Research ArticleEstimating Risk of Natural Gas Portfolios by UsingGARCH-EVT-Copula Model
Jiechen Tang1 Chao Zhou23 Xinyu Yuan4 and Songsak Sriboonchitta1
1Faculty of Economics Chiang Mai University 2397 Suthep A Mueang Chiang Mai 200060 Thailand2School of Economics Northwest Normal University Lanzhou China3The Peoplersquos Bank of China Zhang Ye City Branch Zhangye China4Faculty of Economics and Management Yunnan Normal University Yunnan China
Correspondence should be addressed to Jiechen Tang tangjiechen1002163com
Received 25 July 2014 Revised 9 November 2014 Accepted 30 November 2014
Academic Editor WingKeung Wong
Copyright copy 2015 Jiechen Tang 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
This paper concentrates on estimating the risk of Title Transfer Facility (TTF) Hub natural gas portfolios by using the GARCH-EVT-copula modelWe first use the univariate ARMA-GARCHmodel tomodel each natural gas return series Second the extremevalue distribution (EVT) is fitted to the tails of the residuals to model marginal residual distributionsThird multivariate Gaussiancopula and Student 119905-copula are employed to describe the natural gas portfolio risk dependence structure Finally we simulate Nportfolios and estimate value at risk (VaR) and conditional value at risk (CVaR) Our empirical results show that for an equallyweighted portfolio of five natural gases the VaR and CVaR values obtained from the Student 119905-copula are larger than those obtainedfrom the Gaussian copula Moreover when minimizing the portfolio risk the optimal natural gas portfolio weights are found to besimilar across the multivariate Gaussian copula and Student 119905-copula and different confidence levels
1 Introduction
Natural gas has become an important commodity for globaleconomy today and is commonly regarded as a strategiccommodity Since the momentous arrival of hedge funds andother new players that have their focus on this strategic com-modity the market for natural gas has become increasinglyvolatile and risky Hence managing the risks associated withspot and futures natural gas prices has become a crucial issuefor financial institutions regulators and portfolio managersOne of the most popular and usual tools for measuringmarket risk is VaR VaR is defined as the amount of loss ina portfolio under given probability over a certain time period[1] VaR is considered as a benchmark for measuring marketrisk which is the main reason that it can summarize risksin a single number Additionally CVaR is a supplement toVaR and it measures the expected loss given that the loss isgreater than or equal to the VaR at certain confidence levels[2] The purpose of this study is to apply the GARCH-EVT-copula model to estimate the risk measures (VaR and CVaR)for natural gas portfolios
More and more empirical studies concerned with VaRestimation have been using the copula-GARCHmodelManyresearch literatures have proved that the copula method isa better method to estimate VaR For example Huang et al[3] proposed a new method conditional copula-GARCH tomeasure the VaR of stock portfolios Their empirical resultssuggested that conditional copula-GARCH could be quiterobust in estimating VaR Low et al [4] investigated whetherapplying copula models to forecast returns for portfolioscould produce superior investment performance comparedto traditional modelsThey found that whenmanaging largerportfolios theClayton canonical vine copula outperforms theClayton standard copula Aloui et al [5] used the copula-GARCH approach to investigate the conditional dependencestructure between crude oil prices and US dollar exchangerates The empirical results suggested that using the Student119905-copula improved the accuracy of VaR forecasts Chen andTu [6] estimated the risk (VaR) of hedged portfolio toillustrate the potential model risk They found that copula-based HPVaR outperforms the conventional CCC- andDCC-GARCH estimators Weiszlig and Supper [7] estimated
Hindawi Publishing Corporatione Scientific World JournalVolume 2015 Article ID 125958 7 pageshttpdxdoiorg1011552015125958
2 The Scientific World Journal
the liquidity-adjusted intraday VaR of stock portfolio byusing vine copulas for the dependence structure and theACDP (Autoregressive Conditional Double Poisson) andGARCH processes for the marginal distribution The empir-ical evidences showed that the GARCH-Vine copula modelperformed well in estimating the liquidity-adjusted intradayportfolio profits and losses Berger [8] used time-varyingcopulas combined with EVT-based margins He pointed outthat when forecasting the VaR of a portfolio it is crucialto find the ldquorightrdquo portfolio return distribution They foundthat reliable copulas combined with EVT-based margins cancapture reliable VaR figures Aloui et al [9] investigatedthe conditional dependence structure between the crude oiland natural gas markets and derived implications for riskmanagement issues related to an oil and gas portfolio withina VaR framework
The contributions of this paper are twofold First tothe best of our knowledge there are no previous empir-ical papers estimating the risks of natural gas portfoliosusing the GARCH-EVT-copula model In this sense thisstudy can perfectly contribute to this area of the litera-ture Second risk measures of natural gas portfolios whichare computed by the GARCH-EVT-copula model can beutilized to have implications for natural gas portfolio riskmanagement
The outline of this paper is as follows The next sectiondescribes the GARCH-EVT-copula model used in the paperSection 3 discusses the data and their properties Section 4presents the empirical results of the model and estimatethe VaR and the CVaR five-dimensional natural gas port-folios and the optimal natural gas portfolio weights underminimized portfolio risk The last section delivers the finalconcluding remarks
2 Data Description
