aversionand they hope to get a forecast result with larger probability

Although the conditional distributions of 1199031119879+1 1199032119879+1 119903119901119879+1 can be calculated through the known marginaldistributions it is very difficult to calculate quantile fromtime-varying Copula density function and it is adverse tomeasure and calculate the VaR of risk portfolio Thereforethis paper measures the dynamic risk of portfolio and its esti-mation value approximately through simulating119866-119867CopulaGARCH model Based on the parameters 120579(119899) of the samplethe return series of risk assets [119903

(119899119898)

11+119879 119903(119899119898)

1199011+119879] 119898 =

1 119872 and the one-step measurement and estimationvalues of VaR of their portfolios can be obtained through esti-mating (ℎ(119899)1119879+1 ℎ

(119899+1)119901119879+1) according to the volatility equation

Table 1 Moment estimation results of the daily log return of SSCIHSI TAIEX and SP500

Types of stock index Mean Std Skewness KurtosisSSCI 25078119890 minus 004 00181 minus02144 74467HSI 74505119890 minus 005 00177 minus02765 116866TAIEX 39511119890 minus 005 00140 minus09091 172823SP500 minus97173119890 minus 005 00148 minus03701 120177

of119866-119867 Copula GARCHmodel and calculating 120588(119899)

119879+1 by usingCopula dynamic evolution equation and then repeating thefollowing algorithm for 119872 times (119872 ge 3119901)

Step 1 It is simulating 119872 groups of random vectors[119906(119899119898)

1119879+1 119906(119899119898)

119901119879+1] according to the multivariate 119879-Copuladensity function whose degree of freedom is 120578(119899) and correla-tion matrix is 120588(119899)

119879+1

Step 2 Calculating 119903(119899119898)

119894119879+1 = 120583(119899)

119894+ 119911(119899119898)

119894119879+1radicℎ(119899)

119894119879+1 119894 = 1 119901

Step 3 Firstly one calculates the return rate of risk portfoliothat is equal to sum

119901

119894=1 120582119894119903(119899119898)

119894119879+1 119898 = 1 2 119872 Secondly oneevaluates its 119902-quantile VaR(119899)

119879+1 Thirdly one measures theVaR of the risk portfolio by VaR119879+1 = (1119873)sum

119873

119899=1 VaR(119899)

119879+1

6 Application of the Multivariate Time-Varying 119866-119867 Copula GARCH Model

61 Date Sample and Moment Estimation USA and Chinaas the most developed capitalism country and the largestdeveloping country in the world respectively rank top two ofthe world economy Their stock markets should have strongrepresentation in the world At the same time due to thehistorical reasons there exist several regions with differentpolitical systems such as Mainland China Hong KongTaiwan and Macau in Greater China Macau is similar toHong Kong on the whole For the above-mentioned reasonsthis paper selects the SSCI (China) HSI (HongKong China)TAIEX (Taiwan China) and SP500 (USA) from January3 2000 to June 18 2010 as data samples to estimate theVaR of various index portfolios under different confidencelevels by using the multivariate time-varying 119866-119867 CopulaGARCHmodelThe data comes fromYahoo Financewebsitehttpfinanceyahoocom

The moment estimation results of the daily log return ofSSCI HSI TAIEX and SP500 are shown in Table 1

Table 1 shows that the skewness of daily log returns ofSSCI HSI TAIEX and SP500 is less than 0 and their kurtosisis much larger than that of standard normal distributionwhich is equal to 3 These results demonstrate that thedaily log returns of these indices have the right skew andleptokurtic characteristics Therefore it is appropriate to fitthe daily log return of SSCI HSI TAIEX and SP500 byapplying 119866-119867 distribution which has leptokurtic heavy-tailcharacteristics and it is reasonable to apply the multivariate

Mathematical Problems in Engineering 7

Table 2 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on SSCI index risk asset

Model parameters 1205831

1205961

1205721

1205731

Estimate 000025lowastlowastlowast 00551lowastlowastlowast 00617lowastlowastlowast 08583lowastlowastlowast

119879-statistic 48653 46851 65764 44009Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 3 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on HSI index risk asset

Model parameters 1205832

1205962

1205722

1205732

Estimate 0000075lowastlowastlowast

00342lowastlowastlowast

00586lowastlowastlowast

08987lowastlowastlowast

119879-statistic 46539 48518 61796 66079Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 4 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on TAIEX index risk asset

Model parameters 1205833

1205963

1205723

1205733

Estimate 000040lowastlowastlowast 00343lowastlowastlowast

00517lowastlowastlowast

09054lowastlowastlowast

119879-statistic minus64538 43523 59817 65935Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 5 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on SP500 index risk asset

Model parameters 1205834

1205964

1205724

1205734

Estimate minus00001lowastlowastlowast

00251lowastlowastlowast

00861lowastlowastlowast

08738lowastlowastlowast

119879-statistic minus33873 52684 76324 63786Note lowastlowastlowastin the table denotes that the parameter is significant at 1 level

time-varying 119866-119867 Copula GARCH model to measure theirVaR

62 Parameter Estimates of the Multivariate Time-Varying119866-119867 Copula GARCH Model Based on the parameter esti-mation algorithm proposed in Section 4 the parameters ofthe multivariate time-varying 119866-119867 Copula GARCH modelwith SSCI HSI TAIEX and SP500 can be estimated Theparameter estimation results are shown in Tables 2 3 4 5and 6

From Tables 2 to 6 the following results can be obtained

(1) Consider 1205721 + 1205731 = 092 1205722 + 1205732 = 09573 1205723 +

1205733 = 09571 and 1205724 + 1205734 = 09599 This shows thatthe volatility persistence of Shanghai stock market isthe strongest Taiwan and Hong Kong stock marketrank second and third and the volatility persistenceof USA stock market is minimum It indicates thatthe investorsrsquo expectation of risk compensation inthe emerging markets represented by Chinarsquos stockmarket is stronger than that in the mature marketsrepresented by the USArsquos stock market and the pricediscovery efficiency of innovation in the emergingmarkets represented by Chinarsquos stock market is lowerthan that in the mature markets represented by USArsquosstockmarket In addition the sumof the coefficients120572

and120573 is very close to 1 which indicates that the impactand shock of innovation on the index volatility of eachstock market has a long memory

(2) The degree of freedom 120578 = 1457 and the correlationcoefficients 120588119894119895 of 119879-Copula in Table 6 show thatthere exists the strongest correlation between HongKong stock market and Taiwan stock market andthe correlation between Shanghai stock market andHong Kong stock market is also relatively largeThe above-mentioned facts indicate that the extremeevents probably result in the phenomena that HongKong stock market and Taiwan stock market are upand down synchronously and there exist comovingbehaviors between Shanghai stock market and HongKong stock market

(3) The time-varying coefficient 119887 = 0987 indicatesthat the time-varying correlation coefficient of 119866-119867 Copula GARCH model has a long memory thatis the impact of historical values of each otherrsquoscorrelation coefficient among SSCI HSI TAIEX andSP500 on the expected correlation is relatively large

63 VaR Measurement Based on the Multivariate Time-Varying 119866-119867 Copula GARCH Model Based on the multi-variate time-varying119866-119867 Copula GARCHmodel with SSCIHSI TAIEX and SP500 whose parameters have been esti-mated the VaRs of various index portfolios under differentconfidence levels can be measured The measurement resultsare shown in Table 7

From Table 7 the following results can be obtained

(1) The inequalities VaR (SSCI) lt VaR (HSI) lt VaR(SP500) lt VaR (TAIEX) can be satisfied for anyconfidence level It shows that the risk of extremelosses in Shanghai stock market is higher than that inHong Kong stock market Taiwan stock market andUSA stock market This measurement result is in linewith the actual situation that thematurity of Shanghaistockmarket is far lower than that ofHongKong stockmarket Taiwan stock market and USA stock market

(2) For any confidence level the extreme losses risk ofthe investors who equally allocate their total assetsamong SSCI HSI TAIEX and SP500 is lower thanthat of the investors who put their total assets into oneindex asset The extreme losses risk of the investorsincreases with the concentration of risk asset in theindex portfolios This measurement result is consis-tent with the risk diversification theory of portfolio

7 Conclusion

Considering the ldquoasymmetric leptokurtic and heavy-tailrdquocharacteristics the time-varying volatility characteristicsand extreme-tail dependence characteristics of financialasset return this paper combined the 119866-119867 distributionCopula function and GARCH model to construct a mul-tivariate time-varying 119866-119867 Copula GARCH model whichcan comprehensively describe the ldquoasymmetric leptokurtic

8 Mathematical Problems in Engineering

Table 6 Estimates of correlation coefficients time-varying parameters and degree of freedom of four-variate time-varying G-H CopulaGARCH (1 1) model based on SSCI HSI TAIEX and SP500

Model parameters 12058812

12058813

12058814

12058823

12058824

12058834

a b 120578

Estimate 0256lowastlowastlowast

0139lowastlowastlowast

0021lowastlowastlowast

0528lowastlowastlowast

0193lowastlowastlowast

0128lowastlowast

0008lowastlowastlowast

0987lowastlowastlowast

1457lowastlowastlowast

119879-statistic 1351 1102 3431 5763 9564 2529 5179 3561 6871Note lowastlowastlowast and lowastlowast in the table denote that the parameter is significant at 1 and 5 level respectively

Table 7 VaR estimation results based on the four-variate time-varying G-H Copula GARCHmodel with SSCI HSI TAIEX and SP500

Ratio of index portfolio 1 5 10 15 Ratio of index portfolio 1 5 10 15025 025 025 025 minus00303 minus00165 minus00116 minus00086 025 025 025 025 minus00303 minus00165 minus00116 minus00086050 000 025 025 minus00329 minus00177 minus00128 minus00093 025 050 000 025 minus00361 minus00189 minus00132 minus00100

075 000 000 025 minus00418 minus00226 minus00157 minus00116 025 075 000 000 minus00426 minus00236 minus00165 minus00123

100 000 000 000 minus00536 minus00285 minus00199 minus00150 000 100 000 000 minus00491 minus00269 minus00190 minus00138

025 025 025 025 minus00303 minus00165 minus00116 minus00086 025 025 025 025 minus00303 minus00165 minus00116 minus00086025 025 050 000 minus0332 minus00176 minus00119 minus00093 000 025 050 050 minus00359 minus00181 minus00128 minus00094

000 025 075 000 minus00403 minus00196 minus00127 minus00097 000 000 025 075 minus00402 minus00206 minus00146 minus00108

000 000 100 000 minus00420 minus00211 minus00139 minus00101 000 000 000 100 minus00425 minus00230 minus00157 minus00119

Note the ratio of index portfolio is ranked by the sequence of SSCI HSI TAIEX SP500 in Table 7

and heavy-tailrdquo characteristics the time-varying volatilitycharacteristics and extreme-tail dependence characteristicsof financial asset return It proposed the parameter estimationalgorithm of the multivariate time-varying 119866-119867 CopulaGARCH model by using condition maximum likelihoodmethod and IFM two-step method An algorithm was con-structed to calculate VaR by using the quantile functionand the simulation method based on 119866-119867 Copula GARCHmodel In addition this paper selected the daily log returnof SSCI (China) HSI (Hong Kong China) TAIEX (TaiwanChina) and SP500 (USA) from January 3 2000 to June18 2010 as samples to estimate the parameters of themultivariate time-varying 119866-119867 Copula GARCH model andit also estimated theVaR for various index risk asset portfoliosunder different confidence levelsThe research results showedthat the multivariate time-varying 119866-119867 Copula GARCHmodel constructed in this paper could reasonably estimateand measure the extreme losses of risk portfolios in financialmarket and the measurement results were in line with theactual situation of stock market and the risk diversificationtheory of portfolio The achievement of this paper provideda practical and effective method for measuring the extremelosses of financial market

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees for all help-ful comments and suggestions This work is supported byNatural Science Foundation Project of CQ CSTC (Grant noCSTC 2011BB2088) and Key Project of National NaturalScience Foundation (Grant no 71232004)

References

[1] S J Kon ldquoModels of stock returnsmdasha comparisonrdquoThe Journalof Finance vol 39 no 1 pp 147ndash165 1984

[2] J B Gray and D W French ldquoEmpirical comparisons of dis-tributional models for stock index returnsrdquo Journal of BusinessFinance amp Accounting vol 17 no 3 pp 451ndash459 1990

[3] E Jondeau andMRockinger ldquoTesting for differences in the tailsof stock-market returnsrdquo Journal of Empirical Finance vol 10no 5 pp 559ndash581 2003

[4] C Jiang S Li and S Liang ldquoEmpirical investigation ofdistribution feature of return in china stockmarketrdquoApplicationof Statistics and Management vol 26 no 4 pp 710ndash717 2007

[5] D Dong and W Jin ldquoA subjective model of the distribution ofreturns and empirical analysisrdquo Chinese Journal of ManagementScience vol 15 no 1 pp 112ndash120 2007

[6] J Martinez and B Iglewicz ldquoSome properties of the tukey g andh family of distributionsrdquoCommunications in StatisticsmdashTheoryand Methods vol 13 no 3 pp 353ndash369 1984

[7] T C Mills ldquoModelling skewness and kurtosis in the Londonstock exchange FT-SE index return distributionsrdquo The Statisti-cian vol 44 no 3 pp 323ndash332 1995

[8] H Zhu and Z Pan ldquoA study of portfolios VaRmethod based ong-h distributionrdquo Chinese Journal of Management Science vol13 no 4 pp 7ndash12 2005

[9] K Kuester S Mittnik andM S Paolella ldquoValue-at-risk predic-tion a comparison of alternative strategiesrdquo Journal of FinancialEconometrics vol 4 no 1 pp 53ndash89 2006

[10] M Degen P Embrechts and D Lambrigger ldquoThe quantitativemodeling of operational risk between g-and-h and EVTrdquo AstinBulletin vol 37 no 2 pp 265ndash291 2007

[11] E Jondeau and M Rockinger ldquoThe Copula-GARCH modelof conditional dependencies an international stock marketapplicationrdquo Journal of International Money and Finance vol25 no 5 pp 827ndash853 2006

[12] J C Rodriguez ldquoMeasuring financial contagion a Copulaapproachrdquo Journal of Empirical Finance vol 14 no 3 pp 401ndash423 2007

Mathematical Problems in Engineering 9

[13] M Fischer C Kock S Schluter and F Weigert ldquoAn empiricalanalysis of multivariate copula modelsrdquo Quantitative Financevol 9 no 7 pp 839ndash854 2009

[14] W Sun S Rachev F J Fabozzi and S S Kalev ldquoA newapproach to modeling co-movement of international equitymarkets evidence of unconditional copula-based simulation oftail dependencerdquo Empirical Economics vol 36 no 1 pp 201ndash229 2009

[15] Z Liu T Zhang and F Wen ldquoTime-varying risk attitude andconditional skewnessrdquo Abstract and Applied Analysis vol 2014Article ID 174848 11 pages 2014

[16] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 3 no 48 pp235ndash254 2014

[17] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[18] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics and Economics vol 45 no 3 pp 315ndash324 2009

[19] A Ghorbel and A Trabelsi ldquoMeasure of financial risk usingconditional extreme value copulas with EVT marginsrdquo Journalof Risk vol 11 no 4 pp 51ndash85 2009

[20] L Chollete A Heinen and A Valdesogo ldquoModeling interna-tional financial returns with a multivariate regime-switchingcopulardquo Journal of Financial Econometrics vol 7 no 4 pp 437ndash480 2009

[21] M Huggenberger and T Klett ldquoA g-and-h copula approach torisk measurement in multivariate financial modelsrdquo WorkingPaper 2010 httpssrncomabstract=1677431

[22] J Wang W Bao and J Hu ldquoPortfolios VaR analyses basedon the multi-dimensional gumbel copula functionrdquo Journal ofApplied Statistics and Management vol 29 no 1 pp 137ndash1432010

[23] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis and Applications vol 16 no 1 pp93ndash103 2014

[24] J Liu M Tao C Ma and F Wen ldquoUtility indifference pricingof convertible bondsrdquo International Journal of InformationTechnology and Decision Making vol 13 no 2 pp 429ndash4442014

[25] R B Nelsen An Introduction to Copulas Springer Science ampBusiness Media New York NY USA 2007

[26] H JoeMultivariate Models and Dependence Concepts vol 73 ofMonographs on Statistics and Applied Probability Chapman ampHall London UK 1997

[27] A J Patton ldquoEstimation of multivariate models for time seriesof possibly different lengthsrdquo Journal of Applied Econometricsvol 21 no 2 pp 147ndash173 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article Multivariate Time-Varying Copula GARCH ...downloads.hindawi.com/journals/mpe/2015/286014.pdf · model, they presented an empirical pricing study of China s market.