This study focuses on natural gas portfolios consisting ofspot and futures prices at the Title Transfer Facility (TTF)Hub Our dataset consists of daily series of one-day-aheadgas prices (TTFFD) one-month-ahead cash-settled futuresnatural gas prices (TTFFM) one-quarter-ahead cash-settledfutures natural gas prices (TTFFQ) one-season-ahead cash-settled futures natural gas prices (TTFFS) and one-year-ahead cash-settled futures natural gas prices (denoted byTTFFY) The natural gas price series are expressed in europer megawatt hour (CMWh) The data are sourced fromDatastream and are from January 1 2007 to May 30 2014The data used in this paper are from Monday to Friday withtotal of 1934 observations
Figure 1 is plotted to illustrate the natural gas spot andfutures prices at the TTF Hub The figures of the spot andfutures prices reveal the typical properties of natural gasprices extreme price spikes and volatility clustered Thedescriptive statistics for natural gas spot and futures returnsare reported in Table 1 The natural gas spot and futuresreturns (119903
119905) are computed on a continuous compounding
basis as 119903119905= ln(119875
119905119875119905minus1) where 119875
119905and 119875
119905minus1are current
and one-period lagged daily natural gas spot or futuresprices The mean returns for all spot and futures series
are very small and near zero indicating that there is nosignificant trend in the data All the returns exhibit significantskewness except for TTFFD and kurtosis All the returnsexhibit positive skewness except for TTFFD Second all thereturns strongly reject the null hypotheses for the JB (Jarque-Bera) test of normality Finally augmentedDickey-Fuller [10]and Kwiatkowski-Phillips-Schmidt-Shin [11] tests are usedfor the unit root test Table 1 shows the results indicatingthat all return series are stationary at the 1 significancelevel
3 Methodology
31 Model forMarginal Distribution We adopt the univariatestandard GARCH model for natural gas price to obtain ourmarginal distributions in this study The univariate standardGARCH(1 1) model [12] is defined as follows
where 119904 is TTFFD TTFFM TTFFQ TTFFS and TTFFYrespectively 119903
119904119905denotes the spot or futures return of natural
gas ℎ119904119905
is the conditional variance of volatility 120601119904119894
is the119894th lag AR parameter 120579
119904119895is the 119894th lag MA parameter 120572
119904is
the ARCH parameter and 120573119904is the GARCH parameter The
conditions 120596119904gt 0 120572
119904 120573119904ge 0 and 120572
119904+ 120573119904lt 1 are to ensure
that the conditional variance process remains positive andstationary 119911
119904119905is the standardized error In Section 2 we know
that all series are not normal distribution Hence we assumethat 119911
119904119905is 119905-distribution df
119904is the degree of freedom which
is denoted as 120582119904 In order to better estimate the tails of the
distribution this paper applied EVT to those residuals Wemodel residuals by using the generalized Pareto distribution(GPD) estimate for the upper and lower tails and theGaussiankernel estimate for the remaining part The CDF of a GPDdistribution is parameterized as follows
where 120585 is the shape parameter 120573 is the scale parameter and119898119871(119898119877) is the lower (upper) threshold
There is a trade-off between high precision and low vari-ance the critical step is the choice of the optimal thresholdFollowing DuMouchel [13] this paper chose the exceedancesto be the 10th percentile of the sample
The Scientific World Journal 3
40
30
20
10
0
40
35
30
25
20
15
10
5
50
40
30
20
10
0
2007 2008 2009 2010 2011 2012 2013 2014
TTFFD TTFFM
TTFFQ TTFFS
TTFFY
2007 2008 2009 2010 2011 2012 2013 2014
2007 2008 2009 2010 2011 2012 2013 2014
2007 2008 2009 2010 2011 2012 2013 2014
2007 2008 2009 2010 2011 2012 2013 2014
50
40
30
20
10
0
50
40
30
20
10
Figure 1 The time series plots of daily natural gas spot and futures prices at the Title Transfer Facility (TTF) Hub
Table 1 Descriptive statistics of spot and generic futures returns for natural gas portfolio
is the cumulative distribution function of thestandardized error from the marginal models Φ
sum120588denotes
the standardized multivariate normal distribution with thecorrelation matrix sum120588 Φminus1 is the inverse of the univariatestandard normal distribution and 120588 isin (minus1 1) measures thedependence
The Gaussian copula cannot capture the dependence of afat tail In order to capture the fat tail we also introduced themultivariate Student 119905-copula in this study The multivariateStudent 119905-copula [14] is defined as
where 119905sum120588V is the standardized multivariate Student 119905-
distribution with correlation and degrees of freedom matrixsum120588 V correlation 120588 isin (minus1 1) 119905minus1V denotes the inverse of theunivariate Student 119905-distribution The parameters of V and 120588determine the extent of symmetric extreme dependence bythe upper and lower tail dependences 120582
331 VaR and CVaR In order to analyze risk managementthis paper used VaR and CVaR to measure the risk Weestimate the VaR and the CVaR by taking the 120572-quantilesof the portfolio return forecasts as given in the followingequation
We assume that the weight in each asset is the same thatis 120596 = [(15) (15) (15) (15) (15)]
1015840 As a result wecan compute the VaR and the CVaR of the equally weightedportfolio
332 Optimal Portfolio with Minimum Risk We estimatethe VaR and the CVaR of an equally weighted portfoliobut the major concern for commercial banks and individual
investors is to minimize the risk of the investment portfolioTo address this concern in this paper we also compute theoptimal weights of each asset thatminimize the portfolio riskAccording to Wang et al [2] the algorithm is as follows Weassume that the weight in individual asset within a portfoliois 120596 = [120596
1 1205962 1205963 1205964 1205965]1015840 where 0 le 120596
119904le 1 (119904 = 1 2 5)
and 1205961+ 1205962+ 1205963+ 1205964+ 1205965= 1 Many weight vectors
can satisfy these conditions Given a weight vector 120596 119873returns of the portfolio are obtained this means that returns= 119903119905+1
times 120596 Subsequently the VaR of these portfolios at agiven confidence level can be computed After getting allthe VaR values corresponding to all the weight vectors theminimumVaR can be calculatedThus we can get the weightof each asset that minimizes the VaR that is minVaR
120572=
risk(1198871 1198872 1198873 1198874 1198875)
34 Estimation Procedure We follow a seven-step estimationprocedure in this paper
(1) We first complete the parameter estimation andobtain the standardized residuals by fitting the uni-variate ARMA(119901 119902)-GARCH(1 1) model for eachnatural gas return series
(2) For EVT we chose the exceedances to be the 10thpercentile of the sample and used the sample MEFplot and the Hill plot to determine an appropriatethreshold for the upper and lower tails of the dis-tribution of the residuals We assume that excessresiduals over this threshold follow the generalizedPareto distribution (GPD) and estimate for the upperand lower tails and use theGaussian kernel estimationfor the remaining part
(3) We transform the standardized residuals 119911119904119905from the
GARCHmodel into the variates 119906119904119905 using the ECDFs
(empirical cumulative distribution functions)There-after we perform the goodness-of-fit test for each 119906
119904119905
(4) We fit a copula model and estimate its parameter(5) We simulate 119873 times from the estimated copula
model to convert to 119873 standardized residuals usingthe inverse of the corresponding distribution functionfor each model
(6) We forecast 119873 portfolio returns using the ARMA(119901119902)-GARCH(1 1)model
(7) We estimate the VaR and the CVaR of the equallyweighted portfolio Additionally we compute theoptimal portfolio with the minimum risk
4 Empirical Results
41 Results for Marginal Model Firstly estimate theARMA(119901 119902)-GARCH(1 1) model for each natural gasreturn The AIC BIC and Loglik statistics are adopted toselect the most suitable models The estimated coefficientsand standard errors for the marginal model are presentedin Table 2 Table 2 shows that the AR(1) estimates 120601
1199041are
significant in all the series except for TTFFM and TTFFQwhich implies that the current returns have spillover effects
The Scientific World Journal 5
Table 2 Estimated parameters of agricultural commodity returnsfor marginal model
TTFFD TTFFM TTFFQ TTFFS TTFFY
119888119904119900
minus00002(00004)
minus00007lowastlowast(00003)
minus00008(00003)
minus00005lowast(00002)
minus00003(00002)
1206011199041
minus01350lowastlowast(00234)
00363(00206)
00206(00190)
00668lowastlowast(00218)
00588lowastlowast(00223)
120596119904
00000lowastlowast(00000)
00000(00000)
00000lowastlowast(00000)
00000lowastlowast(00000)
00000lowastlowast(00000)
120572119904
02045lowastlowast(00263)
00560lowastlowast(00169)
00120lowastlowast(00035)
01865lowastlowast(00341)
01661lowastlowast(00243)
120573119904
07945lowastlowast(00238)
09430lowastlowast(00170)
09848lowastlowast(00030)
08125lowastlowast(00305)
08329lowastlowast(00213)
120582119904
43963lowastlowast(03857)
32386lowastlowast(02172)
28659lowastlowast(01965)
31684lowastlowast(01866)
36138lowastlowast(02616)
LL 39100000 49437790 51541850 54091680 57945190AIC minus40372 minus51063 minus53239 minus55876 minus59861ARCH(10) 07074 08258 09988 10000 09996NoteThe table shows the estimates and the standard errors of the parametersfor the marginal distribution model defined in (3) and (4) lowastlowast and lowastdenote rejection of the null hypothesis at the 1 and 5 significance levelsrespectively ARCH(10) is the P value of Englersquos LM test for the ARCH effectin the residuals up to the 10th order
from their returns in one of the previous periods Secondit can be observed that the ARCH parameter 120572
that all the returns experience ARCH and GARCH effectsThe 119875 values of ARCH(10) are more than 10 which showsthat there is no heteroskedasticity effect in the residualsAdditionally the degrees of freedom 120582
119904of the Student
119905-distribution for all the returns are significant suggestingthat the error terms are not normal distribution and thatthe Student 119905-distribution works reasonably well for all thereturn series
As described in Section 31 we apply EVT to thoseresiduals GPD specially is for tail estimation and we choosethe exceedances to be the 10th percentile of the sampleThe Gaussian kernel estimate is for the remaining partThe parameter estimation for each natural gasrsquos residuals isdemonstrated in Table 3
Given the availability of the estimates ofmarginalmodelswe proceed to estimate the copula functions For that wetransform the standardized residuals 119911
119904119905from the GARCH
model into the variates 119906119904119905 using the ECDF Each variate 119906
119904119905
should be uniform (0 1) otherwise the copula model couldbe misspecifiedThis paper follows Patton [15] and Reboredo[16] to carry out the goodness-of-fit test which are Ljung-Box(LB) test and Kolmogorov-Smirnov (KS) test LB test is usedto examine the serial correlation under the null hypothesisof serial independence To do so (119906
119904119905minus 119906119904)119896 is regressed on
the first 10 lags of the variables And KS test is used to testthe null hypothesis that the 119906
119894119905are uniform (0 1) Table 4
reports the 119875 values of these two tests At the 5 significancelevel all the null hypotheses are rejected which implies thatthe marginal distribution model demonstrates that these are
Table 3 Parameter estimation for each natural gasrsquos residuals
KS test 09997 10000 09436 09997 08967Note This table reports the P values from the Ljung-Box (LB) test for serialindependence of the first four moments of the variable 119906
119904119905 We regress