Mathematical Problems in Engineering 3

In this paper we would take full advantage of thestrengths of119866-119867 distribution Copula function andGARCHmodel in depicting the return distribution of financial asset tobuild multivariate time-varying119866-119867Copula GARCHmodelwhich can simultaneously describe ldquoasymmetric leptokurticand heavy-tailrdquo characteristics the time-varying volatilitycharacteristics and the extreme-tail dependence character-istics of financial asset return and propose the estimationmethod of model parameters and the calculation algorithmof VaR Then this paper selects the SSCI (China) HSI(Hong Kong China) TAIEX (Taiwan China) and SP500(USA) from January 3 2000 to June 18 2010 as samples toestimate the parameters and calculate the VaRs of variousindex portfolios under different confidence levels

2 119866-119867 Distribution Copula Functionand Its Tail Dependence Index

21 119866-119867 Distribution

211 119866 Distribution Assuming that random variable 119885

obeys the standard normal distribution and 119892 is a realnumber then the random variable 119884119892 = 119866119892(119911) obeys 119866

distribution Consider

119866119892 (119911) =119890119892119911

minus 1119892

119911 sim 119873 (0 1) (1)

where 119892 controls the skewness of 119866 distribution When 119892 rarr

0 119866119892(119911) rarr 119911 and 119866 distribution tends to be symmetricWith the increase of the absolute value of 119892 the degree ofasymmetry increases Changing the sign of 119892 can changethe asymmetric direction of 119866 distribution but it does notchange its degree of asymmetry

212 119867 Distribution Assuming that random variable 119885

obeys the standard normal distribution and ℎ is a realnumber then the random variable 119884ℎ = 119867ℎ(119911) obeys 119867

distribution Consider

119867ℎ (119911) = 119890ℎ119911

22 119911 sim 119873 (0 1) (2)

119867 distribution stretches the tail of the standard normaldistribution ℎ controls the tail heaviness of 119867 distributionThe larger the ℎ is the heavier the tail is Because 119867ℎ(119911)

is an even function 119867 distribution is symmetric But theheaviness of its tail changes compared to the standard normaldistribution

213 119866-119867 Distribution The random variable 119884119892ℎ can beobtained by introducing both functions 119866119892(119911) and 119867ℎ(119911) torevise standard normal random variable 119885 Consider

119884119892ℎ = 119866119892 (119911)119867ℎ (119911) =119890119892119911

minus 1119892

119890ℎ119911

22 (3)

Then 119883119892ℎ can be obtained through linear transformation of119884119892ℎ Consider

119883119892ℎ = 119860+119861119890119892119911

minus 1119892

119890ℎ119911

22 119911 sim 119873 (0 1) (4)

The distribution of the random variable119883119892ℎ obeys the119866-119867 distribution 119860 119861 119892 and ℎ are real numbers 119892 describesthe asymmetry of 119866-119867 distribution and ℎ describes theheavy-tail characteristics of 119866-119867 distribution Obviously (3)is a special form of (4) The random variable 119884119892ℎ in (3)is the random variable of 119866-119867 distribution after centralstandardization

22 Copula Function and Its Tail Dependence Index Assum-ing that marginal distribution of random vector 119906119894 = 119865119894(119909119894)

(119894 = 1 2 119901) obeys uniformdistribution119880(0 1) accordingto the Sklar theorem the joint distribution function of 119875-dimensional random vectors 119865(1199091 119909119901) can be repre-sented as the following formula

119865 (1199091 119909119901) = 119862 (1198651 (1199091) 119865119901 (119909119901)) (5)

where 119862 is the Copula function of 119865 which is a hyper-cube [0 1]119901 multivariate density function defined on 119875-dimensional space R119901 If the marginal distribution is contin-uous there is a unique Copula function 119862 Then

119862 (1199061 119906119901) = 119865 (119865minus11 (1199061) 119865

minus1119901

(119906119901)) (6)

On the contrary given 119875-dimensional Copula func-tion 119862(1199061 119906119901) and its marginal distribution function1198651(1199091) 119865119901(119909119901) the density function of 119875-dimensionaljoint distribution function is

119891 (1199091 119909119901) = 119888 (1198651 (1199061) 119865119901 (119906119901))

119901

prod

119894=1119891119894 (119909119894) (7)

If 119891119894(119909119894) is the edge density 119888(1199061 119906119901) denotes Copuladensity derived from (6) Thus

119888 (1199061 119906119901) =

119891 (119865minus11 (1199061) 119865

minus1119901

(119906119901))

prod119901

119894=1119891119894 (119865minus1119894

(119906119894)) (8)

Since the joint distribution function of random vari-ables defines the correlation among its components Copulafunction determines the dependent structure among randomvariables uniquely The upper tail index 120582119906 and lower tailindex 120582119897 of tail dependence indicators can be defined asfollows

120582119906 = lim119902rarr 1

1 minus 2119902 + 119888 (119902 119902)

1 minus 119902

120582119897 = lim119902rarr 0

119888 (119902 119902)

119902

(9)

According to Nelsen [25] Gauss Copula function gen-erated by multivariate normal distribution function whosecorrelation matrix is R can be represented as follows

119862119866119906

R (1199061 119906119901)

= int

Φminus11 (1199061)

minusinfin

sdot sdot sdot int

Φminus1119901

(119906119901

)

minusinfin

1

radic(2120587)119901 |R|

expminusu1015840Rminus1u

2119889u

(10)

4 Mathematical Problems in Engineering

where u = (1199061 119906119901) and Φminus1 is the inverse function

of single normal distribution Because the Gauss Copulafunction does not have the characteristics of tail dependencewe often use the119879-Copula function whose degree of freedomis 120578 and correlation matrix is R to measure tail dependencestructure of risk asset in empirical analysis that is

119862119905

120578R (1199061 119906119901)

= int

119905minus1120578

(1199061)

minusinfin

sdot sdot sdot int

119905minus1120578

(119906119901

)

minusinfin

Γ ((120578 + 119901) 2) (1 + u1015840Rminus1u2)minus(120578+119901)2

Γ (1205782)radic(120587120578)119901|R|

119889u(11)

where 119905minus1120578

is the inverse function of simple standard Studentrsquos119905-distribution whose degree of freedom is 120578 When 120578 rarr infin119879-Copula function degenerates to Gauss Copula function Itstail index 120582119906 = 120582119897 = 0 that is the tail is independent The tailindex of 119879-Copula function is

120582119906 = 120582119897 = 2119905120578+1 (minus

radic(120578 + 1) (1 minus 120588)

radic1 + 120588) (12)

where 119905120578+1 is simple standard Studentrsquos 119905-distribution whosedegree of freedom is 120578 + 1 Considering that the innovationimpacts on the price of risk asset in varying degrees at differ-ent times 120578 and 120588 should have time-varying characteristicsFor this reason tail index also has the same characteristics

3 Multivariate Time-Varying 119866-119867 CopulaGARCH Model

Let r119905 = (1199031119905 119903119901119905) denote return time series of 119901 riskassets The prior information set before time 119905 is

I119905minus1 = r119905minus1 h119905minus1 r119905minus2 h119905minus2 =

119901

prod

119894=1I119894119905minus1 (13)

where I119894119905minus1 = 119903119894119905minus1 ℎ119894119905minus1 119903119894119905minus2 ℎ119894119905minus2 ℎ119894119905 is conditionalvolatility of 119903119894119905 about single asset prior information set I119894119905minus1Let 119862(sdot | I119905minus1) denote 119875-dimensional conditional Copulafunction and 119865119894(119903119894119905 | I119894119905minus1) be the conditional distributionof the 119894th component According to Sklar theorem theconditional joint distribution of 119901 risk assets return is

119865 (r119905 | I119905minus1)

= 119862 (1198651 (1199031119905 | I1119905minus1) 119865119901 (119903119901119905 | I119901119905minus1) | I119905minus1)

(14)

Numerous empirical studies show that the risk assetreturn series obey GARCH (1 1) model Based on thisassuming that 119903119894119905 satisfies the GARCH (1 1) model we canget the following 119866-119867 Copula GARCH (1 1) model whichdescribes the time-varying dependence structure of 119901 risk

assets return after filtering the time-varying characteristics ofsingle series

119903119894119905 = 120583119894 + 120576119894119905 119894 = 1 2 119901

120576119894119905 = radicℎ119894119905119911119894119905

ℎ119894119905 = 120603119894 +1205721198941205762119894119905minus1 +120573119894ℎ119894119905minus1

119865 (z119905 | I119905minus1)

= 119862 (1198651 (1199111119905 | I1119905minus1) 119865119901 (119911119901119905 | I119901119905minus1) | I119905minus1)

(15)

where the parameters satisfy the conditions 120603119894 120572119894 120573119894 gt 0and 120572119894 + 120573119894 lt 1 These parameters can ensure the stabilityof conditional volatility series The innovation series z119905obey 119866-119867 distribution whose parameter is (119892 ℎ) in (4) Butin order to simplify the analysis we only consider 119866-119867distribution after central standardization given by (3) and itsdensity function is written as 119891119884

119894

(119910119894) The Copula function119862(sdot | I119905minus1) is given by (11) and its density 119888(sdot) can berepresented as the following time-varying119879-Copula functionwhose degree of freedom is 120578

119888119905

120578120588119905

(1199061119905 119906119901119905)

=

119891119905

120578120588119905

(119891minus1V1 (1199061119905) 119891

minus1V119901

(119906119901119905))

prod119901

119894=1119891120578 (119891minus1V119894

(119906119894119905))

(16)

where 119891119905

120578120588119905

denotes the multivariate Studentrsquos 119905-distributionwhose degree of freedom is 120578 and time-varying correlationmatrix is 120588

119905= (120588119894119895119905)119901times119901 and

120588119894119894119905 = 1

119891V119894

(119906119894119905) =Γ ((V119894 + 1) 2)Γ (V1198942)radicV119894120587

(1+1199062119894119905

V119894)

minus(V119894

+1)2

(17)

The joint density function of 119901 risk assets return is

119891 (y | u h119905) = 119888119905

120578120588119905

(1199091119905 119909119901119905)

119901

prod

119894=1119891119884119894

(119910119894)

= Γ (120578 + 119901

2) Γ (

120578

2)

119901minus1(1+

x1015840119905120588119905x119905

120578)

minus(120578+119901)2

sdot (1003816100381610038161003816120588119905

1003816100381610038161003816)minus12119901

prod

119894=1(1 +

1199092119894119905

120578)

(120578+1)2 119901

prod

119894=1119891119884119894

(119910119894)

(18)

Mathematical Problems in Engineering 5

where x119905 = (1199091119905 119909119901119905) and 119909119894119905 = 119905minus1120578

(119905V119894

(120576119894119905)) Then thelikelihood function of overall samples is

119897 (120579 | y) =

119879

prod

119905=1Γ (

120578 + 119901

2) Γ (

120578

2)

119901minus1

sdot (1+x1015840119905120588119905x119905

120578)

minus(120578+119901)2

(1003816100381610038161003816120588119905

1003816100381610038161003816)minus12

Γ (120578 + 12

)

minus119901

sdot

119901

prod

119894=1(1+

1199092119894119905

120578)

(120578+1)2 119901

prod

119894=1119891119884119894

(119910119894)

(19)

where 120579 = (120583119894 120603119894 120572119894 120573119894 V119894)119901

119894=1 119886 119887120588119905 120578 and x119905 = (1199091119905 119909119901119905)The value of correlationmatrix 120588

119905is similar to the time-

varying correlation matrix of multivariate Copula GARCHmodel proposed by Jondeau and Rockinger [11] That is 120588

119905

satisfies the following evolution equation

120588119905= (1minus 119886minus 119887)120588+ 119886Ψ119905minus1 + 119887120588

119905minus1 (20)

where 0 le 119886 119887 le 1 119886 + 119887 le 1 120588 is a positive definite matrixwhose main diagonal elements are 1 and other elements arestatic correlation coefficientsΨ119905minus1 is a 119901times119901matrix in whichevery element

120595119894119895119905minus1 =sum119898

119897=1 119909119894119905minus119897119909119895119905minus119897

radicsum119898

119897=1 1199092119894119905minus119897

radicsum119898

119897=1 1199092119895119905minus119897

119894 119895 = 1 2 119901 (21)

denotes the correlation coefficients of 119901 risk asset returns(119898 ge 119901 + 2) 119909119905 = (1199091119905 119909119901119905) = (119905

minus1V1 (119891V1(1199111119905))

119905minus1V119901

(119891V119901

(119911119901119905))) Each element 120588119894119895119905 of 120588119905 satisfies minus1 le 120588119894119895119905 le 1

4 Parameter Estimation Algorithm ofthe Multivariate Time-Varying 119866-119867 CopulaGARCH Model

On the basis of Huggenberger and Klett [21] this section willuse dynamic correlation matrix 120588

119905instead of static correla-

tion matrix in the multidimensional discrete-time stochasticprocess to estimate the parameters of multivariate time-varying119866-119867CopulaGARCHmodel established in Section 3Assuming thatΘ denotes the parameter space defined by themodel and (r1 r2 r119879) denotes the log return samples of119875-dimensional risk asset which is generated by multivariateconditional density function119891120588

119905

|I119905minus1

(r119905 | I119905minus1 1205790) where 1205790 isin

Θ I119905minus1 is 120590 algebra of time 119905 minus 1 and before the maximumlikelihood estimation of parameter vector 120579 can be calculatedby the following equation

= argmax120579isinΘ

119879

sum

119905=1log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 120579) (22)

where 119891120588119905

|I119905minus1

(r119905 | I119905minus1 120579) can be obtained by calculating thederivative of (5) Let 119888120579 denote Copula density functionThus

119891120588119905

|I119905minus1

(1199031119905 119903119901119905 | I119905minus1 120579)

= 119888120579 (1198651119905 (1199031119905 120579) 119865119901119905 (119903119901119905 120579))

sdot

119901

prod

119894=1119891119894119905 (119903119905119894 120579)

(23)

The probability density function and distribution func-tion can be obtained in the process ofmodel built in Section 3Using the IFM method proposed by Joe [26] we can convert(22) into an optimization problem Therefore we need todivide the parameter vector 120579 into two subparameter vectors120579119888 and 120579119903 that is 120579 = (120579119888 120579119903) where 120579119903 = (1205791199031

120579119903119901

)120579119903119894

is the parameter vector of 119894th marginal distribution and120579119888 is the parameter vector of Copula function Because IFMmethod is a two-step likelihood estimation method themodel parameters should be estimated through the followingtwo steps

Step 1 Solving the maximum likelihood estimator of theparameter vector of each risk asset return

119903119894

= argmax120579119903

119894

119879

sum

119905=1log119891119894119905 (119903119894119905 | 120579119903

119894

) 119894 = 1 2 119901 (24)

Thismeans that we need to estimate parameters vector 119903119894

of 119901 distributions continuously

Step 2 Taking each 119903119894

into the likelihood equation (22) wecan obtain the parameter vector 120579119888 of Copula function and itsmaximum likelihood estimator 119888 Consider

119888

= argmax120579119888

119879

sum

119905=1log 119888120579

119888

(1198651119905 (10038161003816100381610038161199031119905

1003816100381610038161003816 1199031) 119865119901119905 (

10038161003816100381610038161003816119903119901119905

10038161003816100381610038161003816119903119901

))

(25)

In themaximum likelihood estimationweneed to use thederivative function of the density function of 119866-119867 marginaldistribution with respect to the component of parametervector Because the density function of 119866-119867 marginal distri-bution is very complex this paper uses the implicit functiondifferentiation rule to take its derivative The estimator 2119904of parameter vector 120579 obtained by the above-mentionedtwo-step method obeys normal distribution consistentlyand asymptotically under the standard regularity conditionsproposed inHuggenberger andKlett [21] Joe [26] andPatton[27] that is

radic119879(2119904119879 minus 1205790)119889

997888rarr119879rarrinfin

119873(0Ωminus1ΣΩ) (26)

6 Mathematical Problems in Engineering

where

Ω = minus119864(1205972

1205971205791205971205791015840log119891120588

119905

|I119905minus1

(120588119905| I119905minus1 1205790))

Σ = 119864(120597

120597120579(log119891120588

119905

|I119905minus1

(120588119905| I119905minus1 1205790))

sdot120597

120597120579(log119891120588

119905

|I119905minus1

(120588119905| I119905minus1 1205790))

1015840

)

(27)

Because the matrixes Σ and Ω can be estimated by theestimated parameter vector consistently

Ω119879 = minus119879minus1119879

sum

119905=1

1205972

1205971205791205971205791015840log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 2119904119879)

Σ119879 = 119879minus1119879

sum

119905=1

120597

120597120579(log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 2119904119879))

sdot120597

120597120579(log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 2119904119879))1015840

(28)

Thus (26) can be used to calculate the standard deviationof the estimator 2119904119879

5 VaR Algorithm Based on the MultivariateTime-Varying 119866-119867 Copula GARCH Model

After estimating the parameters of multivariate time-varying119866-119867 Copula GARCH model VaR of the risk portfolio canbe measured VaR of risk portfolio indicates the expectedmaximum losses of risk portfolio held by investors within agiven confidence level and a certain period of time Assumingthat r119905 = (1199031119905 119903119901119905) (119905 = 1 2 119879) are the return samplesof 119901 risk assets which satisfy themultivariate time-varying119866-119867 Copula GARCH (1 1)model in Section 3 andsum

119901

119894=1 120582119894119903119894119905 isthe portfolio of 119901 risk assets in which the weight of the riskasset 119894 is 120582119894 (119894 = 1 2 119901) that can be less than 0 because ofpermitting short-purchasing and short-selling the risk assetsand meet sum

119901

119894=1 120582119894 = 1 the VaR of risk portfolio underconfidence level 119902 at time 119905 should satisfy Pr(sum119901

119894=1 120582119894119903119894119905 le

VaR119905) = 119902 The confidence level 119902 can reflect the differentrisk preferences of investors or financial institutions to acertain extent Choosing a larger confidence level means thatinvestors or financial institutions have greater risk aversionand they hope to get a forecast result with larger probability

Although the conditional distributions of 1199031119879+1 1199032119879+1 119903119901119879+1 can be calculated through the known marginaldistributions it is very difficult to calculate quantile fromtime-varying Copula density function and it is adverse tomeasure and calculate the VaR of risk portfolio Thereforethis paper measures the dynamic risk of portfolio and its esti-mation value approximately through simulating119866-119867CopulaGARCH model Based on the parameters 120579(119899) of the samplethe return series of risk assets [119903

(119899119898)

11+119879 119903(119899119898)

1199011+119879] 119898 =

1 119872 and the one-step measurement and estimationvalues of VaR of their portfolios can be obtained through esti-mating (ℎ(119899)1119879+1 ℎ

(119899+1)119901119879+1) according to the volatility equation

Table 1 Moment estimation results of the daily log return of SSCIHSI TAIEX and SP500

Types of stock index Mean Std Skewness KurtosisSSCI 25078119890 minus 004 00181 minus02144 74467HSI 74505119890 minus 005 00177 minus02765 116866TAIEX 39511119890 minus 005 00140 minus09091 172823SP500 minus97173119890 minus 005 00148 minus03701 120177

of119866-119867 Copula GARCHmodel and calculating 120588(119899)

119879+1 by usingCopula dynamic evolution equation and then repeating thefollowing algorithm for 119872 times (119872 ge 3119901)

Step 1 It is simulating 119872 groups of random vectors[119906(119899119898)

1119879+1 119906(119899119898)

119901119879+1] according to the multivariate 119879-Copuladensity function whose degree of freedom is 120578(119899) and correla-tion matrix is 120588(119899)

119879+1

Step 2 Calculating 119903(119899119898)

119894119879+1 = 120583(119899)

119894+ 119911(119899119898)

119894119879+1radicℎ(119899)

119894119879+1 119894 = 1 119901

Step 3 Firstly one calculates the return rate of risk portfoliothat is equal to sum

119901

119894=1 120582119894119903(119899119898)

119894119879+1 119898 = 1 2 119872 Secondly oneevaluates its 119902-quantile VaR(119899)

119879+1 Thirdly one measures theVaR of the risk portfolio by VaR119879+1 = (1119873)sum

119873

119899=1 VaR(119899)