(119906119904119905minus 119906119904)119896 on the first five lags of the variables for 119896 = 1 2 3 4 In
addition we present the P values of the Kolmogorov-Smirnov (KS) test forthe adequacy of the distribution model
not misspecified Hence it can be concluded that the copulamodel can correctly capture the dependence
42 Results for Copula After estimating the parameters ofthe marginal distribution we proceed further to estimatethe copula parameter The multivariate Gaussian copula andStudent 119905-copula are applied in this study The dependencematrix is estimated by the maximum-likelihood estimation(MLE) method The results of the dependence matrix forthe multivariate Gaussian copula model and the Student 119905-copula model are reported in Tables 5 and 6 respectivelyThedependence parameters are positive and strongly significantat 10 level for theGaussian and the Student 119905-copulamodelsThese findings point out the presence of significantly strongpositive relationships between the TTF natural gas portfoliosSecond the degree of freedom for the multivariate Student119905-copula V is reasonably low (49837) which suggests theexistence of substantially extreme comovements and taildependence for all the pairs of the TTF natural gas portfoliosThis finding indicates that extreme events or crises are likelyto spread from one series to another during the bull and bearperiods
43 Portfolio Risk Analysis After modeling the dependencestructure we simulate one-month natural gas portfolioreturns based on the dependence structure specified in boththe multivariate Gaussian copula and the Student 119905-copulaand estimate the risk of a five-dimensional natural gasportfolio With an equally weighted portfolio of five natural
6 The Scientific World Journal
Table 5 Estimated parameters of natural gas portfolio for multi-variate Gaussian copula
Note lowastlowast denotes rejection of the null hypothesis at the 1 and 5 signifi-cance levels respectively
gas returns (TTFFD TTFFM TTFFQ TTFFS and TTFFY)we can calculate the VaR and the CVaR of the portfolio Theresults of the VaR and the CVaR calculations are presentedin Table 7 From Table 7 we can perceive that the VaR andthe CVaR calculated from the Student 119905-copula are more thanthose from the Gaussian copula under the same confidencelevel The reason is that the Student 119905-copula considers theextreme dependence whereas the Gaussian copula does not
Likewise in this paper we also compute the optimalweights of each asset that minimize the portfolio risk Theoptimal portfolio weight under the minimum portfolio riskand the VaR (CVaR) under different copulas and confidencelevels are presented in Table 8 From Table 8 we can draw thefollowing interesting conclusions
(1) The optimal portfolio weights are similar acrossthe multivariate Gaussian copula and Student 119905-copula and different confidence levels The optimalinvestment tends to concentrate on TTFFM TTFFQTTFFS and TTFFY Moreover the optimal portfolioweights of TTFFM TTFFQ TTFFS and TTFFY arealmost equal However the optimal portfolio weightof TTFFD is nearly zero This is because TTFFD isthe spot (one-day-ahead) price and TTFFM TTFFQTTFFS and TTFFY are the futures prices Futures
Table 7 Portfolio risk under equally weighted natural gas portfolio
Confidence level Risk value Normal copula 119905-copula
090 VaR 00746 00755CVaR 01094 01112
095 VaR 00977 00991CVaR 01344 01365
099 VaR 01565 01624CVaR 01989 01994
markets have risk aversion functionUnderminimumportfolio risk investors prefer the futures market Onthe other hand with higher confidence levels theweight of TTFFD investment becomes larger Thereason is that with higher confidence levels investorswill be more risk tolerant and therefore willing totake more risk to achieve higher expected returns
(2) Comparing the risk value calculated by the Gaussiancopula and Student 119905-copula we observe that mostof the VaR and all the CVaR from the Gaussiancopula are less than those from the Student 119905-copulaThis result is consistent with the conclusions whencomputing the VaR and the CVaR of the equallyweighted foreign exchange portfolio This is becausethe Student 119905-copula on account of its heavy tail isable to capture the extreme dependence The extremeevents of the asset returns have higher dependenceand hence larger VaR and CVaR
5 Conclusion
We employed the GARCH-EVT-copula model to estimatethe risk of a multidimensional natural gas portfolio Thisstudy focuses on the natural gas portfolio at the Title TransferFacility (TTF) Hub The data consist of daily series ofone-day-ahead gas prices (TTFFD) one-month-ahead cash-settled futures natural gas prices (TTFFM) one-quarter-ahead cash-settled futures natural gas prices (TTFFQ) one-season-ahead cash-settled futures natural gas prices (TTFFS)and one-year-ahead cash-settled futures natural gas prices(denoted by TTFFY) Our dataset is from January 1 2007 toMay 30 2014 The multivariate Gaussian copula and Student119905-copula have been employed to model the dependencestructure
The empirical results show that for an equally weightedportfolio of five natural gases theVaR and theCVaR obtainedfrom the Student 119905-copula are larger than those obtained fromthe Gaussian copula The reason is that the Gaussian copulacannot capture the extreme dependence but the Student119905-copula can Second the optimal portfolio weights arefound to be similar across the multivariate Gaussian copulaand Student 119905-copula and different confidence levels uponminimizing the portfolio risk The optimal investment tendsto concentrate on TTFFM TTFFQ TTFFS and TTFFY Inaddition the optimal portfolio weights of TTFFM TTFFQTTFFS and TTFFY are almost equal At the same timethe optimal portfolio weight of TTFFD is nearly zero With
The Scientific World Journal 7
Table 8 Optimal investment proportion of natural gas portfolio with minimum risk
higher confidence levels theweight of the TTFFD investmentbecomes larger
Further studies on estimating the risk of natural gasportfolios should concentrate on modeling dependenceSpecifically we should employ more copulas multivariateClaytonmultivariateGumbel and vine copula Furthermorethe risk of natural gas portfolio in different Hubs should beconsidered Third we should work on backtesting to checkthe performance of the approach compared to other popularapproaches of VaR estimation We intend to address thesematters in our future research
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A Ghorbel and A Trabelsi ldquoEnergy portfolio risk managementusing time-varying extreme value copula methodsrdquo EconomicModelling vol 38 pp 470ndash485 2014
[2] Z-R Wang X-H Chen Y-B Jin and Y-J Zhou ldquoEstimatingrisk of foreign exchange portfolio using VaR and CVaR basedonGARCH-EVT-CopulamodelrdquoPhysicaA StatisticalMechan-ics and Its Applications vol 389 no 21 pp 4918ndash4928 2010
[3] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics amp Economics vol 45 no 3 pp 315ndash3242009
[4] R K Y Low J Alcock R Faff and T Brailsford ldquoCanonicalvine copulas in the context of modern portfolio managementare they worth itrdquo Journal of Banking and Finance vol 37 no8 pp 3085ndash3099 2013
[5] R Aloui M S Aıssa and D K Nguyen ldquoConditional depen-dence structure between oil prices and exchange rates a copula-GARCHapproachrdquo Journal of InternationalMoney and Financevol 32 no 1 pp 719ndash738 2013
[6] YH Chen andAH Tu ldquoEstimating hedged portfolio value-at-risk using the conditional copula an illustration of model riskrdquoInternational Review of Economics and Finance vol 27 pp 514ndash528 2013
[7] G N F Weiszlig and H Supper ldquoForecasting liquidity-adjustedintraday value-at-riskwith vine copulasrdquo Journal of Banking andFinance vol 37 no 9 pp 3334ndash3350 2013
[8] T Berger ldquoForecasting value-at-risk using time varying copulasandEVT return distributionsrdquo International Economics vol 133pp 93ndash106 2013
[9] R Aloui M S B Aıssa S Hammoudeh and D K NguyenldquoDependence and extreme dependence of crude oil and naturalgas prices with applications to risk managementrdquo Energy Eco-nomics vol 42 pp 332ndash342 2014
[10] D A Dickey and W A Fuller ldquoDistribution of the estimatorsfor autoregressive time series with a unit rootrdquo Journal of theAmerican Statistical Association vol 74 no 366 pp 427ndash4311979
[11] D Kwiatkowski P C B Phillips P Schmidt and Y Shin ldquoTest-ing the null hypothesis of stationarity against the alternative ofa unit root How sure are we that economic time series have aunit rootrdquo Journal of Econometrics vol 54 no 1ndash3 pp 159ndash1781992
[12] T Bollerslev ldquoGeneralized autoregressive conditional het-eroskedasticityrdquo Journal of Econometrics vol 31 no 3 pp 307ndash327 1986
[13] W H DuMouchel ldquoEstimating the stable index 120572 in order tomeasure tail thickness a critiquerdquo The Annals of Statistics vol11 no 4 pp 1019ndash1031 1983
[14] P Embrechts F Lindskog and A McNeil ldquoModelling depen-dence with copulas and applications to risk managementrdquo inHandbook of Heavy Tailed Distributions in Finance S RachevEd pp 25ndash26 Elsevier Amsterdam The Netherlands 2003
[15] A J Patton ldquoModelling asymmetric exchange rate depen-dencerdquo International Economic Review vol 47 no 2 pp 527ndash556 2006
[16] J C Reboredo ldquoHow do crude oil prices co-move A copulaapproachrdquo Energy Economics vol 33 no 5 pp 948ndash955 2011
the liquidity-adjusted intraday VaR of stock portfolio byusing vine copulas for the dependence structure and theACDP (Autoregressive Conditional Double Poisson) andGARCH processes for the marginal distribution The empir-ical evidences showed that the GARCH-Vine copula modelperformed well in estimating the liquidity-adjusted intradayportfolio profits and losses Berger [8] used time-varyingcopulas combined with EVT-based margins He pointed outthat when forecasting the VaR of a portfolio it is crucialto find the ldquorightrdquo portfolio return distribution They foundthat reliable copulas combined with EVT-based margins cancapture reliable VaR figures Aloui et al [9] investigatedthe conditional dependence structure between the crude oiland natural gas markets and derived implications for riskmanagement issues related to an oil and gas portfolio withina VaR framework
The contributions of this paper are twofold First tothe best of our knowledge there are no previous empir-ical papers estimating the risks of natural gas portfoliosusing the GARCH-EVT-copula model In this sense thisstudy can perfectly contribute to this area of the litera-ture Second risk measures of natural gas portfolios whichare computed by the GARCH-EVT-copula model can beutilized to have implications for natural gas portfolio riskmanagement
The outline of this paper is as follows The next sectiondescribes the GARCH-EVT-copula model used in the paperSection 3 discusses the data and their properties Section 4presents the empirical results of the model and estimatethe VaR and the CVaR five-dimensional natural gas port-folios and the optimal natural gas portfolio weights underminimized portfolio risk The last section delivers the finalconcluding remarks
2 Data Description
This study focuses on natural gas portfolios consisting ofspot and futures prices at the Title Transfer Facility (TTF)Hub Our dataset consists of daily series of one-day-aheadgas prices (TTFFD) one-month-ahead cash-settled futuresnatural gas prices (TTFFM) one-quarter-ahead cash-settledfutures natural gas prices (TTFFQ) one-season-ahead cash-settled futures natural gas prices (TTFFS) and one-year-ahead cash-settled futures natural gas prices (denoted byTTFFY) The natural gas price series are expressed in europer megawatt hour (CMWh) The data are sourced fromDatastream and are from January 1 2007 to May 30 2014The data used in this paper are from Monday to Friday withtotal of 1934 observations