119879+1

6 Application of the Multivariate Time-Varying 119866-119867 Copula GARCH Model

61 Date Sample and Moment Estimation USA and Chinaas the most developed capitalism country and the largestdeveloping country in the world respectively rank top two ofthe world economy Their stock markets should have strongrepresentation in the world At the same time due to thehistorical reasons there exist several regions with differentpolitical systems such as Mainland China Hong KongTaiwan and Macau in Greater China Macau is similar toHong Kong on the whole For the above-mentioned reasonsthis paper selects the SSCI (China) HSI (HongKong China)TAIEX (Taiwan China) and SP500 (USA) from January3 2000 to June 18 2010 as data samples to estimate theVaR of various index portfolios under different confidencelevels by using the multivariate time-varying 119866-119867 CopulaGARCHmodelThe data comes fromYahoo Financewebsitehttpfinanceyahoocom

The moment estimation results of the daily log return ofSSCI HSI TAIEX and SP500 are shown in Table 1

Table 1 shows that the skewness of daily log returns ofSSCI HSI TAIEX and SP500 is less than 0 and their kurtosisis much larger than that of standard normal distributionwhich is equal to 3 These results demonstrate that thedaily log returns of these indices have the right skew andleptokurtic characteristics Therefore it is appropriate to fitthe daily log return of SSCI HSI TAIEX and SP500 byapplying 119866-119867 distribution which has leptokurtic heavy-tailcharacteristics and it is reasonable to apply the multivariate

Mathematical Problems in Engineering 7

Table 2 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on SSCI index risk asset

Model parameters 1205831

1205961

1205721

1205731

Estimate 000025lowastlowastlowast 00551lowastlowastlowast 00617lowastlowastlowast 08583lowastlowastlowast

119879-statistic 48653 46851 65764 44009Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 3 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on HSI index risk asset

Model parameters 1205832

1205962

1205722

1205732

Estimate 0000075lowastlowastlowast

00342lowastlowastlowast

00586lowastlowastlowast

08987lowastlowastlowast

119879-statistic 46539 48518 61796 66079Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 4 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on TAIEX index risk asset

Model parameters 1205833

1205963

1205723

1205733

Estimate 000040lowastlowastlowast 00343lowastlowastlowast

00517lowastlowastlowast

09054lowastlowastlowast

119879-statistic minus64538 43523 59817 65935Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 5 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on SP500 index risk asset

Model parameters 1205834

1205964

1205724

1205734

Estimate minus00001lowastlowastlowast

00251lowastlowastlowast

00861lowastlowastlowast

08738lowastlowastlowast

119879-statistic minus33873 52684 76324 63786Note lowastlowastlowastin the table denotes that the parameter is significant at 1 level

time-varying 119866-119867 Copula GARCH model to measure theirVaR

62 Parameter Estimates of the Multivariate Time-Varying119866-119867 Copula GARCH Model Based on the parameter esti-mation algorithm proposed in Section 4 the parameters ofthe multivariate time-varying 119866-119867 Copula GARCH modelwith SSCI HSI TAIEX and SP500 can be estimated Theparameter estimation results are shown in Tables 2 3 4 5and 6

From Tables 2 to 6 the following results can be obtained

(1) Consider 1205721 + 1205731 = 092 1205722 + 1205732 = 09573 1205723 +

1205733 = 09571 and 1205724 + 1205734 = 09599 This shows thatthe volatility persistence of Shanghai stock market isthe strongest Taiwan and Hong Kong stock marketrank second and third and the volatility persistenceof USA stock market is minimum It indicates thatthe investorsrsquo expectation of risk compensation inthe emerging markets represented by Chinarsquos stockmarket is stronger than that in the mature marketsrepresented by the USArsquos stock market and the pricediscovery efficiency of innovation in the emergingmarkets represented by Chinarsquos stock market is lowerthan that in the mature markets represented by USArsquosstockmarket In addition the sumof the coefficients120572

and120573 is very close to 1 which indicates that the impactand shock of innovation on the index volatility of eachstock market has a long memory

(2) The degree of freedom 120578 = 1457 and the correlationcoefficients 120588119894119895 of 119879-Copula in Table 6 show thatthere exists the strongest correlation between HongKong stock market and Taiwan stock market andthe correlation between Shanghai stock market andHong Kong stock market is also relatively largeThe above-mentioned facts indicate that the extremeevents probably result in the phenomena that HongKong stock market and Taiwan stock market are upand down synchronously and there exist comovingbehaviors between Shanghai stock market and HongKong stock market

(3) The time-varying coefficient 119887 = 0987 indicatesthat the time-varying correlation coefficient of 119866-119867 Copula GARCH model has a long memory thatis the impact of historical values of each otherrsquoscorrelation coefficient among SSCI HSI TAIEX andSP500 on the expected correlation is relatively large

63 VaR Measurement Based on the Multivariate Time-Varying 119866-119867 Copula GARCH Model Based on the multi-variate time-varying119866-119867 Copula GARCHmodel with SSCIHSI TAIEX and SP500 whose parameters have been esti-mated the VaRs of various index portfolios under differentconfidence levels can be measured The measurement resultsare shown in Table 7

From Table 7 the following results can be obtained

(1) The inequalities VaR (SSCI) lt VaR (HSI) lt VaR(SP500) lt VaR (TAIEX) can be satisfied for anyconfidence level It shows that the risk of extremelosses in Shanghai stock market is higher than that inHong Kong stock market Taiwan stock market andUSA stock market This measurement result is in linewith the actual situation that thematurity of Shanghaistockmarket is far lower than that ofHongKong stockmarket Taiwan stock market and USA stock market

(2) For any confidence level the extreme losses risk ofthe investors who equally allocate their total assetsamong SSCI HSI TAIEX and SP500 is lower thanthat of the investors who put their total assets into oneindex asset The extreme losses risk of the investorsincreases with the concentration of risk asset in theindex portfolios This measurement result is consis-tent with the risk diversification theory of portfolio

7 Conclusion

Considering the ldquoasymmetric leptokurtic and heavy-tailrdquocharacteristics the time-varying volatility characteristicsand extreme-tail dependence characteristics of financialasset return this paper combined the 119866-119867 distributionCopula function and GARCH model to construct a mul-tivariate time-varying 119866-119867 Copula GARCH model whichcan comprehensively describe the ldquoasymmetric leptokurtic

8 Mathematical Problems in Engineering

Table 6 Estimates of correlation coefficients time-varying parameters and degree of freedom of four-variate time-varying G-H CopulaGARCH (1 1) model based on SSCI HSI TAIEX and SP500

Model parameters 12058812

12058813

12058814

12058823

12058824

12058834

a b 120578

Estimate 0256lowastlowastlowast

0139lowastlowastlowast

0021lowastlowastlowast

0528lowastlowastlowast

0193lowastlowastlowast

0128lowastlowast

0008lowastlowastlowast

0987lowastlowastlowast

1457lowastlowastlowast

119879-statistic 1351 1102 3431 5763 9564 2529 5179 3561 6871Note lowastlowastlowast and lowastlowast in the table denote that the parameter is significant at 1 and 5 level respectively

Table 7 VaR estimation results based on the four-variate time-varying G-H Copula GARCHmodel with SSCI HSI TAIEX and SP500

Ratio of index portfolio 1 5 10 15 Ratio of index portfolio 1 5 10 15025 025 025 025 minus00303 minus00165 minus00116 minus00086 025 025 025 025 minus00303 minus00165 minus00116 minus00086050 000 025 025 minus00329 minus00177 minus00128 minus00093 025 050 000 025 minus00361 minus00189 minus00132 minus00100

075 000 000 025 minus00418 minus00226 minus00157 minus00116 025 075 000 000 minus00426 minus00236 minus00165 minus00123

100 000 000 000 minus00536 minus00285 minus00199 minus00150 000 100 000 000 minus00491 minus00269 minus00190 minus00138

025 025 025 025 minus00303 minus00165 minus00116 minus00086 025 025 025 025 minus00303 minus00165 minus00116 minus00086025 025 050 000 minus0332 minus00176 minus00119 minus00093 000 025 050 050 minus00359 minus00181 minus00128 minus00094

000 025 075 000 minus00403 minus00196 minus00127 minus00097 000 000 025 075 minus00402 minus00206 minus00146 minus00108

000 000 100 000 minus00420 minus00211 minus00139 minus00101 000 000 000 100 minus00425 minus00230 minus00157 minus00119

Note the ratio of index portfolio is ranked by the sequence of SSCI HSI TAIEX SP500 in Table 7

and heavy-tailrdquo characteristics the time-varying volatilitycharacteristics and extreme-tail dependence characteristicsof financial asset return It proposed the parameter estimationalgorithm of the multivariate time-varying 119866-119867 CopulaGARCH model by using condition maximum likelihoodmethod and IFM two-step method An algorithm was con-structed to calculate VaR by using the quantile functionand the simulation method based on 119866-119867 Copula GARCHmodel In addition this paper selected the daily log returnof SSCI (China) HSI (Hong Kong China) TAIEX (TaiwanChina) and SP500 (USA) from January 3 2000 to June18 2010 as samples to estimate the parameters of themultivariate time-varying 119866-119867 Copula GARCH model andit also estimated theVaR for various index risk asset portfoliosunder different confidence levelsThe research results showedthat the multivariate time-varying 119866-119867 Copula GARCHmodel constructed in this paper could reasonably estimateand measure the extreme losses of risk portfolios in financialmarket and the measurement results were in line with theactual situation of stock market and the risk diversificationtheory of portfolio The achievement of this paper provideda practical and effective method for measuring the extremelosses of financial market

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees for all help-ful comments and suggestions This work is supported byNatural Science Foundation Project of CQ CSTC (Grant noCSTC 2011BB2088) and Key Project of National NaturalScience Foundation (Grant no 71232004)

References

[1] S J Kon ldquoModels of stock returnsmdasha comparisonrdquoThe Journalof Finance vol 39 no 1 pp 147ndash165 1984

[2] J B Gray and D W French ldquoEmpirical comparisons of dis-tributional models for stock index returnsrdquo Journal of BusinessFinance amp Accounting vol 17 no 3 pp 451ndash459 1990

[3] E Jondeau andMRockinger ldquoTesting for differences in the tailsof stock-market returnsrdquo Journal of Empirical Finance vol 10no 5 pp 559ndash581 2003

[4] C Jiang S Li and S Liang ldquoEmpirical investigation ofdistribution feature of return in china stockmarketrdquoApplicationof Statistics and Management vol 26 no 4 pp 710ndash717 2007

[5] D Dong and W Jin ldquoA subjective model of the distribution ofreturns and empirical analysisrdquo Chinese Journal of ManagementScience vol 15 no 1 pp 112ndash120 2007

[6] J Martinez and B Iglewicz ldquoSome properties of the tukey g andh family of distributionsrdquoCommunications in StatisticsmdashTheoryand Methods vol 13 no 3 pp 353ndash369 1984

[7] T C Mills ldquoModelling skewness and kurtosis in the Londonstock exchange FT-SE index return distributionsrdquo The Statisti-cian vol 44 no 3 pp 323ndash332 1995

[8] H Zhu and Z Pan ldquoA study of portfolios VaRmethod based ong-h distributionrdquo Chinese Journal of Management Science vol13 no 4 pp 7ndash12 2005

[9] K Kuester S Mittnik andM S Paolella ldquoValue-at-risk predic-tion a comparison of alternative strategiesrdquo Journal of FinancialEconometrics vol 4 no 1 pp 53ndash89 2006

[10] M Degen P Embrechts and D Lambrigger ldquoThe quantitativemodeling of operational risk between g-and-h and EVTrdquo AstinBulletin vol 37 no 2 pp 265ndash291 2007

[11] E Jondeau and M Rockinger ldquoThe Copula-GARCH modelof conditional dependencies an international stock marketapplicationrdquo Journal of International Money and Finance vol25 no 5 pp 827ndash853 2006

[12] J C Rodriguez ldquoMeasuring financial contagion a Copulaapproachrdquo Journal of Empirical Finance vol 14 no 3 pp 401ndash423 2007

Mathematical Problems in Engineering 9

[13] M Fischer C Kock S Schluter and F Weigert ldquoAn empiricalanalysis of multivariate copula modelsrdquo Quantitative Financevol 9 no 7 pp 839ndash854 2009

[14] W Sun S Rachev F J Fabozzi and S S Kalev ldquoA newapproach to modeling co-movement of international equitymarkets evidence of unconditional copula-based simulation oftail dependencerdquo Empirical Economics vol 36 no 1 pp 201ndash229 2009

[15] Z Liu T Zhang and F Wen ldquoTime-varying risk attitude andconditional skewnessrdquo Abstract and Applied Analysis vol 2014Article ID 174848 11 pages 2014

[16] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 3 no 48 pp235ndash254 2014

[17] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[18] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics and Economics vol 45 no 3 pp 315ndash324 2009

[19] A Ghorbel and A Trabelsi ldquoMeasure of financial risk usingconditional extreme value copulas with EVT marginsrdquo Journalof Risk vol 11 no 4 pp 51ndash85 2009

[20] L Chollete A Heinen and A Valdesogo ldquoModeling interna-tional financial returns with a multivariate regime-switchingcopulardquo Journal of Financial Econometrics vol 7 no 4 pp 437ndash480 2009

[21] M Huggenberger and T Klett ldquoA g-and-h copula approach torisk measurement in multivariate financial modelsrdquo WorkingPaper 2010 httpssrncomabstract=1677431

[22] J Wang W Bao and J Hu ldquoPortfolios VaR analyses basedon the multi-dimensional gumbel copula functionrdquo Journal ofApplied Statistics and Management vol 29 no 1 pp 137ndash1432010

[23] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis and Applications vol 16 no 1 pp93ndash103 2014

[24] J Liu M Tao C Ma and F Wen ldquoUtility indifference pricingof convertible bondsrdquo International Journal of InformationTechnology and Decision Making vol 13 no 2 pp 429ndash4442014

[25] R B Nelsen An Introduction to Copulas Springer Science ampBusiness Media New York NY USA 2007

[26] H JoeMultivariate Models and Dependence Concepts vol 73 ofMonographs on Statistics and Applied Probability Chapman ampHall London UK 1997

[27] A J Patton ldquoEstimation of multivariate models for time seriesof possibly different lengthsrdquo Journal of Applied Econometricsvol 21 no 2 pp 147ndash173 2006

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article Multivariate Time-Varying Copula GARCH ...downloads.hindawi.com/journals/mpe/2015/286014.pdf · model, they presented an empirical pricing study of China s market.

4 Mathematical Problems in Engineering

where u = (1199061 119906119901) and Φminus1 is the inverse function

of single normal distribution Because the Gauss Copulafunction does not have the characteristics of tail dependencewe often use the119879-Copula function whose degree of freedomis 120578 and correlation matrix is R to measure tail dependencestructure of risk asset in empirical analysis that is

119862119905

120578R (1199061 119906119901)

= int

119905minus1120578

(1199061)

minusinfin

sdot sdot sdot int

119905minus1120578

(119906119901

)

minusinfin

Γ ((120578 + 119901) 2) (1 + u1015840Rminus1u2)minus(120578+119901)2

Γ (1205782)radic(120587120578)119901|R|

119889u(11)

where 119905minus1120578

is the inverse function of simple standard Studentrsquos119905-distribution whose degree of freedom is 120578 When 120578 rarr infin119879-Copula function degenerates to Gauss Copula function Itstail index 120582119906 = 120582119897 = 0 that is the tail is independent The tailindex of 119879-Copula function is

120582119906 = 120582119897 = 2119905120578+1 (minus

radic(120578 + 1) (1 minus 120588)

radic1 + 120588) (12)

where 119905120578+1 is simple standard Studentrsquos 119905-distribution whosedegree of freedom is 120578 + 1 Considering that the innovationimpacts on the price of risk asset in varying degrees at differ-ent times 120578 and 120588 should have time-varying characteristicsFor this reason tail index also has the same characteristics

3 Multivariate Time-Varying 119866-119867 CopulaGARCH Model

Let r119905 = (1199031119905 119903119901119905) denote return time series of 119901 riskassets The prior information set before time 119905 is

I119905minus1 = r119905minus1 h119905minus1 r119905minus2 h119905minus2 =

119901

prod

119894=1I119894119905minus1 (13)

where I119894119905minus1 = 119903119894119905minus1 ℎ119894119905minus1 119903119894119905minus2 ℎ119894119905minus2 ℎ119894119905 is conditionalvolatility of 119903119894119905 about single asset prior information set I119894119905minus1Let 119862(sdot | I119905minus1) denote 119875-dimensional conditional Copulafunction and 119865119894(119903119894119905 | I119894119905minus1) be the conditional distributionof the 119894th component According to Sklar theorem theconditional joint distribution of 119901 risk assets return is

119865 (r119905 | I119905minus1)

= 119862 (1198651 (1199031119905 | I1119905minus1) 119865119901 (119903119901119905 | I119901119905minus1) | I119905minus1)

(14)

Numerous empirical studies show that the risk assetreturn series obey GARCH (1 1) model Based on thisassuming that 119903119894119905 satisfies the GARCH (1 1) model we canget the following 119866-119867 Copula GARCH (1 1) model whichdescribes the time-varying dependence structure of 119901 risk

assets return after filtering the time-varying characteristics ofsingle series

119903119894119905 = 120583119894 + 120576119894119905 119894 = 1 2 119901

120576119894119905 = radicℎ119894119905119911119894119905

ℎ119894119905 = 120603119894 +1205721198941205762119894119905minus1 +120573119894ℎ119894119905minus1

119865 (z119905 | I119905minus1)

= 119862 (1198651 (1199111119905 | I1119905minus1) 119865119901 (119911119901119905 | I119901119905minus1) | I119905minus1)

(15)

where the parameters satisfy the conditions 120603119894 120572119894 120573119894 gt 0and 120572119894 + 120573119894 lt 1 These parameters can ensure the stabilityof conditional volatility series The innovation series z119905obey 119866-119867 distribution whose parameter is (119892 ℎ) in (4) Butin order to simplify the analysis we only consider 119866-119867distribution after central standardization given by (3) and itsdensity function is written as 119891119884

119894

(119910119894) The Copula function119862(sdot | I119905minus1) is given by (11) and its density 119888(sdot) can berepresented as the following time-varying119879-Copula functionwhose degree of freedom is 120578

119888119905

120578120588119905

(1199061119905 119906119901119905)

=

119891119905

120578120588119905

(119891minus1V1 (1199061119905) 119891

minus1V119901

(119906119901119905))

prod119901

119894=1119891120578 (119891minus1V119894

(119906119894119905))

(16)

where 119891119905

120578120588119905

denotes the multivariate Studentrsquos 119905-distributionwhose degree of freedom is 120578 and time-varying correlationmatrix is 120588

119905= (120588119894119895119905)119901times119901 and

120588119894119894119905 = 1

119891V119894

(119906119894119905) =Γ ((V119894 + 1) 2)Γ (V1198942)radicV119894120587

(1+1199062119894119905

V119894)

minus(V119894

+1)2

(17)

The joint density function of 119901 risk assets return is

119891 (y | u h119905) = 119888119905

120578120588119905

(1199091119905 119909119901119905)

119901

prod

119894=1119891119884119894

(119910119894)

= Γ (120578 + 119901

2) Γ (

120578

2)

119901minus1(1+

x1015840119905120588119905x119905

120578)

minus(120578+119901)2

sdot (1003816100381610038161003816120588119905

1003816100381610038161003816)minus12119901

prod

119894=1(1 +

1199092119894119905

120578)

(120578+1)2 119901

prod

119894=1119891119884119894

(119910119894)

(18)

Mathematical Problems in Engineering 5

where x119905 = (1199091119905 119909119901119905) and 119909119894119905 = 119905minus1120578

(119905V119894

(120576119894119905)) Then thelikelihood function of overall samples is

119897 (120579 | y) =

119879

prod

119905=1Γ (

120578 + 119901

2) Γ (

120578

2)

119901minus1

sdot (1+x1015840119905120588119905x119905

120578)

minus(120578+119901)2

(1003816100381610038161003816120588119905

1003816100381610038161003816)minus12

Γ (120578 + 12

)

minus119901

sdot

119901

prod

119894=1(1+

1199092119894119905

120578)

(120578+1)2 119901

prod

119894=1119891119884119894

(119910119894)

(19)

where 120579 = (120583119894 120603119894 120572119894 120573119894 V119894)119901

119894=1 119886 119887120588119905 120578 and x119905 = (1199091119905 119909119901119905)The value of correlationmatrix 120588

119905is similar to the time-

varying correlation matrix of multivariate Copula GARCHmodel proposed by Jondeau and Rockinger [11] That is 120588

119905

satisfies the following evolution equation

120588119905= (1minus 119886minus 119887)120588+ 119886Ψ119905minus1 + 119887120588

119905minus1 (20)

where 0 le 119886 119887 le 1 119886 + 119887 le 1 120588 is a positive definite matrixwhose main diagonal elements are 1 and other elements arestatic correlation coefficientsΨ119905minus1 is a 119901times119901matrix in whichevery element

120595119894119895119905minus1 =sum119898

119897=1 119909119894119905minus119897119909119895119905minus119897

radicsum119898

119897=1 1199092119894119905minus119897

radicsum119898

119897=1 1199092119895119905minus119897

119894 119895 = 1 2 119901 (21)

denotes the correlation coefficients of 119901 risk asset returns(119898 ge 119901 + 2) 119909119905 = (1199091119905 119909119901119905) = (119905

minus1V1 (119891V1(1199111119905))

119905minus1V119901

(119891V119901

(119911119901119905))) Each element 120588119894119895119905 of 120588119905 satisfies minus1 le 120588119894119895119905 le 1

4 Parameter Estimation Algorithm ofthe Multivariate Time-Varying 119866-119867 CopulaGARCH Model

On the basis of Huggenberger and Klett [21] this section willuse dynamic correlation matrix 120588

119905instead of static correla-

tion matrix in the multidimensional discrete-time stochasticprocess to estimate the parameters of multivariate time-varying119866-119867CopulaGARCHmodel established in Section 3Assuming thatΘ denotes the parameter space defined by themodel and (r1 r2 r119879) denotes the log return samples of119875-dimensional risk asset which is generated by multivariateconditional density function119891120588

119905

|I119905minus1

(r119905 | I119905minus1 1205790) where 1205790 isin

Θ I119905minus1 is 120590 algebra of time 119905 minus 1 and before the maximumlikelihood estimation of parameter vector 120579 can be calculatedby the following equation

= argmax120579isinΘ

119879

sum

119905=1log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 120579) (22)

where 119891120588119905

|I119905minus1

(r119905 | I119905minus1 120579) can be obtained by calculating thederivative of (5) Let 119888120579 denote Copula density functionThus

119891120588119905

|I119905minus1

(1199031119905 119903119901119905 | I119905minus1 120579)

= 119888120579 (1198651119905 (1199031119905 120579) 119865119901119905 (119903119901119905 120579))

sdot

119901

prod

119894=1119891119894119905 (119903119905119894 120579)

(23)

The probability density function and distribution func-tion can be obtained in the process ofmodel built in Section 3Using the IFM method proposed by Joe [26] we can convert(22) into an optimization problem Therefore we need todivide the parameter vector 120579 into two subparameter vectors120579119888 and 120579119903 that is 120579 = (120579119888 120579119903) where 120579119903 = (1205791199031

120579119903119901

)120579119903119894

is the parameter vector of 119894th marginal distribution and120579119888 is the parameter vector of Copula function Because IFMmethod is a two-step likelihood estimation method themodel parameters should be estimated through the followingtwo steps

Step 1 Solving the maximum likelihood estimator of theparameter vector of each risk asset return

119903119894

= argmax120579119903

119894

119879

sum

119905=1log119891119894119905 (119903119894119905 | 120579119903

119894

) 119894 = 1 2 119901 (24)

Thismeans that we need to estimate parameters vector 119903119894

of 119901 distributions continuously

Step 2 Taking each 119903119894

into the likelihood equation (22) wecan obtain the parameter vector 120579119888 of Copula function and itsmaximum likelihood estimator 119888 Consider

119888

= argmax120579119888

119879

sum

119905=1log 119888120579

119888

(1198651119905 (10038161003816100381610038161199031119905

1003816100381610038161003816 1199031) 119865119901119905 (

10038161003816100381610038161003816119903119901119905

10038161003816100381610038161003816119903119901

))

(25)

In themaximum likelihood estimationweneed to use thederivative function of the density function of 119866-119867 marginaldistribution with respect to the component of parametervector Because the density function of 119866-119867 marginal distri-bution is very complex this paper uses the implicit functiondifferentiation rule to take its derivative The estimator 2119904of parameter vector 120579 obtained by the above-mentionedtwo-step method obeys normal distribution consistentlyand asymptotically under the standard regularity conditionsproposed inHuggenberger andKlett [21] Joe [26] andPatton[27] that is

radic119879(2119904119879 minus 1205790)119889

997888rarr119879rarrinfin

119873(0Ωminus1ΣΩ) (26)

6 Mathematical Problems in Engineering

where

Ω = minus119864(1205972

1205971205791205971205791015840log119891120588

119905

|I119905minus1

(120588119905| I119905minus1 1205790))

Σ = 119864(120597

120597120579(log119891120588

119905

|I119905minus1

(120588119905| I119905minus1 1205790))

sdot120597

120597120579(log119891120588

119905

|I119905minus1

(120588119905| I119905minus1 1205790))

1015840

)

(27)

Because the matrixes Σ and Ω can be estimated by theestimated parameter vector consistently

Ω119879 = minus119879minus1119879

sum

119905=1

1205972

1205971205791205971205791015840log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 2119904119879)

Σ119879 = 119879minus1119879

sum

119905=1

120597

120597120579(log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 2119904119879))

sdot120597

120597120579(log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 2119904119879))1015840

(28)

Thus (26) can be used to calculate the standard deviationof the estimator 2119904119879

5 VaR Algorithm Based on the MultivariateTime-Varying 119866-119867 Copula GARCH Model

After estimating the parameters of multivariate time-varying119866-119867 Copula GARCH model VaR of the risk portfolio canbe measured VaR of risk portfolio indicates the expectedmaximum losses of risk portfolio held by investors within agiven confidence level and a certain period of time Assumingthat r119905 = (1199031119905 119903119901119905) (119905 = 1 2 119879) are the return samplesof 119901 risk assets which satisfy themultivariate time-varying119866-119867 Copula GARCH (1 1)model in Section 3 andsum

119901

119894=1 120582119894119903119894119905 isthe portfolio of 119901 risk assets in which the weight of the riskasset 119894 is 120582119894 (119894 = 1 2 119901) that can be less than 0 because ofpermitting short-purchasing and short-selling the risk assetsand meet sum

119901

119894=1 120582119894 = 1 the VaR of risk portfolio underconfidence level 119902 at time 119905 should satisfy Pr(sum119901

119894=1 120582119894119903119894119905 le

VaR119905) = 119902 The confidence level 119902 can reflect the differentrisk preferences of investors or financial institutions to acertain extent Choosing a larger confidence level means thatinvestors or financial institutions have greater risk aversionand they hope to get a forecast result with larger probability

Although the conditional distributions of 1199031119879+1 1199032119879+1 119903119901119879+1 can be calculated through the known marginaldistributions it is very difficult to calculate quantile fromtime-varying Copula density function and it is adverse tomeasure and calculate the VaR of risk portfolio Thereforethis paper measures the dynamic risk of portfolio and its esti-mation value approximately through simulating119866-119867CopulaGARCH model Based on the parameters 120579(119899) of the samplethe return series of risk assets [119903

(119899119898)

11+119879 119903(119899119898)

1199011+119879] 119898 =

1 119872 and the one-step measurement and estimationvalues of VaR of their portfolios can be obtained through esti-mating (ℎ(119899)1119879+1 ℎ

(119899+1)119901119879+1) according to the volatility equation

Table 1 Moment estimation results of the daily log return of SSCIHSI TAIEX and SP500

Types of stock index Mean Std Skewness KurtosisSSCI 25078119890 minus 004 00181 minus02144 74467HSI 74505119890 minus 005 00177 minus02765 116866TAIEX 39511119890 minus 005 00140 minus09091 172823SP500 minus97173119890 minus 005 00148 minus03701 120177

of119866-119867 Copula GARCHmodel and calculating 120588(119899)

119879+1 by usingCopula dynamic evolution equation and then repeating thefollowing algorithm for 119872 times (119872 ge 3119901)

Step 1 It is simulating 119872 groups of random vectors[119906(119899119898)

1119879+1 119906(119899119898)

119901119879+1] according to the multivariate 119879-Copuladensity function whose degree of freedom is 120578(119899) and correla-tion matrix is 120588(119899)

119879+1

Step 2 Calculating 119903(119899119898)

119894119879+1 = 120583(119899)

119894+ 119911(119899119898)

119894119879+1radicℎ(119899)

119894119879+1 119894 = 1 119901

Step 3 Firstly one calculates the return rate of risk portfoliothat is equal to sum

119901

119894=1 120582119894119903(119899119898)

119894119879+1 119898 = 1 2 119872 Secondly oneevaluates its 119902-quantile VaR(119899)

119879+1 Thirdly one measures theVaR of the risk portfolio by VaR119879+1 = (1119873)sum

119873

119899=1 VaR(119899)

119879+1

6 Application of the Multivariate Time-Varying 119866-119867 Copula GARCH Model

61 Date Sample and Moment Estimation USA and Chinaas the most developed capitalism country and the largestdeveloping country in the world respectively rank top two ofthe world economy Their stock markets should have strongrepresentation in the world At the same time due to thehistorical reasons there exist several regions with differentpolitical systems such as Mainland China Hong KongTaiwan and Macau in Greater China Macau is similar toHong Kong on the whole For the above-mentioned reasonsthis paper selects the SSCI (China) HSI (HongKong China)TAIEX (Taiwan China) and SP500 (USA) from January3 2000 to June 18 2010 as data samples to estimate theVaR of various index portfolios under different confidencelevels by using the multivariate time-varying 119866-119867 CopulaGARCHmodelThe data comes fromYahoo Financewebsitehttpfinanceyahoocom

The moment estimation results of the daily log return ofSSCI HSI TAIEX and SP500 are shown in Table 1

Table 1 shows that the skewness of daily log returns ofSSCI HSI TAIEX and SP500 is less than 0 and their kurtosisis much larger than that of standard normal distributionwhich is equal to 3 These results demonstrate that thedaily log returns of these indices have the right skew andleptokurtic characteristics Therefore it is appropriate to fitthe daily log return of SSCI HSI TAIEX and SP500 byapplying 119866-119867 distribution which has leptokurtic heavy-tailcharacteristics and it is reasonable to apply the multivariate

Mathematical Problems in Engineering 7

Table 2 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on SSCI index risk asset

Model parameters 1205831

1205961

1205721

1205731

Estimate 000025lowastlowastlowast 00551lowastlowastlowast 00617lowastlowastlowast 08583lowastlowastlowast

119879-statistic 48653 46851 65764 44009Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 3 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on HSI index risk asset

Model parameters 1205832

1205962

1205722

1205732

Estimate 0000075lowastlowastlowast

00342lowastlowastlowast

00586lowastlowastlowast

08987lowastlowastlowast

119879-statistic 46539 48518 61796 66079Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 4 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on TAIEX index risk asset

Model parameters 1205833

1205963

1205723

1205733

Estimate 000040lowastlowastlowast 00343lowastlowastlowast

00517lowastlowastlowast

09054lowastlowastlowast

119879-statistic minus64538 43523 59817 65935Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 5 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on SP500 index risk asset

Model parameters 1205834

1205964

1205724

1205734

Estimate minus00001lowastlowastlowast

00251lowastlowastlowast

00861lowastlowastlowast

08738lowastlowastlowast

119879-statistic minus33873 52684 76324 63786Note lowastlowastlowastin the table denotes that the parameter is significant at 1 level

time-varying 119866-119867 Copula GARCH model to measure theirVaR

62 Parameter Estimates of the Multivariate Time-Varying119866-119867 Copula GARCH Model Based on the parameter esti-mation algorithm proposed in Section 4 the parameters ofthe multivariate time-varying 119866-119867 Copula GARCH modelwith SSCI HSI TAIEX and SP500 can be estimated Theparameter estimation results are shown in Tables 2 3 4 5and 6

From Tables 2 to 6 the following results can be obtained

(1) Consider 1205721 + 1205731 = 092 1205722 + 1205732 = 09573 1205723 +

1205733 = 09571 and 1205724 + 1205734 = 09599 This shows thatthe volatility persistence of Shanghai stock market isthe strongest Taiwan and Hong Kong stock marketrank second and third and the volatility persistenceof USA stock market is minimum It indicates thatthe investorsrsquo expectation of risk compensation inthe emerging markets represented by Chinarsquos stockmarket is stronger than that in the mature marketsrepresented by the USArsquos stock market and the pricediscovery efficiency of innovation in the emergingmarkets represented by Chinarsquos stock market is lowerthan that in the mature markets represented by USArsquosstockmarket In addition the sumof the coefficients120572

and120573 is very close to 1 which indicates that the impactand shock of innovation on the index volatility of eachstock market has a long memory

(2) The degree of freedom 120578 = 1457 and the correlationcoefficients 120588119894119895 of 119879-Copula in Table 6 show thatthere exists the strongest correlation between HongKong stock market and Taiwan stock market andthe correlation between Shanghai stock market andHong Kong stock market is also relatively largeThe above-mentioned facts indicate that the extremeevents probably result in the phenomena that HongKong stock market and Taiwan stock market are upand down synchronously and there exist comovingbehaviors between Shanghai stock market and HongKong stock market

(3) The time-varying coefficient 119887 = 0987 indicatesthat the time-varying correlation coefficient of 119866-119867 Copula GARCH model has a long memory thatis the impact of historical values of each otherrsquoscorrelation coefficient among SSCI HSI TAIEX andSP500 on the expected correlation is relatively large

63 VaR Measurement Based on the Multivariate Time-Varying 119866-119867 Copula GARCH Model Based on the multi-variate time-varying119866-119867 Copula GARCHmodel with SSCIHSI TAIEX and SP500 whose parameters have been esti-mated the VaRs of various index portfolios under differentconfidence levels can be measured The measurement resultsare shown in Table 7

From Table 7 the following results can be obtained

(1) The inequalities VaR (SSCI) lt VaR (HSI) lt VaR(SP500) lt VaR (TAIEX) can be satisfied for anyconfidence level It shows that the risk of extremelosses in Shanghai stock market is higher than that inHong Kong stock market Taiwan stock market andUSA stock market This measurement result is in linewith the actual situation that thematurity of Shanghaistockmarket is far lower than that ofHongKong stockmarket Taiwan stock market and USA stock market

(2) For any confidence level the extreme losses risk ofthe investors who equally allocate their total assetsamong SSCI HSI TAIEX and SP500 is lower thanthat of the investors who put their total assets into oneindex asset The extreme losses risk of the investorsincreases with the concentration of risk asset in theindex portfolios This measurement result is consis-tent with the risk diversification theory of portfolio

7 Conclusion

Considering the ldquoasymmetric leptokurtic and heavy-tailrdquocharacteristics the time-varying volatility characteristicsand extreme-tail dependence characteristics of financialasset return this paper combined the 119866-119867 distributionCopula function and GARCH model to construct a mul-tivariate time-varying 119866-119867 Copula GARCH model whichcan comprehensively describe the ldquoasymmetric leptokurtic

8 Mathematical Problems in Engineering

Table 6 Estimates of correlation coefficients time-varying parameters and degree of freedom of four-variate time-varying G-H CopulaGARCH (1 1) model based on SSCI HSI TAIEX and SP500

Model parameters 12058812

12058813

12058814

12058823

12058824

12058834

a b 120578

Estimate 0256lowastlowastlowast

0139lowastlowastlowast

0021lowastlowastlowast

0528lowastlowastlowast

0193lowastlowastlowast

0128lowastlowast

0008lowastlowastlowast

0987lowastlowastlowast

1457lowastlowastlowast

119879-statistic 1351 1102 3431 5763 9564 2529 5179 3561 6871Note lowastlowastlowast and lowastlowast in the table denote that the parameter is significant at 1 and 5 level respectively

Table 7 VaR estimation results based on the four-variate time-varying G-H Copula GARCHmodel with SSCI HSI TAIEX and SP500

Ratio of index portfolio 1 5 10 15 Ratio of index portfolio 1 5 10 15025 025 025 025 minus00303 minus00165 minus00116 minus00086 025 025 025 025 minus00303 minus00165 minus00116 minus00086050 000 025 025 minus00329 minus00177 minus00128 minus00093 025 050 000 025 minus00361 minus00189 minus00132 minus00100

075 000 000 025 minus00418 minus00226 minus00157 minus00116 025 075 000 000 minus00426 minus00236 minus00165 minus00123