Figure 1 is plotted to illustrate the natural gas spot andfutures prices at the TTF Hub The figures of the spot andfutures prices reveal the typical properties of natural gasprices extreme price spikes and volatility clustered Thedescriptive statistics for natural gas spot and futures returnsare reported in Table 1 The natural gas spot and futuresreturns (119903
119905) are computed on a continuous compounding
basis as 119903119905= ln(119875
119905119875119905minus1) where 119875
119905and 119875
119905minus1are current
and one-period lagged daily natural gas spot or futuresprices The mean returns for all spot and futures series
are very small and near zero indicating that there is nosignificant trend in the data All the returns exhibit significantskewness except for TTFFD and kurtosis All the returnsexhibit positive skewness except for TTFFD Second all thereturns strongly reject the null hypotheses for the JB (Jarque-Bera) test of normality Finally augmentedDickey-Fuller [10]and Kwiatkowski-Phillips-Schmidt-Shin [11] tests are usedfor the unit root test Table 1 shows the results indicatingthat all return series are stationary at the 1 significancelevel
3 Methodology
31 Model forMarginal Distribution We adopt the univariatestandard GARCH model for natural gas price to obtain ourmarginal distributions in this study The univariate standardGARCH(1 1) model [12] is defined as follows
where 119904 is TTFFD TTFFM TTFFQ TTFFS and TTFFYrespectively 119903
119904119905denotes the spot or futures return of natural
gas ℎ119904119905
is the conditional variance of volatility 120601119904119894
is the119894th lag AR parameter 120579
119904119895is the 119894th lag MA parameter 120572
119904is
the ARCH parameter and 120573119904is the GARCH parameter The
conditions 120596119904gt 0 120572
119904 120573119904ge 0 and 120572
119904+ 120573119904lt 1 are to ensure
that the conditional variance process remains positive andstationary 119911
119904119905is the standardized error In Section 2 we know
that all series are not normal distribution Hence we assumethat 119911
119904119905is 119905-distribution df
119904is the degree of freedom which
is denoted as 120582119904 In order to better estimate the tails of the
distribution this paper applied EVT to those residuals Wemodel residuals by using the generalized Pareto distribution(GPD) estimate for the upper and lower tails and theGaussiankernel estimate for the remaining part The CDF of a GPDdistribution is parameterized as follows
where 120585 is the shape parameter 120573 is the scale parameter and119898119871(119898119877) is the lower (upper) threshold
There is a trade-off between high precision and low vari-ance the critical step is the choice of the optimal thresholdFollowing DuMouchel [13] this paper chose the exceedancesto be the 10th percentile of the sample
The Scientific World Journal 3
40
30
20
10
0
40
35
30
25
20
15
10
5
50
40
30
20
10
0
2007 2008 2009 2010 2011 2012 2013 2014
TTFFD TTFFM
TTFFQ TTFFS
TTFFY
2007 2008 2009 2010 2011 2012 2013 2014
2007 2008 2009 2010 2011 2012 2013 2014
2007 2008 2009 2010 2011 2012 2013 2014
2007 2008 2009 2010 2011 2012 2013 2014
50
40
30
20
10
0
50
40
30
20
10
Figure 1 The time series plots of daily natural gas spot and futures prices at the Title Transfer Facility (TTF) Hub
Table 1 Descriptive statistics of spot and generic futures returns for natural gas portfolio
is the cumulative distribution function of thestandardized error from the marginal models Φ
sum120588denotes
the standardized multivariate normal distribution with thecorrelation matrix sum120588 Φminus1 is the inverse of the univariatestandard normal distribution and 120588 isin (minus1 1) measures thedependence
The Gaussian copula cannot capture the dependence of afat tail In order to capture the fat tail we also introduced themultivariate Student 119905-copula in this study The multivariateStudent 119905-copula [14] is defined as
where 119905sum120588V is the standardized multivariate Student 119905-
distribution with correlation and degrees of freedom matrixsum120588 V correlation 120588 isin (minus1 1) 119905minus1V denotes the inverse of theunivariate Student 119905-distribution The parameters of V and 120588determine the extent of symmetric extreme dependence bythe upper and lower tail dependences 120582
331 VaR and CVaR In order to analyze risk managementthis paper used VaR and CVaR to measure the risk Weestimate the VaR and the CVaR by taking the 120572-quantilesof the portfolio return forecasts as given in the followingequation
We assume that the weight in each asset is the same thatis 120596 = [(15) (15) (15) (15) (15)]
1015840 As a result wecan compute the VaR and the CVaR of the equally weightedportfolio
332 Optimal Portfolio with Minimum Risk We estimatethe VaR and the CVaR of an equally weighted portfoliobut the major concern for commercial banks and individual
investors is to minimize the risk of the investment portfolioTo address this concern in this paper we also compute theoptimal weights of each asset thatminimize the portfolio riskAccording to Wang et al [2] the algorithm is as follows Weassume that the weight in individual asset within a portfoliois 120596 = [120596
1 1205962 1205963 1205964 1205965]1015840 where 0 le 120596
119904le 1 (119904 = 1 2 5)
and 1205961+ 1205962+ 1205963+ 1205964+ 1205965= 1 Many weight vectors
can satisfy these conditions Given a weight vector 120596 119873returns of the portfolio are obtained this means that returns= 119903119905+1
times 120596 Subsequently the VaR of these portfolios at agiven confidence level can be computed After getting allthe VaR values corresponding to all the weight vectors theminimumVaR can be calculatedThus we can get the weightof each asset that minimizes the VaR that is minVaR
120572=
risk(1198871 1198872 1198873 1198874 1198875)
34 Estimation Procedure We follow a seven-step estimationprocedure in this paper