100 000 000 000 minus00536 minus00285 minus00199 minus00150 000 100 000 000 minus00491 minus00269 minus00190 minus00138

025 025 025 025 minus00303 minus00165 minus00116 minus00086 025 025 025 025 minus00303 minus00165 minus00116 minus00086025 025 050 000 minus0332 minus00176 minus00119 minus00093 000 025 050 050 minus00359 minus00181 minus00128 minus00094

000 025 075 000 minus00403 minus00196 minus00127 minus00097 000 000 025 075 minus00402 minus00206 minus00146 minus00108

000 000 100 000 minus00420 minus00211 minus00139 minus00101 000 000 000 100 minus00425 minus00230 minus00157 minus00119

Note the ratio of index portfolio is ranked by the sequence of SSCI HSI TAIEX SP500 in Table 7

and heavy-tailrdquo characteristics the time-varying volatilitycharacteristics and extreme-tail dependence characteristicsof financial asset return It proposed the parameter estimationalgorithm of the multivariate time-varying 119866-119867 CopulaGARCH model by using condition maximum likelihoodmethod and IFM two-step method An algorithm was con-structed to calculate VaR by using the quantile functionand the simulation method based on 119866-119867 Copula GARCHmodel In addition this paper selected the daily log returnof SSCI (China) HSI (Hong Kong China) TAIEX (TaiwanChina) and SP500 (USA) from January 3 2000 to June18 2010 as samples to estimate the parameters of themultivariate time-varying 119866-119867 Copula GARCH model andit also estimated theVaR for various index risk asset portfoliosunder different confidence levelsThe research results showedthat the multivariate time-varying 119866-119867 Copula GARCHmodel constructed in this paper could reasonably estimateand measure the extreme losses of risk portfolios in financialmarket and the measurement results were in line with theactual situation of stock market and the risk diversificationtheory of portfolio The achievement of this paper provideda practical and effective method for measuring the extremelosses of financial market

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees for all help-ful comments and suggestions This work is supported byNatural Science Foundation Project of CQ CSTC (Grant noCSTC 2011BB2088) and Key Project of National NaturalScience Foundation (Grant no 71232004)

References

[1] S J Kon ldquoModels of stock returnsmdasha comparisonrdquoThe Journalof Finance vol 39 no 1 pp 147ndash165 1984

[2] J B Gray and D W French ldquoEmpirical comparisons of dis-tributional models for stock index returnsrdquo Journal of BusinessFinance amp Accounting vol 17 no 3 pp 451ndash459 1990

[3] E Jondeau andMRockinger ldquoTesting for differences in the tailsof stock-market returnsrdquo Journal of Empirical Finance vol 10no 5 pp 559ndash581 2003

[4] C Jiang S Li and S Liang ldquoEmpirical investigation ofdistribution feature of return in china stockmarketrdquoApplicationof Statistics and Management vol 26 no 4 pp 710ndash717 2007

[5] D Dong and W Jin ldquoA subjective model of the distribution ofreturns and empirical analysisrdquo Chinese Journal of ManagementScience vol 15 no 1 pp 112ndash120 2007

[6] J Martinez and B Iglewicz ldquoSome properties of the tukey g andh family of distributionsrdquoCommunications in StatisticsmdashTheoryand Methods vol 13 no 3 pp 353ndash369 1984

[7] T C Mills ldquoModelling skewness and kurtosis in the Londonstock exchange FT-SE index return distributionsrdquo The Statisti-cian vol 44 no 3 pp 323ndash332 1995

[8] H Zhu and Z Pan ldquoA study of portfolios VaRmethod based ong-h distributionrdquo Chinese Journal of Management Science vol13 no 4 pp 7ndash12 2005

[9] K Kuester S Mittnik andM S Paolella ldquoValue-at-risk predic-tion a comparison of alternative strategiesrdquo Journal of FinancialEconometrics vol 4 no 1 pp 53ndash89 2006

[10] M Degen P Embrechts and D Lambrigger ldquoThe quantitativemodeling of operational risk between g-and-h and EVTrdquo AstinBulletin vol 37 no 2 pp 265ndash291 2007

[11] E Jondeau and M Rockinger ldquoThe Copula-GARCH modelof conditional dependencies an international stock marketapplicationrdquo Journal of International Money and Finance vol25 no 5 pp 827ndash853 2006

[12] J C Rodriguez ldquoMeasuring financial contagion a Copulaapproachrdquo Journal of Empirical Finance vol 14 no 3 pp 401ndash423 2007

Mathematical Problems in Engineering 9

[13] M Fischer C Kock S Schluter and F Weigert ldquoAn empiricalanalysis of multivariate copula modelsrdquo Quantitative Financevol 9 no 7 pp 839ndash854 2009

[14] W Sun S Rachev F J Fabozzi and S S Kalev ldquoA newapproach to modeling co-movement of international equitymarkets evidence of unconditional copula-based simulation oftail dependencerdquo Empirical Economics vol 36 no 1 pp 201ndash229 2009

[15] Z Liu T Zhang and F Wen ldquoTime-varying risk attitude andconditional skewnessrdquo Abstract and Applied Analysis vol 2014Article ID 174848 11 pages 2014

[16] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 3 no 48 pp235ndash254 2014

[17] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[18] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics and Economics vol 45 no 3 pp 315ndash324 2009

[19] A Ghorbel and A Trabelsi ldquoMeasure of financial risk usingconditional extreme value copulas with EVT marginsrdquo Journalof Risk vol 11 no 4 pp 51ndash85 2009

[20] L Chollete A Heinen and A Valdesogo ldquoModeling interna-tional financial returns with a multivariate regime-switchingcopulardquo Journal of Financial Econometrics vol 7 no 4 pp 437ndash480 2009

[21] M Huggenberger and T Klett ldquoA g-and-h copula approach torisk measurement in multivariate financial modelsrdquo WorkingPaper 2010 httpssrncomabstract=1677431

[22] J Wang W Bao and J Hu ldquoPortfolios VaR analyses basedon the multi-dimensional gumbel copula functionrdquo Journal ofApplied Statistics and Management vol 29 no 1 pp 137ndash1432010

[23] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis and Applications vol 16 no 1 pp93ndash103 2014

[24] J Liu M Tao C Ma and F Wen ldquoUtility indifference pricingof convertible bondsrdquo International Journal of InformationTechnology and Decision Making vol 13 no 2 pp 429ndash4442014

[25] R B Nelsen An Introduction to Copulas Springer Science ampBusiness Media New York NY USA 2007

[26] H JoeMultivariate Models and Dependence Concepts vol 73 ofMonographs on Statistics and Applied Probability Chapman ampHall London UK 1997

[27] A J Patton ldquoEstimation of multivariate models for time seriesof possibly different lengthsrdquo Journal of Applied Econometricsvol 21 no 2 pp 147ndash173 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article Multivariate Time-Varying Copula GARCH ...downloads.hindawi.com/journals/mpe/2015/286014.pdf · model, they presented an empirical pricing study of China s market.

Mathematical Problems in Engineering 5

where x119905 = (1199091119905 119909119901119905) and 119909119894119905 = 119905minus1120578

(119905V119894

(120576119894119905)) Then thelikelihood function of overall samples is

119897 (120579 | y) =

119879

prod

119905=1Γ (

120578 + 119901

2) Γ (

120578

2)

119901minus1

sdot (1+x1015840119905120588119905x119905

120578)

minus(120578+119901)2

(1003816100381610038161003816120588119905

1003816100381610038161003816)minus12

Γ (120578 + 12

)

minus119901

sdot

119901

prod

119894=1(1+

1199092119894119905

120578)

(120578+1)2 119901

prod

119894=1119891119884119894

(119910119894)

(19)

where 120579 = (120583119894 120603119894 120572119894 120573119894 V119894)119901

119894=1 119886 119887120588119905 120578 and x119905 = (1199091119905 119909119901119905)The value of correlationmatrix 120588

119905is similar to the time-

varying correlation matrix of multivariate Copula GARCHmodel proposed by Jondeau and Rockinger [11] That is 120588

119905

satisfies the following evolution equation

120588119905= (1minus 119886minus 119887)120588+ 119886Ψ119905minus1 + 119887120588

119905minus1 (20)

where 0 le 119886 119887 le 1 119886 + 119887 le 1 120588 is a positive definite matrixwhose main diagonal elements are 1 and other elements arestatic correlation coefficientsΨ119905minus1 is a 119901times119901matrix in whichevery element

120595119894119895119905minus1 =sum119898

119897=1 119909119894119905minus119897119909119895119905minus119897

radicsum119898

119897=1 1199092119894119905minus119897

radicsum119898

119897=1 1199092119895119905minus119897

119894 119895 = 1 2 119901 (21)

denotes the correlation coefficients of 119901 risk asset returns(119898 ge 119901 + 2) 119909119905 = (1199091119905 119909119901119905) = (119905

minus1V1 (119891V1(1199111119905))

119905minus1V119901

(119891V119901

(119911119901119905))) Each element 120588119894119895119905 of 120588119905 satisfies minus1 le 120588119894119895119905 le 1

4 Parameter Estimation Algorithm ofthe Multivariate Time-Varying 119866-119867 CopulaGARCH Model

On the basis of Huggenberger and Klett [21] this section willuse dynamic correlation matrix 120588

119905instead of static correla-

tion matrix in the multidimensional discrete-time stochasticprocess to estimate the parameters of multivariate time-varying119866-119867CopulaGARCHmodel established in Section 3Assuming thatΘ denotes the parameter space defined by themodel and (r1 r2 r119879) denotes the log return samples of119875-dimensional risk asset which is generated by multivariateconditional density function119891120588

119905

|I119905minus1

(r119905 | I119905minus1 1205790) where 1205790 isin

Θ I119905minus1 is 120590 algebra of time 119905 minus 1 and before the maximumlikelihood estimation of parameter vector 120579 can be calculatedby the following equation

= argmax120579isinΘ

119879

sum

119905=1log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 120579) (22)

where 119891120588119905

|I119905minus1

(r119905 | I119905minus1 120579) can be obtained by calculating thederivative of (5) Let 119888120579 denote Copula density functionThus

119891120588119905

|I119905minus1

(1199031119905 119903119901119905 | I119905minus1 120579)

= 119888120579 (1198651119905 (1199031119905 120579) 119865119901119905 (119903119901119905 120579))

sdot

119901

prod

119894=1119891119894119905 (119903119905119894 120579)

(23)

The probability density function and distribution func-tion can be obtained in the process ofmodel built in Section 3Using the IFM method proposed by Joe [26] we can convert(22) into an optimization problem Therefore we need todivide the parameter vector 120579 into two subparameter vectors120579119888 and 120579119903 that is 120579 = (120579119888 120579119903) where 120579119903 = (1205791199031

120579119903119901

)120579119903119894

is the parameter vector of 119894th marginal distribution and120579119888 is the parameter vector of Copula function Because IFMmethod is a two-step likelihood estimation method themodel parameters should be estimated through the followingtwo steps

Step 1 Solving the maximum likelihood estimator of theparameter vector of each risk asset return

119903119894

= argmax120579119903

119894

119879

sum

119905=1log119891119894119905 (119903119894119905 | 120579119903

119894

) 119894 = 1 2 119901 (24)

Thismeans that we need to estimate parameters vector 119903119894

of 119901 distributions continuously

Step 2 Taking each 119903119894

into the likelihood equation (22) wecan obtain the parameter vector 120579119888 of Copula function and itsmaximum likelihood estimator 119888 Consider

119888

= argmax120579119888

119879

sum

119905=1log 119888120579

119888

(1198651119905 (10038161003816100381610038161199031119905

1003816100381610038161003816 1199031) 119865119901119905 (

10038161003816100381610038161003816119903119901119905

10038161003816100381610038161003816119903119901

))

(25)

In themaximum likelihood estimationweneed to use thederivative function of the density function of 119866-119867 marginaldistribution with respect to the component of parametervector Because the density function of 119866-119867 marginal distri-bution is very complex this paper uses the implicit functiondifferentiation rule to take its derivative The estimator 2119904of parameter vector 120579 obtained by the above-mentionedtwo-step method obeys normal distribution consistentlyand asymptotically under the standard regularity conditionsproposed inHuggenberger andKlett [21] Joe [26] andPatton[27] that is

radic119879(2119904119879 minus 1205790)119889

997888rarr119879rarrinfin

119873(0Ωminus1ΣΩ) (26)

6 Mathematical Problems in Engineering

where

Ω = minus119864(1205972

1205971205791205971205791015840log119891120588

119905

|I119905minus1

(120588119905| I119905minus1 1205790))

Σ = 119864(120597

120597120579(log119891120588

119905

|I119905minus1

(120588119905| I119905minus1 1205790))

sdot120597

120597120579(log119891120588

119905

|I119905minus1

(120588119905| I119905minus1 1205790))

1015840

)

(27)

Because the matrixes Σ and Ω can be estimated by theestimated parameter vector consistently

Ω119879 = minus119879minus1119879

sum

119905=1

1205972

1205971205791205971205791015840log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 2119904119879)

Σ119879 = 119879minus1119879

sum

119905=1

120597

120597120579(log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 2119904119879))

sdot120597

120597120579(log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 2119904119879))1015840

(28)

Thus (26) can be used to calculate the standard deviationof the estimator 2119904119879

5 VaR Algorithm Based on the MultivariateTime-Varying 119866-119867 Copula GARCH Model

After estimating the parameters of multivariate time-varying119866-119867 Copula GARCH model VaR of the risk portfolio canbe measured VaR of risk portfolio indicates the expectedmaximum losses of risk portfolio held by investors within agiven confidence level and a certain period of time Assumingthat r119905 = (1199031119905 119903119901119905) (119905 = 1 2 119879) are the return samplesof 119901 risk assets which satisfy themultivariate time-varying119866-119867 Copula GARCH (1 1)model in Section 3 andsum

119901

119894=1 120582119894119903119894119905 isthe portfolio of 119901 risk assets in which the weight of the riskasset 119894 is 120582119894 (119894 = 1 2 119901) that can be less than 0 because ofpermitting short-purchasing and short-selling the risk assetsand meet sum

119901

119894=1 120582119894 = 1 the VaR of risk portfolio underconfidence level 119902 at time 119905 should satisfy Pr(sum119901

119894=1 120582119894119903119894119905 le

VaR119905) = 119902 The confidence level 119902 can reflect the differentrisk preferences of investors or financial institutions to acertain extent Choosing a larger confidence level means thatinvestors or financial institutions have greater risk aversionand they hope to get a forecast result with larger probability

Although the conditional distributions of 1199031119879+1 1199032119879+1 119903119901119879+1 can be calculated through the known marginaldistributions it is very difficult to calculate quantile fromtime-varying Copula density function and it is adverse tomeasure and calculate the VaR of risk portfolio Thereforethis paper measures the dynamic risk of portfolio and its esti-mation value approximately through simulating119866-119867CopulaGARCH model Based on the parameters 120579(119899) of the samplethe return series of risk assets [119903

(119899119898)

11+119879 119903(119899119898)

1199011+119879] 119898 =

1 119872 and the one-step measurement and estimationvalues of VaR of their portfolios can be obtained through esti-mating (ℎ(119899)1119879+1 ℎ

(119899+1)119901119879+1) according to the volatility equation

Table 1 Moment estimation results of the daily log return of SSCIHSI TAIEX and SP500

Types of stock index Mean Std Skewness KurtosisSSCI 25078119890 minus 004 00181 minus02144 74467HSI 74505119890 minus 005 00177 minus02765 116866TAIEX 39511119890 minus 005 00140 minus09091 172823SP500 minus97173119890 minus 005 00148 minus03701 120177

of119866-119867 Copula GARCHmodel and calculating 120588(119899)

119879+1 by usingCopula dynamic evolution equation and then repeating thefollowing algorithm for 119872 times (119872 ge 3119901)

Step 1 It is simulating 119872 groups of random vectors[119906(119899119898)

1119879+1 119906(119899119898)

119901119879+1] according to the multivariate 119879-Copuladensity function whose degree of freedom is 120578(119899) and correla-tion matrix is 120588(119899)

119879+1

Step 2 Calculating 119903(119899119898)

119894119879+1 = 120583(119899)

119894+ 119911(119899119898)

119894119879+1radicℎ(119899)

119894119879+1 119894 = 1 119901

Step 3 Firstly one calculates the return rate of risk portfoliothat is equal to sum

119901

119894=1 120582119894119903(119899119898)

119894119879+1 119898 = 1 2 119872 Secondly oneevaluates its 119902-quantile VaR(119899)

119879+1 Thirdly one measures theVaR of the risk portfolio by VaR119879+1 = (1119873)sum

119873

119899=1 VaR(119899)

119879+1

6 Application of the Multivariate Time-Varying 119866-119867 Copula GARCH Model

61 Date Sample and Moment Estimation USA and Chinaas the most developed capitalism country and the largestdeveloping country in the world respectively rank top two ofthe world economy Their stock markets should have strongrepresentation in the world At the same time due to thehistorical reasons there exist several regions with differentpolitical systems such as Mainland China Hong KongTaiwan and Macau in Greater China Macau is similar toHong Kong on the whole For the above-mentioned reasonsthis paper selects the SSCI (China) HSI (HongKong China)TAIEX (Taiwan China) and SP500 (USA) from January3 2000 to June 18 2010 as data samples to estimate theVaR of various index portfolios under different confidencelevels by using the multivariate time-varying 119866-119867 CopulaGARCHmodelThe data comes fromYahoo Financewebsitehttpfinanceyahoocom

The moment estimation results of the daily log return ofSSCI HSI TAIEX and SP500 are shown in Table 1

Table 1 shows that the skewness of daily log returns ofSSCI HSI TAIEX and SP500 is less than 0 and their kurtosisis much larger than that of standard normal distributionwhich is equal to 3 These results demonstrate that thedaily log returns of these indices have the right skew andleptokurtic characteristics Therefore it is appropriate to fitthe daily log return of SSCI HSI TAIEX and SP500 byapplying 119866-119867 distribution which has leptokurtic heavy-tailcharacteristics and it is reasonable to apply the multivariate

Mathematical Problems in Engineering 7

Table 2 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on SSCI index risk asset

Model parameters 1205831

1205961

1205721

1205731

Estimate 000025lowastlowastlowast 00551lowastlowastlowast 00617lowastlowastlowast 08583lowastlowastlowast

119879-statistic 48653 46851 65764 44009Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 3 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on HSI index risk asset

Model parameters 1205832

1205962

1205722

1205732

Estimate 0000075lowastlowastlowast

00342lowastlowastlowast

00586lowastlowastlowast

08987lowastlowastlowast

119879-statistic 46539 48518 61796 66079Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 4 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on TAIEX index risk asset

Model parameters 1205833

1205963

1205723

1205733

Estimate 000040lowastlowastlowast 00343lowastlowastlowast

00517lowastlowastlowast

09054lowastlowastlowast

119879-statistic minus64538 43523 59817 65935Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 5 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on SP500 index risk asset

Model parameters 1205834

1205964

1205724

1205734

Estimate minus00001lowastlowastlowast

00251lowastlowastlowast

00861lowastlowastlowast

08738lowastlowastlowast

119879-statistic minus33873 52684 76324 63786Note lowastlowastlowastin the table denotes that the parameter is significant at 1 level

time-varying 119866-119867 Copula GARCH model to measure theirVaR

62 Parameter Estimates of the Multivariate Time-Varying119866-119867 Copula GARCH Model Based on the parameter esti-mation algorithm proposed in Section 4 the parameters ofthe multivariate time-varying 119866-119867 Copula GARCH modelwith SSCI HSI TAIEX and SP500 can be estimated Theparameter estimation results are shown in Tables 2 3 4 5and 6

From Tables 2 to 6 the following results can be obtained

(1) Consider 1205721 + 1205731 = 092 1205722 + 1205732 = 09573 1205723 +

1205733 = 09571 and 1205724 + 1205734 = 09599 This shows thatthe volatility persistence of Shanghai stock market isthe strongest Taiwan and Hong Kong stock marketrank second and third and the volatility persistenceof USA stock market is minimum It indicates thatthe investorsrsquo expectation of risk compensation inthe emerging markets represented by Chinarsquos stockmarket is stronger than that in the mature marketsrepresented by the USArsquos stock market and the pricediscovery efficiency of innovation in the emergingmarkets represented by Chinarsquos stock market is lowerthan that in the mature markets represented by USArsquosstockmarket In addition the sumof the coefficients120572

and120573 is very close to 1 which indicates that the impactand shock of innovation on the index volatility of eachstock market has a long memory

(2) The degree of freedom 120578 = 1457 and the correlationcoefficients 120588119894119895 of 119879-Copula in Table 6 show thatthere exists the strongest correlation between HongKong stock market and Taiwan stock market andthe correlation between Shanghai stock market andHong Kong stock market is also relatively largeThe above-mentioned facts indicate that the extremeevents probably result in the phenomena that HongKong stock market and Taiwan stock market are upand down synchronously and there exist comovingbehaviors between Shanghai stock market and HongKong stock market

(3) The time-varying coefficient 119887 = 0987 indicatesthat the time-varying correlation coefficient of 119866-119867 Copula GARCH model has a long memory thatis the impact of historical values of each otherrsquoscorrelation coefficient among SSCI HSI TAIEX andSP500 on the expected correlation is relatively large

63 VaR Measurement Based on the Multivariate Time-Varying 119866-119867 Copula GARCH Model Based on the multi-variate time-varying119866-119867 Copula GARCHmodel with SSCIHSI TAIEX and SP500 whose parameters have been esti-mated the VaRs of various index portfolios under differentconfidence levels can be measured The measurement resultsare shown in Table 7

From Table 7 the following results can be obtained

(1) The inequalities VaR (SSCI) lt VaR (HSI) lt VaR(SP500) lt VaR (TAIEX) can be satisfied for anyconfidence level It shows that the risk of extremelosses in Shanghai stock market is higher than that inHong Kong stock market Taiwan stock market andUSA stock market This measurement result is in linewith the actual situation that thematurity of Shanghaistockmarket is far lower than that ofHongKong stockmarket Taiwan stock market and USA stock market

(2) For any confidence level the extreme losses risk ofthe investors who equally allocate their total assetsamong SSCI HSI TAIEX and SP500 is lower thanthat of the investors who put their total assets into oneindex asset The extreme losses risk of the investorsincreases with the concentration of risk asset in theindex portfolios This measurement result is consis-tent with the risk diversification theory of portfolio

7 Conclusion

Considering the ldquoasymmetric leptokurtic and heavy-tailrdquocharacteristics the time-varying volatility characteristicsand extreme-tail dependence characteristics of financialasset return this paper combined the 119866-119867 distributionCopula function and GARCH model to construct a mul-tivariate time-varying 119866-119867 Copula GARCH model whichcan comprehensively describe the ldquoasymmetric leptokurtic

8 Mathematical Problems in Engineering

Table 6 Estimates of correlation coefficients time-varying parameters and degree of freedom of four-variate time-varying G-H CopulaGARCH (1 1) model based on SSCI HSI TAIEX and SP500

Model parameters 12058812

12058813

12058814

12058823

12058824

12058834

a b 120578

Estimate 0256lowastlowastlowast

0139lowastlowastlowast

0021lowastlowastlowast

0528lowastlowastlowast

0193lowastlowastlowast

0128lowastlowast

0008lowastlowastlowast

0987lowastlowastlowast

1457lowastlowastlowast

119879-statistic 1351 1102 3431 5763 9564 2529 5179 3561 6871Note lowastlowastlowast and lowastlowast in the table denote that the parameter is significant at 1 and 5 level respectively

Table 7 VaR estimation results based on the four-variate time-varying G-H Copula GARCHmodel with SSCI HSI TAIEX and SP500

Ratio of index portfolio 1 5 10 15 Ratio of index portfolio 1 5 10 15025 025 025 025 minus00303 minus00165 minus00116 minus00086 025 025 025 025 minus00303 minus00165 minus00116 minus00086050 000 025 025 minus00329 minus00177 minus00128 minus00093 025 050 000 025 minus00361 minus00189 minus00132 minus00100

075 000 000 025 minus00418 minus00226 minus00157 minus00116 025 075 000 000 minus00426 minus00236 minus00165 minus00123

100 000 000 000 minus00536 minus00285 minus00199 minus00150 000 100 000 000 minus00491 minus00269 minus00190 minus00138

025 025 025 025 minus00303 minus00165 minus00116 minus00086 025 025 025 025 minus00303 minus00165 minus00116 minus00086025 025 050 000 minus0332 minus00176 minus00119 minus00093 000 025 050 050 minus00359 minus00181 minus00128 minus00094

000 025 075 000 minus00403 minus00196 minus00127 minus00097 000 000 025 075 minus00402 minus00206 minus00146 minus00108

000 000 100 000 minus00420 minus00211 minus00139 minus00101 000 000 000 100 minus00425 minus00230 minus00157 minus00119

Note the ratio of index portfolio is ranked by the sequence of SSCI HSI TAIEX SP500 in Table 7

and heavy-tailrdquo characteristics the time-varying volatilitycharacteristics and extreme-tail dependence characteristicsof financial asset return It proposed the parameter estimationalgorithm of the multivariate time-varying 119866-119867 CopulaGARCH model by using condition maximum likelihoodmethod and IFM two-step method An algorithm was con-structed to calculate VaR by using the quantile functionand the simulation method based on 119866-119867 Copula GARCHmodel In addition this paper selected the daily log returnof SSCI (China) HSI (Hong Kong China) TAIEX (TaiwanChina) and SP500 (USA) from January 3 2000 to June18 2010 as samples to estimate the parameters of themultivariate time-varying 119866-119867 Copula GARCH model andit also estimated theVaR for various index risk asset portfoliosunder different confidence levelsThe research results showedthat the multivariate time-varying 119866-119867 Copula GARCHmodel constructed in this paper could reasonably estimateand measure the extreme losses of risk portfolios in financialmarket and the measurement results were in line with theactual situation of stock market and the risk diversificationtheory of portfolio The achievement of this paper provideda practical and effective method for measuring the extremelosses of financial market

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees for all help-ful comments and suggestions This work is supported byNatural Science Foundation Project of CQ CSTC (Grant noCSTC 2011BB2088) and Key Project of National NaturalScience Foundation (Grant no 71232004)

References

[1] S J Kon ldquoModels of stock returnsmdasha comparisonrdquoThe Journalof Finance vol 39 no 1 pp 147ndash165 1984

[2] J B Gray and D W French ldquoEmpirical comparisons of dis-tributional models for stock index returnsrdquo Journal of BusinessFinance amp Accounting vol 17 no 3 pp 451ndash459 1990

[3] E Jondeau andMRockinger ldquoTesting for differences in the tailsof stock-market returnsrdquo Journal of Empirical Finance vol 10no 5 pp 559ndash581 2003

[4] C Jiang S Li and S Liang ldquoEmpirical investigation ofdistribution feature of return in china stockmarketrdquoApplicationof Statistics and Management vol 26 no 4 pp 710ndash717 2007

[5] D Dong and W Jin ldquoA subjective model of the distribution ofreturns and empirical analysisrdquo Chinese Journal of ManagementScience vol 15 no 1 pp 112ndash120 2007

[6] J Martinez and B Iglewicz ldquoSome properties of the tukey g andh family of distributionsrdquoCommunications in StatisticsmdashTheoryand Methods vol 13 no 3 pp 353ndash369 1984

[7] T C Mills ldquoModelling skewness and kurtosis in the Londonstock exchange FT-SE index return distributionsrdquo The Statisti-cian vol 44 no 3 pp 323ndash332 1995

[8] H Zhu and Z Pan ldquoA study of portfolios VaRmethod based ong-h distributionrdquo Chinese Journal of Management Science vol13 no 4 pp 7ndash12 2005

[9] K Kuester S Mittnik andM S Paolella ldquoValue-at-risk predic-tion a comparison of alternative strategiesrdquo Journal of FinancialEconometrics vol 4 no 1 pp 53ndash89 2006

[10] M Degen P Embrechts and D Lambrigger ldquoThe quantitativemodeling of operational risk between g-and-h and EVTrdquo AstinBulletin vol 37 no 2 pp 265ndash291 2007

[11] E Jondeau and M Rockinger ldquoThe Copula-GARCH modelof conditional dependencies an international stock marketapplicationrdquo Journal of International Money and Finance vol25 no 5 pp 827ndash853 2006

[12] J C Rodriguez ldquoMeasuring financial contagion a Copulaapproachrdquo Journal of Empirical Finance vol 14 no 3 pp 401ndash423 2007

Mathematical Problems in Engineering 9

[13] M Fischer C Kock S Schluter and F Weigert ldquoAn empiricalanalysis of multivariate copula modelsrdquo Quantitative Financevol 9 no 7 pp 839ndash854 2009

[14] W Sun S Rachev F J Fabozzi and S S Kalev ldquoA newapproach to modeling co-movement of international equitymarkets evidence of unconditional copula-based simulation oftail dependencerdquo Empirical Economics vol 36 no 1 pp 201ndash229 2009

[15] Z Liu T Zhang and F Wen ldquoTime-varying risk attitude andconditional skewnessrdquo Abstract and Applied Analysis vol 2014Article ID 174848 11 pages 2014

[16] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 3 no 48 pp235ndash254 2014

[17] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[18] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics and Economics vol 45 no 3 pp 315ndash324 2009

[19] A Ghorbel and A Trabelsi ldquoMeasure of financial risk usingconditional extreme value copulas with EVT marginsrdquo Journalof Risk vol 11 no 4 pp 51ndash85 2009

[20] L Chollete A Heinen and A Valdesogo ldquoModeling interna-tional financial returns with a multivariate regime-switchingcopulardquo Journal of Financial Econometrics vol 7 no 4 pp 437ndash480 2009

[21] M Huggenberger and T Klett ldquoA g-and-h copula approach torisk measurement in multivariate financial modelsrdquo WorkingPaper 2010 httpssrncomabstract=1677431

[22] J Wang W Bao and J Hu ldquoPortfolios VaR analyses basedon the multi-dimensional gumbel copula functionrdquo Journal ofApplied Statistics and Management vol 29 no 1 pp 137ndash1432010

[23] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis and Applications vol 16 no 1 pp93ndash103 2014

[24] J Liu M Tao C Ma and F Wen ldquoUtility indifference pricingof convertible bondsrdquo International Journal of InformationTechnology and Decision Making vol 13 no 2 pp 429ndash4442014

[25] R B Nelsen An Introduction to Copulas Springer Science ampBusiness Media New York NY USA 2007

[26] H JoeMultivariate Models and Dependence Concepts vol 73 ofMonographs on Statistics and Applied Probability Chapman ampHall London UK 1997

[27] A J Patton ldquoEstimation of multivariate models for time seriesof possibly different lengthsrdquo Journal of Applied Econometricsvol 21 no 2 pp 147ndash173 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Multivariate Time-Varying Copula GARCH ...downloads.hindawi.com/journals/mpe/2015/286014.pdf · model, they presented an empirical pricing study of China s market.

6 Mathematical Problems in Engineering

where

Ω = minus119864(1205972

1205971205791205971205791015840log119891120588

119905

|I119905minus1

(120588119905| I119905minus1 1205790))

Σ = 119864(120597

120597120579(log119891120588

119905

|I119905minus1

(120588119905| I119905minus1 1205790))

sdot120597

120597120579(log119891120588

119905

|I119905minus1

(120588119905| I119905minus1 1205790))

1015840

)

(27)

Because the matrixes Σ and Ω can be estimated by theestimated parameter vector consistently

Ω119879 = minus119879minus1119879

sum

119905=1

1205972

1205971205791205971205791015840log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 2119904119879)

Σ119879 = 119879minus1119879

sum

119905=1

120597

120597120579(log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 2119904119879))

sdot120597

120597120579(log119891120588

119905

|I119905minus1

(r119905 | I119905minus1 2119904119879))1015840

(28)

Thus (26) can be used to calculate the standard deviationof the estimator 2119904119879

5 VaR Algorithm Based on the MultivariateTime-Varying 119866-119867 Copula GARCH Model

After estimating the parameters of multivariate time-varying119866-119867 Copula GARCH model VaR of the risk portfolio canbe measured VaR of risk portfolio indicates the expectedmaximum losses of risk portfolio held by investors within agiven confidence level and a certain period of time Assumingthat r119905 = (1199031119905 119903119901119905) (119905 = 1 2 119879) are the return samplesof 119901 risk assets which satisfy themultivariate time-varying119866-119867 Copula GARCH (1 1)model in Section 3 andsum

119901

119894=1 120582119894119903119894119905 isthe portfolio of 119901 risk assets in which the weight of the riskasset 119894 is 120582119894 (119894 = 1 2 119901) that can be less than 0 because ofpermitting short-purchasing and short-selling the risk assetsand meet sum

119901

119894=1 120582119894 = 1 the VaR of risk portfolio underconfidence level 119902 at time 119905 should satisfy Pr(sum119901

119894=1 120582119894119903119894119905 le

VaR119905) = 119902 The confidence level 119902 can reflect the differentrisk preferences of investors or financial institutions to acertain extent Choosing a larger confidence level means thatinvestors or financial institutions have greater risk aversionand they hope to get a forecast result with larger probability

Although the conditional distributions of 1199031119879+1 1199032119879+1 119903119901119879+1 can be calculated through the known marginaldistributions it is very difficult to calculate quantile fromtime-varying Copula density function and it is adverse tomeasure and calculate the VaR of risk portfolio Thereforethis paper measures the dynamic risk of portfolio and its esti-mation value approximately through simulating119866-119867CopulaGARCH model Based on the parameters 120579(119899) of the samplethe return series of risk assets [119903

(119899119898)

11+119879 119903(119899119898)

1199011+119879] 119898 =

1 119872 and the one-step measurement and estimationvalues of VaR of their portfolios can be obtained through esti-mating (ℎ(119899)1119879+1 ℎ

(119899+1)119901119879+1) according to the volatility equation

Table 1 Moment estimation results of the daily log return of SSCIHSI TAIEX and SP500

Types of stock index Mean Std Skewness KurtosisSSCI 25078119890 minus 004 00181 minus02144 74467HSI 74505119890 minus 005 00177 minus02765 116866TAIEX 39511119890 minus 005 00140 minus09091 172823SP500 minus97173119890 minus 005 00148 minus03701 120177

of119866-119867 Copula GARCHmodel and calculating 120588(119899)

119879+1 by usingCopula dynamic evolution equation and then repeating thefollowing algorithm for 119872 times (119872 ge 3119901)

Step 1 It is simulating 119872 groups of random vectors[119906(119899119898)

1119879+1 119906(119899119898)

119901119879+1] according to the multivariate 119879-Copuladensity function whose degree of freedom is 120578(119899) and correla-tion matrix is 120588(119899)

119879+1

Step 2 Calculating 119903(119899119898)

119894119879+1 = 120583(119899)

119894+ 119911(119899119898)

119894119879+1radicℎ(119899)

119894119879+1 119894 = 1 119901

Step 3 Firstly one calculates the return rate of risk portfoliothat is equal to sum

119901

119894=1 120582119894119903(119899119898)

119894119879+1 119898 = 1 2 119872 Secondly oneevaluates its 119902-quantile VaR(119899)

119879+1 Thirdly one measures theVaR of the risk portfolio by VaR119879+1 = (1119873)sum

119873

119899=1 VaR(119899)

119879+1

6 Application of the Multivariate Time-Varying 119866-119867 Copula GARCH Model

61 Date Sample and Moment Estimation USA and Chinaas the most developed capitalism country and the largestdeveloping country in the world respectively rank top two ofthe world economy Their stock markets should have strongrepresentation in the world At the same time due to thehistorical reasons there exist several regions with differentpolitical systems such as Mainland China Hong KongTaiwan and Macau in Greater China Macau is similar toHong Kong on the whole For the above-mentioned reasonsthis paper selects the SSCI (China) HSI (HongKong China)TAIEX (Taiwan China) and SP500 (USA) from January3 2000 to June 18 2010 as data samples to estimate theVaR of various index portfolios under different confidencelevels by using the multivariate time-varying 119866-119867 CopulaGARCHmodelThe data comes fromYahoo Financewebsitehttpfinanceyahoocom

The moment estimation results of the daily log return ofSSCI HSI TAIEX and SP500 are shown in Table 1

Table 1 shows that the skewness of daily log returns ofSSCI HSI TAIEX and SP500 is less than 0 and their kurtosisis much larger than that of standard normal distributionwhich is equal to 3 These results demonstrate that thedaily log returns of these indices have the right skew andleptokurtic characteristics Therefore it is appropriate to fitthe daily log return of SSCI HSI TAIEX and SP500 byapplying 119866-119867 distribution which has leptokurtic heavy-tailcharacteristics and it is reasonable to apply the multivariate

Mathematical Problems in Engineering 7

Table 2 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on SSCI index risk asset