(1) We first complete the parameter estimation andobtain the standardized residuals by fitting the uni-variate ARMA(119901 119902)-GARCH(1 1) model for eachnatural gas return series
(2) For EVT we chose the exceedances to be the 10thpercentile of the sample and used the sample MEFplot and the Hill plot to determine an appropriatethreshold for the upper and lower tails of the dis-tribution of the residuals We assume that excessresiduals over this threshold follow the generalizedPareto distribution (GPD) and estimate for the upperand lower tails and use theGaussian kernel estimationfor the remaining part
(3) We transform the standardized residuals 119911119904119905from the
GARCHmodel into the variates 119906119904119905 using the ECDFs
(empirical cumulative distribution functions)There-after we perform the goodness-of-fit test for each 119906
119904119905
(4) We fit a copula model and estimate its parameter(5) We simulate 119873 times from the estimated copula
model to convert to 119873 standardized residuals usingthe inverse of the corresponding distribution functionfor each model
(6) We forecast 119873 portfolio returns using the ARMA(119901119902)-GARCH(1 1)model
(7) We estimate the VaR and the CVaR of the equallyweighted portfolio Additionally we compute theoptimal portfolio with the minimum risk
4 Empirical Results
41 Results for Marginal Model Firstly estimate theARMA(119901 119902)-GARCH(1 1) model for each natural gasreturn The AIC BIC and Loglik statistics are adopted toselect the most suitable models The estimated coefficientsand standard errors for the marginal model are presentedin Table 2 Table 2 shows that the AR(1) estimates 120601
1199041are
significant in all the series except for TTFFM and TTFFQwhich implies that the current returns have spillover effects
The Scientific World Journal 5
Table 2 Estimated parameters of agricultural commodity returnsfor marginal model
TTFFD TTFFM TTFFQ TTFFS TTFFY
119888119904119900
minus00002(00004)
minus00007lowastlowast(00003)
minus00008(00003)
minus00005lowast(00002)
minus00003(00002)
1206011199041
minus01350lowastlowast(00234)
00363(00206)
00206(00190)
00668lowastlowast(00218)
00588lowastlowast(00223)
120596119904
00000lowastlowast(00000)
00000(00000)
00000lowastlowast(00000)
00000lowastlowast(00000)
00000lowastlowast(00000)
120572119904
02045lowastlowast(00263)
00560lowastlowast(00169)
00120lowastlowast(00035)
01865lowastlowast(00341)
01661lowastlowast(00243)
120573119904
07945lowastlowast(00238)
09430lowastlowast(00170)
09848lowastlowast(00030)
08125lowastlowast(00305)
08329lowastlowast(00213)
120582119904
43963lowastlowast(03857)
32386lowastlowast(02172)
28659lowastlowast(01965)
31684lowastlowast(01866)
36138lowastlowast(02616)
LL 39100000 49437790 51541850 54091680 57945190AIC minus40372 minus51063 minus53239 minus55876 minus59861ARCH(10) 07074 08258 09988 10000 09996NoteThe table shows the estimates and the standard errors of the parametersfor the marginal distribution model defined in (3) and (4) lowastlowast and lowastdenote rejection of the null hypothesis at the 1 and 5 significance levelsrespectively ARCH(10) is the P value of Englersquos LM test for the ARCH effectin the residuals up to the 10th order
from their returns in one of the previous periods Secondit can be observed that the ARCH parameter 120572
that all the returns experience ARCH and GARCH effectsThe 119875 values of ARCH(10) are more than 10 which showsthat there is no heteroskedasticity effect in the residualsAdditionally the degrees of freedom 120582
119904of the Student
119905-distribution for all the returns are significant suggestingthat the error terms are not normal distribution and thatthe Student 119905-distribution works reasonably well for all thereturn series
As described in Section 31 we apply EVT to thoseresiduals GPD specially is for tail estimation and we choosethe exceedances to be the 10th percentile of the sampleThe Gaussian kernel estimate is for the remaining partThe parameter estimation for each natural gasrsquos residuals isdemonstrated in Table 3
Given the availability of the estimates ofmarginalmodelswe proceed to estimate the copula functions For that wetransform the standardized residuals 119911
119904119905from the GARCH
model into the variates 119906119904119905 using the ECDF Each variate 119906
119904119905
should be uniform (0 1) otherwise the copula model couldbe misspecifiedThis paper follows Patton [15] and Reboredo[16] to carry out the goodness-of-fit test which are Ljung-Box(LB) test and Kolmogorov-Smirnov (KS) test LB test is usedto examine the serial correlation under the null hypothesisof serial independence To do so (119906
119904119905minus 119906119904)119896 is regressed on
the first 10 lags of the variables And KS test is used to testthe null hypothesis that the 119906
119894119905are uniform (0 1) Table 4
reports the 119875 values of these two tests At the 5 significancelevel all the null hypotheses are rejected which implies thatthe marginal distribution model demonstrates that these are
Table 3 Parameter estimation for each natural gasrsquos residuals
KS test 09997 10000 09436 09997 08967Note This table reports the P values from the Ljung-Box (LB) test for serialindependence of the first four moments of the variable 119906
119904119905 We regress
(119906119904119905minus 119906119904)119896 on the first five lags of the variables for 119896 = 1 2 3 4 In
addition we present the P values of the Kolmogorov-Smirnov (KS) test forthe adequacy of the distribution model
not misspecified Hence it can be concluded that the copulamodel can correctly capture the dependence