Model parameters 1205831

1205961

1205721

1205731

Estimate 000025lowastlowastlowast 00551lowastlowastlowast 00617lowastlowastlowast 08583lowastlowastlowast

119879-statistic 48653 46851 65764 44009Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 3 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on HSI index risk asset

Model parameters 1205832

1205962

1205722

1205732

Estimate 0000075lowastlowastlowast

00342lowastlowastlowast

00586lowastlowastlowast

08987lowastlowastlowast

119879-statistic 46539 48518 61796 66079Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 4 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on TAIEX index risk asset

Model parameters 1205833

1205963

1205723

1205733

Estimate 000040lowastlowastlowast 00343lowastlowastlowast

00517lowastlowastlowast

09054lowastlowastlowast

119879-statistic minus64538 43523 59817 65935Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 5 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on SP500 index risk asset

Model parameters 1205834

1205964

1205724

1205734

Estimate minus00001lowastlowastlowast

00251lowastlowastlowast

00861lowastlowastlowast

08738lowastlowastlowast

119879-statistic minus33873 52684 76324 63786Note lowastlowastlowastin the table denotes that the parameter is significant at 1 level

time-varying 119866-119867 Copula GARCH model to measure theirVaR

62 Parameter Estimates of the Multivariate Time-Varying119866-119867 Copula GARCH Model Based on the parameter esti-mation algorithm proposed in Section 4 the parameters ofthe multivariate time-varying 119866-119867 Copula GARCH modelwith SSCI HSI TAIEX and SP500 can be estimated Theparameter estimation results are shown in Tables 2 3 4 5and 6

From Tables 2 to 6 the following results can be obtained

(1) Consider 1205721 + 1205731 = 092 1205722 + 1205732 = 09573 1205723 +

1205733 = 09571 and 1205724 + 1205734 = 09599 This shows thatthe volatility persistence of Shanghai stock market isthe strongest Taiwan and Hong Kong stock marketrank second and third and the volatility persistenceof USA stock market is minimum It indicates thatthe investorsrsquo expectation of risk compensation inthe emerging markets represented by Chinarsquos stockmarket is stronger than that in the mature marketsrepresented by the USArsquos stock market and the pricediscovery efficiency of innovation in the emergingmarkets represented by Chinarsquos stock market is lowerthan that in the mature markets represented by USArsquosstockmarket In addition the sumof the coefficients120572

and120573 is very close to 1 which indicates that the impactand shock of innovation on the index volatility of eachstock market has a long memory

(2) The degree of freedom 120578 = 1457 and the correlationcoefficients 120588119894119895 of 119879-Copula in Table 6 show thatthere exists the strongest correlation between HongKong stock market and Taiwan stock market andthe correlation between Shanghai stock market andHong Kong stock market is also relatively largeThe above-mentioned facts indicate that the extremeevents probably result in the phenomena that HongKong stock market and Taiwan stock market are upand down synchronously and there exist comovingbehaviors between Shanghai stock market and HongKong stock market

(3) The time-varying coefficient 119887 = 0987 indicatesthat the time-varying correlation coefficient of 119866-119867 Copula GARCH model has a long memory thatis the impact of historical values of each otherrsquoscorrelation coefficient among SSCI HSI TAIEX andSP500 on the expected correlation is relatively large

63 VaR Measurement Based on the Multivariate Time-Varying 119866-119867 Copula GARCH Model Based on the multi-variate time-varying119866-119867 Copula GARCHmodel with SSCIHSI TAIEX and SP500 whose parameters have been esti-mated the VaRs of various index portfolios under differentconfidence levels can be measured The measurement resultsare shown in Table 7

From Table 7 the following results can be obtained

(1) The inequalities VaR (SSCI) lt VaR (HSI) lt VaR(SP500) lt VaR (TAIEX) can be satisfied for anyconfidence level It shows that the risk of extremelosses in Shanghai stock market is higher than that inHong Kong stock market Taiwan stock market andUSA stock market This measurement result is in linewith the actual situation that thematurity of Shanghaistockmarket is far lower than that ofHongKong stockmarket Taiwan stock market and USA stock market

(2) For any confidence level the extreme losses risk ofthe investors who equally allocate their total assetsamong SSCI HSI TAIEX and SP500 is lower thanthat of the investors who put their total assets into oneindex asset The extreme losses risk of the investorsincreases with the concentration of risk asset in theindex portfolios This measurement result is consis-tent with the risk diversification theory of portfolio

7 Conclusion

Considering the ldquoasymmetric leptokurtic and heavy-tailrdquocharacteristics the time-varying volatility characteristicsand extreme-tail dependence characteristics of financialasset return this paper combined the 119866-119867 distributionCopula function and GARCH model to construct a mul-tivariate time-varying 119866-119867 Copula GARCH model whichcan comprehensively describe the ldquoasymmetric leptokurtic

8 Mathematical Problems in Engineering

Table 6 Estimates of correlation coefficients time-varying parameters and degree of freedom of four-variate time-varying G-H CopulaGARCH (1 1) model based on SSCI HSI TAIEX and SP500

Model parameters 12058812

12058813

12058814

12058823

12058824

12058834

a b 120578

Estimate 0256lowastlowastlowast

0139lowastlowastlowast

0021lowastlowastlowast

0528lowastlowastlowast

0193lowastlowastlowast

0128lowastlowast

0008lowastlowastlowast

0987lowastlowastlowast

1457lowastlowastlowast

119879-statistic 1351 1102 3431 5763 9564 2529 5179 3561 6871Note lowastlowastlowast and lowastlowast in the table denote that the parameter is significant at 1 and 5 level respectively

Table 7 VaR estimation results based on the four-variate time-varying G-H Copula GARCHmodel with SSCI HSI TAIEX and SP500

Ratio of index portfolio 1 5 10 15 Ratio of index portfolio 1 5 10 15025 025 025 025 minus00303 minus00165 minus00116 minus00086 025 025 025 025 minus00303 minus00165 minus00116 minus00086050 000 025 025 minus00329 minus00177 minus00128 minus00093 025 050 000 025 minus00361 minus00189 minus00132 minus00100

075 000 000 025 minus00418 minus00226 minus00157 minus00116 025 075 000 000 minus00426 minus00236 minus00165 minus00123

100 000 000 000 minus00536 minus00285 minus00199 minus00150 000 100 000 000 minus00491 minus00269 minus00190 minus00138

025 025 025 025 minus00303 minus00165 minus00116 minus00086 025 025 025 025 minus00303 minus00165 minus00116 minus00086025 025 050 000 minus0332 minus00176 minus00119 minus00093 000 025 050 050 minus00359 minus00181 minus00128 minus00094

000 025 075 000 minus00403 minus00196 minus00127 minus00097 000 000 025 075 minus00402 minus00206 minus00146 minus00108

000 000 100 000 minus00420 minus00211 minus00139 minus00101 000 000 000 100 minus00425 minus00230 minus00157 minus00119

Note the ratio of index portfolio is ranked by the sequence of SSCI HSI TAIEX SP500 in Table 7

and heavy-tailrdquo characteristics the time-varying volatilitycharacteristics and extreme-tail dependence characteristicsof financial asset return It proposed the parameter estimationalgorithm of the multivariate time-varying 119866-119867 CopulaGARCH model by using condition maximum likelihoodmethod and IFM two-step method An algorithm was con-structed to calculate VaR by using the quantile functionand the simulation method based on 119866-119867 Copula GARCHmodel In addition this paper selected the daily log returnof SSCI (China) HSI (Hong Kong China) TAIEX (TaiwanChina) and SP500 (USA) from January 3 2000 to June18 2010 as samples to estimate the parameters of themultivariate time-varying 119866-119867 Copula GARCH model andit also estimated theVaR for various index risk asset portfoliosunder different confidence levelsThe research results showedthat the multivariate time-varying 119866-119867 Copula GARCHmodel constructed in this paper could reasonably estimateand measure the extreme losses of risk portfolios in financialmarket and the measurement results were in line with theactual situation of stock market and the risk diversificationtheory of portfolio The achievement of this paper provideda practical and effective method for measuring the extremelosses of financial market

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees for all help-ful comments and suggestions This work is supported byNatural Science Foundation Project of CQ CSTC (Grant noCSTC 2011BB2088) and Key Project of National NaturalScience Foundation (Grant no 71232004)

References

[1] S J Kon ldquoModels of stock returnsmdasha comparisonrdquoThe Journalof Finance vol 39 no 1 pp 147ndash165 1984

[2] J B Gray and D W French ldquoEmpirical comparisons of dis-tributional models for stock index returnsrdquo Journal of BusinessFinance amp Accounting vol 17 no 3 pp 451ndash459 1990

[3] E Jondeau andMRockinger ldquoTesting for differences in the tailsof stock-market returnsrdquo Journal of Empirical Finance vol 10no 5 pp 559ndash581 2003

[4] C Jiang S Li and S Liang ldquoEmpirical investigation ofdistribution feature of return in china stockmarketrdquoApplicationof Statistics and Management vol 26 no 4 pp 710ndash717 2007

[5] D Dong and W Jin ldquoA subjective model of the distribution ofreturns and empirical analysisrdquo Chinese Journal of ManagementScience vol 15 no 1 pp 112ndash120 2007

[6] J Martinez and B Iglewicz ldquoSome properties of the tukey g andh family of distributionsrdquoCommunications in StatisticsmdashTheoryand Methods vol 13 no 3 pp 353ndash369 1984

[7] T C Mills ldquoModelling skewness and kurtosis in the Londonstock exchange FT-SE index return distributionsrdquo The Statisti-cian vol 44 no 3 pp 323ndash332 1995

[8] H Zhu and Z Pan ldquoA study of portfolios VaRmethod based ong-h distributionrdquo Chinese Journal of Management Science vol13 no 4 pp 7ndash12 2005

[9] K Kuester S Mittnik andM S Paolella ldquoValue-at-risk predic-tion a comparison of alternative strategiesrdquo Journal of FinancialEconometrics vol 4 no 1 pp 53ndash89 2006

[10] M Degen P Embrechts and D Lambrigger ldquoThe quantitativemodeling of operational risk between g-and-h and EVTrdquo AstinBulletin vol 37 no 2 pp 265ndash291 2007

[11] E Jondeau and M Rockinger ldquoThe Copula-GARCH modelof conditional dependencies an international stock marketapplicationrdquo Journal of International Money and Finance vol25 no 5 pp 827ndash853 2006

[12] J C Rodriguez ldquoMeasuring financial contagion a Copulaapproachrdquo Journal of Empirical Finance vol 14 no 3 pp 401ndash423 2007

Mathematical Problems in Engineering 9

[13] M Fischer C Kock S Schluter and F Weigert ldquoAn empiricalanalysis of multivariate copula modelsrdquo Quantitative Financevol 9 no 7 pp 839ndash854 2009

[14] W Sun S Rachev F J Fabozzi and S S Kalev ldquoA newapproach to modeling co-movement of international equitymarkets evidence of unconditional copula-based simulation oftail dependencerdquo Empirical Economics vol 36 no 1 pp 201ndash229 2009

[15] Z Liu T Zhang and F Wen ldquoTime-varying risk attitude andconditional skewnessrdquo Abstract and Applied Analysis vol 2014Article ID 174848 11 pages 2014

[16] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 3 no 48 pp235ndash254 2014

[17] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[18] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics and Economics vol 45 no 3 pp 315ndash324 2009

[19] A Ghorbel and A Trabelsi ldquoMeasure of financial risk usingconditional extreme value copulas with EVT marginsrdquo Journalof Risk vol 11 no 4 pp 51ndash85 2009

[20] L Chollete A Heinen and A Valdesogo ldquoModeling interna-tional financial returns with a multivariate regime-switchingcopulardquo Journal of Financial Econometrics vol 7 no 4 pp 437ndash480 2009

[21] M Huggenberger and T Klett ldquoA g-and-h copula approach torisk measurement in multivariate financial modelsrdquo WorkingPaper 2010 httpssrncomabstract=1677431

[22] J Wang W Bao and J Hu ldquoPortfolios VaR analyses basedon the multi-dimensional gumbel copula functionrdquo Journal ofApplied Statistics and Management vol 29 no 1 pp 137ndash1432010

[23] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis and Applications vol 16 no 1 pp93ndash103 2014

[24] J Liu M Tao C Ma and F Wen ldquoUtility indifference pricingof convertible bondsrdquo International Journal of InformationTechnology and Decision Making vol 13 no 2 pp 429ndash4442014

[25] R B Nelsen An Introduction to Copulas Springer Science ampBusiness Media New York NY USA 2007

[26] H JoeMultivariate Models and Dependence Concepts vol 73 ofMonographs on Statistics and Applied Probability Chapman ampHall London UK 1997

[27] A J Patton ldquoEstimation of multivariate models for time seriesof possibly different lengthsrdquo Journal of Applied Econometricsvol 21 no 2 pp 147ndash173 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Multivariate Time-Varying Copula GARCH ...downloads.hindawi.com/journals/mpe/2015/286014.pdf · model, they presented an empirical pricing study of China s market.

Mathematical Problems in Engineering 7

Table 2 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on SSCI index risk asset

Model parameters 1205831

1205961

1205721

1205731

Estimate 000025lowastlowastlowast 00551lowastlowastlowast 00617lowastlowastlowast 08583lowastlowastlowast

119879-statistic 48653 46851 65764 44009Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 3 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on HSI index risk asset

Model parameters 1205832

1205962

1205722

1205732

Estimate 0000075lowastlowastlowast

00342lowastlowastlowast

00586lowastlowastlowast

08987lowastlowastlowast

119879-statistic 46539 48518 61796 66079Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 4 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on TAIEX index risk asset

Model parameters 1205833

1205963

1205723

1205733

Estimate 000040lowastlowastlowast 00343lowastlowastlowast

00517lowastlowastlowast

09054lowastlowastlowast

119879-statistic minus64538 43523 59817 65935Note lowastlowastlowast in the table denotes that the parameter is significant at 1 level

Table 5 Parameter estimates of the four-variate time-varying G-HCopula GARCH (1 1) model based on SP500 index risk asset

Model parameters 1205834

1205964

1205724

1205734

Estimate minus00001lowastlowastlowast

00251lowastlowastlowast

00861lowastlowastlowast

08738lowastlowastlowast

119879-statistic minus33873 52684 76324 63786Note lowastlowastlowastin the table denotes that the parameter is significant at 1 level

time-varying 119866-119867 Copula GARCH model to measure theirVaR

62 Parameter Estimates of the Multivariate Time-Varying119866-119867 Copula GARCH Model Based on the parameter esti-mation algorithm proposed in Section 4 the parameters ofthe multivariate time-varying 119866-119867 Copula GARCH modelwith SSCI HSI TAIEX and SP500 can be estimated Theparameter estimation results are shown in Tables 2 3 4 5and 6

From Tables 2 to 6 the following results can be obtained

(1) Consider 1205721 + 1205731 = 092 1205722 + 1205732 = 09573 1205723 +

1205733 = 09571 and 1205724 + 1205734 = 09599 This shows thatthe volatility persistence of Shanghai stock market isthe strongest Taiwan and Hong Kong stock marketrank second and third and the volatility persistenceof USA stock market is minimum It indicates thatthe investorsrsquo expectation of risk compensation inthe emerging markets represented by Chinarsquos stockmarket is stronger than that in the mature marketsrepresented by the USArsquos stock market and the pricediscovery efficiency of innovation in the emergingmarkets represented by Chinarsquos stock market is lowerthan that in the mature markets represented by USArsquosstockmarket In addition the sumof the coefficients120572

and120573 is very close to 1 which indicates that the impactand shock of innovation on the index volatility of eachstock market has a long memory

(2) The degree of freedom 120578 = 1457 and the correlationcoefficients 120588119894119895 of 119879-Copula in Table 6 show thatthere exists the strongest correlation between HongKong stock market and Taiwan stock market andthe correlation between Shanghai stock market andHong Kong stock market is also relatively largeThe above-mentioned facts indicate that the extremeevents probably result in the phenomena that HongKong stock market and Taiwan stock market are upand down synchronously and there exist comovingbehaviors between Shanghai stock market and HongKong stock market

(3) The time-varying coefficient 119887 = 0987 indicatesthat the time-varying correlation coefficient of 119866-119867 Copula GARCH model has a long memory thatis the impact of historical values of each otherrsquoscorrelation coefficient among SSCI HSI TAIEX andSP500 on the expected correlation is relatively large

63 VaR Measurement Based on the Multivariate Time-Varying 119866-119867 Copula GARCH Model Based on the multi-variate time-varying119866-119867 Copula GARCHmodel with SSCIHSI TAIEX and SP500 whose parameters have been esti-mated the VaRs of various index portfolios under differentconfidence levels can be measured The measurement resultsare shown in Table 7

From Table 7 the following results can be obtained

(1) The inequalities VaR (SSCI) lt VaR (HSI) lt VaR(SP500) lt VaR (TAIEX) can be satisfied for anyconfidence level It shows that the risk of extremelosses in Shanghai stock market is higher than that inHong Kong stock market Taiwan stock market andUSA stock market This measurement result is in linewith the actual situation that thematurity of Shanghaistockmarket is far lower than that ofHongKong stockmarket Taiwan stock market and USA stock market

(2) For any confidence level the extreme losses risk ofthe investors who equally allocate their total assetsamong SSCI HSI TAIEX and SP500 is lower thanthat of the investors who put their total assets into oneindex asset The extreme losses risk of the investorsincreases with the concentration of risk asset in theindex portfolios This measurement result is consis-tent with the risk diversification theory of portfolio

7 Conclusion

Considering the ldquoasymmetric leptokurtic and heavy-tailrdquocharacteristics the time-varying volatility characteristicsand extreme-tail dependence characteristics of financialasset return this paper combined the 119866-119867 distributionCopula function and GARCH model to construct a mul-tivariate time-varying 119866-119867 Copula GARCH model whichcan comprehensively describe the ldquoasymmetric leptokurtic

8 Mathematical Problems in Engineering

Table 6 Estimates of correlation coefficients time-varying parameters and degree of freedom of four-variate time-varying G-H CopulaGARCH (1 1) model based on SSCI HSI TAIEX and SP500

Model parameters 12058812

12058813

12058814

12058823

12058824

12058834

a b 120578

Estimate 0256lowastlowastlowast

0139lowastlowastlowast

0021lowastlowastlowast

0528lowastlowastlowast

0193lowastlowastlowast

0128lowastlowast

0008lowastlowastlowast

0987lowastlowastlowast

1457lowastlowastlowast

119879-statistic 1351 1102 3431 5763 9564 2529 5179 3561 6871Note lowastlowastlowast and lowastlowast in the table denote that the parameter is significant at 1 and 5 level respectively

Table 7 VaR estimation results based on the four-variate time-varying G-H Copula GARCHmodel with SSCI HSI TAIEX and SP500

Ratio of index portfolio 1 5 10 15 Ratio of index portfolio 1 5 10 15025 025 025 025 minus00303 minus00165 minus00116 minus00086 025 025 025 025 minus00303 minus00165 minus00116 minus00086050 000 025 025 minus00329 minus00177 minus00128 minus00093 025 050 000 025 minus00361 minus00189 minus00132 minus00100

075 000 000 025 minus00418 minus00226 minus00157 minus00116 025 075 000 000 minus00426 minus00236 minus00165 minus00123

100 000 000 000 minus00536 minus00285 minus00199 minus00150 000 100 000 000 minus00491 minus00269 minus00190 minus00138

025 025 025 025 minus00303 minus00165 minus00116 minus00086 025 025 025 025 minus00303 minus00165 minus00116 minus00086025 025 050 000 minus0332 minus00176 minus00119 minus00093 000 025 050 050 minus00359 minus00181 minus00128 minus00094

000 025 075 000 minus00403 minus00196 minus00127 minus00097 000 000 025 075 minus00402 minus00206 minus00146 minus00108

000 000 100 000 minus00420 minus00211 minus00139 minus00101 000 000 000 100 minus00425 minus00230 minus00157 minus00119

Note the ratio of index portfolio is ranked by the sequence of SSCI HSI TAIEX SP500 in Table 7

and heavy-tailrdquo characteristics the time-varying volatilitycharacteristics and extreme-tail dependence characteristicsof financial asset return It proposed the parameter estimationalgorithm of the multivariate time-varying 119866-119867 CopulaGARCH model by using condition maximum likelihoodmethod and IFM two-step method An algorithm was con-structed to calculate VaR by using the quantile functionand the simulation method based on 119866-119867 Copula GARCHmodel In addition this paper selected the daily log returnof SSCI (China) HSI (Hong Kong China) TAIEX (TaiwanChina) and SP500 (USA) from January 3 2000 to June18 2010 as samples to estimate the parameters of themultivariate time-varying 119866-119867 Copula GARCH model andit also estimated theVaR for various index risk asset portfoliosunder different confidence levelsThe research results showedthat the multivariate time-varying 119866-119867 Copula GARCHmodel constructed in this paper could reasonably estimateand measure the extreme losses of risk portfolios in financialmarket and the measurement results were in line with theactual situation of stock market and the risk diversificationtheory of portfolio The achievement of this paper provideda practical and effective method for measuring the extremelosses of financial market

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees for all help-ful comments and suggestions This work is supported byNatural Science Foundation Project of CQ CSTC (Grant noCSTC 2011BB2088) and Key Project of National NaturalScience Foundation (Grant no 71232004)

References

[1] S J Kon ldquoModels of stock returnsmdasha comparisonrdquoThe Journalof Finance vol 39 no 1 pp 147ndash165 1984

[2] J B Gray and D W French ldquoEmpirical comparisons of dis-tributional models for stock index returnsrdquo Journal of BusinessFinance amp Accounting vol 17 no 3 pp 451ndash459 1990

[3] E Jondeau andMRockinger ldquoTesting for differences in the tailsof stock-market returnsrdquo Journal of Empirical Finance vol 10no 5 pp 559ndash581 2003

[4] C Jiang S Li and S Liang ldquoEmpirical investigation ofdistribution feature of return in china stockmarketrdquoApplicationof Statistics and Management vol 26 no 4 pp 710ndash717 2007

[5] D Dong and W Jin ldquoA subjective model of the distribution ofreturns and empirical analysisrdquo Chinese Journal of ManagementScience vol 15 no 1 pp 112ndash120 2007

[6] J Martinez and B Iglewicz ldquoSome properties of the tukey g andh family of distributionsrdquoCommunications in StatisticsmdashTheoryand Methods vol 13 no 3 pp 353ndash369 1984

[7] T C Mills ldquoModelling skewness and kurtosis in the Londonstock exchange FT-SE index return distributionsrdquo The Statisti-cian vol 44 no 3 pp 323ndash332 1995

[8] H Zhu and Z Pan ldquoA study of portfolios VaRmethod based ong-h distributionrdquo Chinese Journal of Management Science vol13 no 4 pp 7ndash12 2005

[9] K Kuester S Mittnik andM S Paolella ldquoValue-at-risk predic-tion a comparison of alternative strategiesrdquo Journal of FinancialEconometrics vol 4 no 1 pp 53ndash89 2006

[10] M Degen P Embrechts and D Lambrigger ldquoThe quantitativemodeling of operational risk between g-and-h and EVTrdquo AstinBulletin vol 37 no 2 pp 265ndash291 2007

[11] E Jondeau and M Rockinger ldquoThe Copula-GARCH modelof conditional dependencies an international stock marketapplicationrdquo Journal of International Money and Finance vol25 no 5 pp 827ndash853 2006

[12] J C Rodriguez ldquoMeasuring financial contagion a Copulaapproachrdquo Journal of Empirical Finance vol 14 no 3 pp 401ndash423 2007

Mathematical Problems in Engineering 9

[13] M Fischer C Kock S Schluter and F Weigert ldquoAn empiricalanalysis of multivariate copula modelsrdquo Quantitative Financevol 9 no 7 pp 839ndash854 2009

[14] W Sun S Rachev F J Fabozzi and S S Kalev ldquoA newapproach to modeling co-movement of international equitymarkets evidence of unconditional copula-based simulation oftail dependencerdquo Empirical Economics vol 36 no 1 pp 201ndash229 2009

[15] Z Liu T Zhang and F Wen ldquoTime-varying risk attitude andconditional skewnessrdquo Abstract and Applied Analysis vol 2014Article ID 174848 11 pages 2014

[16] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 3 no 48 pp235ndash254 2014

[17] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[18] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics and Economics vol 45 no 3 pp 315ndash324 2009

[19] A Ghorbel and A Trabelsi ldquoMeasure of financial risk usingconditional extreme value copulas with EVT marginsrdquo Journalof Risk vol 11 no 4 pp 51ndash85 2009

[20] L Chollete A Heinen and A Valdesogo ldquoModeling interna-tional financial returns with a multivariate regime-switchingcopulardquo Journal of Financial Econometrics vol 7 no 4 pp 437ndash480 2009

[21] M Huggenberger and T Klett ldquoA g-and-h copula approach torisk measurement in multivariate financial modelsrdquo WorkingPaper 2010 httpssrncomabstract=1677431

[22] J Wang W Bao and J Hu ldquoPortfolios VaR analyses basedon the multi-dimensional gumbel copula functionrdquo Journal ofApplied Statistics and Management vol 29 no 1 pp 137ndash1432010

[23] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis and Applications vol 16 no 1 pp93ndash103 2014

[24] J Liu M Tao C Ma and F Wen ldquoUtility indifference pricingof convertible bondsrdquo International Journal of InformationTechnology and Decision Making vol 13 no 2 pp 429ndash4442014

[25] R B Nelsen An Introduction to Copulas Springer Science ampBusiness Media New York NY USA 2007

[26] H JoeMultivariate Models and Dependence Concepts vol 73 ofMonographs on Statistics and Applied Probability Chapman ampHall London UK 1997

[27] A J Patton ldquoEstimation of multivariate models for time seriesof possibly different lengthsrdquo Journal of Applied Econometricsvol 21 no 2 pp 147ndash173 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article Multivariate Time-Varying Copula GARCH ...downloads.hindawi.com/journals/mpe/2015/286014.pdf · model, they presented an empirical pricing study of China s market.

8 Mathematical Problems in Engineering

Table 6 Estimates of correlation coefficients time-varying parameters and degree of freedom of four-variate time-varying G-H CopulaGARCH (1 1) model based on SSCI HSI TAIEX and SP500

Model parameters 12058812

12058813

12058814

12058823

12058824

12058834

a b 120578

Estimate 0256lowastlowastlowast

0139lowastlowastlowast

0021lowastlowastlowast

0528lowastlowastlowast

0193lowastlowastlowast

0128lowastlowast

0008lowastlowastlowast

0987lowastlowastlowast

1457lowastlowastlowast

119879-statistic 1351 1102 3431 5763 9564 2529 5179 3561 6871Note lowastlowastlowast and lowastlowast in the table denote that the parameter is significant at 1 and 5 level respectively

Table 7 VaR estimation results based on the four-variate time-varying G-H Copula GARCHmodel with SSCI HSI TAIEX and SP500

Ratio of index portfolio 1 5 10 15 Ratio of index portfolio 1 5 10 15025 025 025 025 minus00303 minus00165 minus00116 minus00086 025 025 025 025 minus00303 minus00165 minus00116 minus00086050 000 025 025 minus00329 minus00177 minus00128 minus00093 025 050 000 025 minus00361 minus00189 minus00132 minus00100

075 000 000 025 minus00418 minus00226 minus00157 minus00116 025 075 000 000 minus00426 minus00236 minus00165 minus00123

100 000 000 000 minus00536 minus00285 minus00199 minus00150 000 100 000 000 minus00491 minus00269 minus00190 minus00138

025 025 025 025 minus00303 minus00165 minus00116 minus00086 025 025 025 025 minus00303 minus00165 minus00116 minus00086025 025 050 000 minus0332 minus00176 minus00119 minus00093 000 025 050 050 minus00359 minus00181 minus00128 minus00094

000 025 075 000 minus00403 minus00196 minus00127 minus00097 000 000 025 075 minus00402 minus00206 minus00146 minus00108

000 000 100 000 minus00420 minus00211 minus00139 minus00101 000 000 000 100 minus00425 minus00230 minus00157 minus00119

Note the ratio of index portfolio is ranked by the sequence of SSCI HSI TAIEX SP500 in Table 7

and heavy-tailrdquo characteristics the time-varying volatilitycharacteristics and extreme-tail dependence characteristicsof financial asset return It proposed the parameter estimationalgorithm of the multivariate time-varying 119866-119867 CopulaGARCH model by using condition maximum likelihoodmethod and IFM two-step method An algorithm was con-structed to calculate VaR by using the quantile functionand the simulation method based on 119866-119867 Copula GARCHmodel In addition this paper selected the daily log returnof SSCI (China) HSI (Hong Kong China) TAIEX (TaiwanChina) and SP500 (USA) from January 3 2000 to June18 2010 as samples to estimate the parameters of themultivariate time-varying 119866-119867 Copula GARCH model andit also estimated theVaR for various index risk asset portfoliosunder different confidence levelsThe research results showedthat the multivariate time-varying 119866-119867 Copula GARCHmodel constructed in this paper could reasonably estimateand measure the extreme losses of risk portfolios in financialmarket and the measurement results were in line with theactual situation of stock market and the risk diversificationtheory of portfolio The achievement of this paper provideda practical and effective method for measuring the extremelosses of financial market

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees for all help-ful comments and suggestions This work is supported byNatural Science Foundation Project of CQ CSTC (Grant noCSTC 2011BB2088) and Key Project of National NaturalScience Foundation (Grant no 71232004)

References

[1] S J Kon ldquoModels of stock returnsmdasha comparisonrdquoThe Journalof Finance vol 39 no 1 pp 147ndash165 1984

[2] J B Gray and D W French ldquoEmpirical comparisons of dis-tributional models for stock index returnsrdquo Journal of BusinessFinance amp Accounting vol 17 no 3 pp 451ndash459 1990

[3] E Jondeau andMRockinger ldquoTesting for differences in the tailsof stock-market returnsrdquo Journal of Empirical Finance vol 10no 5 pp 559ndash581 2003

[4] C Jiang S Li and S Liang ldquoEmpirical investigation ofdistribution feature of return in china stockmarketrdquoApplicationof Statistics and Management vol 26 no 4 pp 710ndash717 2007

[5] D Dong and W Jin ldquoA subjective model of the distribution ofreturns and empirical analysisrdquo Chinese Journal of ManagementScience vol 15 no 1 pp 112ndash120 2007

[6] J Martinez and B Iglewicz ldquoSome properties of the tukey g andh family of distributionsrdquoCommunications in StatisticsmdashTheoryand Methods vol 13 no 3 pp 353ndash369 1984

[7] T C Mills ldquoModelling skewness and kurtosis in the Londonstock exchange FT-SE index return distributionsrdquo The Statisti-cian vol 44 no 3 pp 323ndash332 1995

[8] H Zhu and Z Pan ldquoA study of portfolios VaRmethod based ong-h distributionrdquo Chinese Journal of Management Science vol13 no 4 pp 7ndash12 2005

[9] K Kuester S Mittnik andM S Paolella ldquoValue-at-risk predic-tion a comparison of alternative strategiesrdquo Journal of FinancialEconometrics vol 4 no 1 pp 53ndash89 2006

[10] M Degen P Embrechts and D Lambrigger ldquoThe quantitativemodeling of operational risk between g-and-h and EVTrdquo AstinBulletin vol 37 no 2 pp 265ndash291 2007

[11] E Jondeau and M Rockinger ldquoThe Copula-GARCH modelof conditional dependencies an international stock marketapplicationrdquo Journal of International Money and Finance vol25 no 5 pp 827ndash853 2006

[12] J C Rodriguez ldquoMeasuring financial contagion a Copulaapproachrdquo Journal of Empirical Finance vol 14 no 3 pp 401ndash423 2007

Mathematical Problems in Engineering 9

[13] M Fischer C Kock S Schluter and F Weigert ldquoAn empiricalanalysis of multivariate copula modelsrdquo Quantitative Financevol 9 no 7 pp 839ndash854 2009

[14] W Sun S Rachev F J Fabozzi and S S Kalev ldquoA newapproach to modeling co-movement of international equitymarkets evidence of unconditional copula-based simulation oftail dependencerdquo Empirical Economics vol 36 no 1 pp 201ndash229 2009

[15] Z Liu T Zhang and F Wen ldquoTime-varying risk attitude andconditional skewnessrdquo Abstract and Applied Analysis vol 2014Article ID 174848 11 pages 2014

[16] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 3 no 48 pp235ndash254 2014

[17] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[18] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics and Economics vol 45 no 3 pp 315ndash324 2009

[19] A Ghorbel and A Trabelsi ldquoMeasure of financial risk usingconditional extreme value copulas with EVT marginsrdquo Journalof Risk vol 11 no 4 pp 51ndash85 2009

[20] L Chollete A Heinen and A Valdesogo ldquoModeling interna-tional financial returns with a multivariate regime-switchingcopulardquo Journal of Financial Econometrics vol 7 no 4 pp 437ndash480 2009

[21] M Huggenberger and T Klett ldquoA g-and-h copula approach torisk measurement in multivariate financial modelsrdquo WorkingPaper 2010 httpssrncomabstract=1677431

[22] J Wang W Bao and J Hu ldquoPortfolios VaR analyses basedon the multi-dimensional gumbel copula functionrdquo Journal ofApplied Statistics and Management vol 29 no 1 pp 137ndash1432010

[23] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis and Applications vol 16 no 1 pp93ndash103 2014

[24] J Liu M Tao C Ma and F Wen ldquoUtility indifference pricingof convertible bondsrdquo International Journal of InformationTechnology and Decision Making vol 13 no 2 pp 429ndash4442014

[25] R B Nelsen An Introduction to Copulas Springer Science ampBusiness Media New York NY USA 2007

[26] H JoeMultivariate Models and Dependence Concepts vol 73 ofMonographs on Statistics and Applied Probability Chapman ampHall London UK 1997

[27] A J Patton ldquoEstimation of multivariate models for time seriesof possibly different lengthsrdquo Journal of Applied Econometricsvol 21 no 2 pp 147ndash173 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Multivariate Time-Varying Copula GARCH ...downloads.hindawi.com/journals/mpe/2015/286014.pdf · model, they presented an empirical pricing study of China s market.

Mathematical Problems in Engineering 9

[13] M Fischer C Kock S Schluter and F Weigert ldquoAn empiricalanalysis of multivariate copula modelsrdquo Quantitative Financevol 9 no 7 pp 839ndash854 2009

[14] W Sun S Rachev F J Fabozzi and S S Kalev ldquoA newapproach to modeling co-movement of international equitymarkets evidence of unconditional copula-based simulation oftail dependencerdquo Empirical Economics vol 36 no 1 pp 201ndash229 2009

[15] Z Liu T Zhang and F Wen ldquoTime-varying risk attitude andconditional skewnessrdquo Abstract and Applied Analysis vol 2014Article ID 174848 11 pages 2014

[16] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 3 no 48 pp235ndash254 2014

[17] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[18] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics and Economics vol 45 no 3 pp 315ndash324 2009

[19] A Ghorbel and A Trabelsi ldquoMeasure of financial risk usingconditional extreme value copulas with EVT marginsrdquo Journalof Risk vol 11 no 4 pp 51ndash85 2009

[20] L Chollete A Heinen and A Valdesogo ldquoModeling interna-tional financial returns with a multivariate regime-switchingcopulardquo Journal of Financial Econometrics vol 7 no 4 pp 437ndash480 2009

[21] M Huggenberger and T Klett ldquoA g-and-h copula approach torisk measurement in multivariate financial modelsrdquo WorkingPaper 2010 httpssrncomabstract=1677431

[22] J Wang W Bao and J Hu ldquoPortfolios VaR analyses basedon the multi-dimensional gumbel copula functionrdquo Journal ofApplied Statistics and Management vol 29 no 1 pp 137ndash1432010

[23] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis and Applications vol 16 no 1 pp93ndash103 2014

[24] J Liu M Tao C Ma and F Wen ldquoUtility indifference pricingof convertible bondsrdquo International Journal of InformationTechnology and Decision Making vol 13 no 2 pp 429ndash4442014

[25] R B Nelsen An Introduction to Copulas Springer Science ampBusiness Media New York NY USA 2007

[26] H JoeMultivariate Models and Dependence Concepts vol 73 ofMonographs on Statistics and Applied Probability Chapman ampHall London UK 1997

[27] A J Patton ldquoEstimation of multivariate models for time seriesof possibly different lengthsrdquo Journal of Applied Econometricsvol 21 no 2 pp 147ndash173 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article Multivariate Time-Varying Copula GARCH ...downloads.hindawi.com/journals/mpe/2015/286014.pdf · model, they presented an empirical pricing study of China s market.

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


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