42 Results for Copula After estimating the parameters ofthe marginal distribution we proceed further to estimatethe copula parameter The multivariate Gaussian copula andStudent 119905-copula are applied in this study The dependencematrix is estimated by the maximum-likelihood estimation(MLE) method The results of the dependence matrix forthe multivariate Gaussian copula model and the Student 119905-copula model are reported in Tables 5 and 6 respectivelyThedependence parameters are positive and strongly significantat 10 level for theGaussian and the Student 119905-copulamodelsThese findings point out the presence of significantly strongpositive relationships between the TTF natural gas portfoliosSecond the degree of freedom for the multivariate Student119905-copula V is reasonably low (49837) which suggests theexistence of substantially extreme comovements and taildependence for all the pairs of the TTF natural gas portfoliosThis finding indicates that extreme events or crises are likelyto spread from one series to another during the bull and bearperiods
43 Portfolio Risk Analysis After modeling the dependencestructure we simulate one-month natural gas portfolioreturns based on the dependence structure specified in boththe multivariate Gaussian copula and the Student 119905-copulaand estimate the risk of a five-dimensional natural gasportfolio With an equally weighted portfolio of five natural
6 The Scientific World Journal
Table 5 Estimated parameters of natural gas portfolio for multi-variate Gaussian copula
Note lowastlowast denotes rejection of the null hypothesis at the 1 and 5 signifi-cance levels respectively
gas returns (TTFFD TTFFM TTFFQ TTFFS and TTFFY)we can calculate the VaR and the CVaR of the portfolio Theresults of the VaR and the CVaR calculations are presentedin Table 7 From Table 7 we can perceive that the VaR andthe CVaR calculated from the Student 119905-copula are more thanthose from the Gaussian copula under the same confidencelevel The reason is that the Student 119905-copula considers theextreme dependence whereas the Gaussian copula does not
Likewise in this paper we also compute the optimalweights of each asset that minimize the portfolio risk Theoptimal portfolio weight under the minimum portfolio riskand the VaR (CVaR) under different copulas and confidencelevels are presented in Table 8 From Table 8 we can draw thefollowing interesting conclusions
(1) The optimal portfolio weights are similar acrossthe multivariate Gaussian copula and Student 119905-copula and different confidence levels The optimalinvestment tends to concentrate on TTFFM TTFFQTTFFS and TTFFY Moreover the optimal portfolioweights of TTFFM TTFFQ TTFFS and TTFFY arealmost equal However the optimal portfolio weightof TTFFD is nearly zero This is because TTFFD isthe spot (one-day-ahead) price and TTFFM TTFFQTTFFS and TTFFY are the futures prices Futures
Table 7 Portfolio risk under equally weighted natural gas portfolio
Confidence level Risk value Normal copula 119905-copula
090 VaR 00746 00755CVaR 01094 01112
095 VaR 00977 00991CVaR 01344 01365
099 VaR 01565 01624CVaR 01989 01994
markets have risk aversion functionUnderminimumportfolio risk investors prefer the futures market Onthe other hand with higher confidence levels theweight of TTFFD investment becomes larger Thereason is that with higher confidence levels investorswill be more risk tolerant and therefore willing totake more risk to achieve higher expected returns
(2) Comparing the risk value calculated by the Gaussiancopula and Student 119905-copula we observe that mostof the VaR and all the CVaR from the Gaussiancopula are less than those from the Student 119905-copulaThis result is consistent with the conclusions whencomputing the VaR and the CVaR of the equallyweighted foreign exchange portfolio This is becausethe Student 119905-copula on account of its heavy tail isable to capture the extreme dependence The extremeevents of the asset returns have higher dependenceand hence larger VaR and CVaR
5 Conclusion
We employed the GARCH-EVT-copula model to estimatethe risk of a multidimensional natural gas portfolio Thisstudy focuses on the natural gas portfolio at the Title TransferFacility (TTF) Hub The data consist of daily series ofone-day-ahead gas prices (TTFFD) one-month-ahead cash-settled futures natural gas prices (TTFFM) one-quarter-ahead cash-settled futures natural gas prices (TTFFQ) one-season-ahead cash-settled futures natural gas prices (TTFFS)and one-year-ahead cash-settled futures natural gas prices(denoted by TTFFY) Our dataset is from January 1 2007 toMay 30 2014 The multivariate Gaussian copula and Student119905-copula have been employed to model the dependencestructure
The empirical results show that for an equally weightedportfolio of five natural gases theVaR and theCVaR obtainedfrom the Student 119905-copula are larger than those obtained fromthe Gaussian copula The reason is that the Gaussian copulacannot capture the extreme dependence but the Student119905-copula can Second the optimal portfolio weights arefound to be similar across the multivariate Gaussian copulaand Student 119905-copula and different confidence levels uponminimizing the portfolio risk The optimal investment tendsto concentrate on TTFFM TTFFQ TTFFS and TTFFY Inaddition the optimal portfolio weights of TTFFM TTFFQTTFFS and TTFFY are almost equal At the same timethe optimal portfolio weight of TTFFD is nearly zero With
The Scientific World Journal 7
Table 8 Optimal investment proportion of natural gas portfolio with minimum risk
higher confidence levels theweight of the TTFFD investmentbecomes larger
Further studies on estimating the risk of natural gasportfolios should concentrate on modeling dependenceSpecifically we should employ more copulas multivariateClaytonmultivariateGumbel and vine copula Furthermorethe risk of natural gas portfolio in different Hubs should beconsidered Third we should work on backtesting to checkthe performance of the approach compared to other popularapproaches of VaR estimation We intend to address thesematters in our future research
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A Ghorbel and A Trabelsi ldquoEnergy portfolio risk managementusing time-varying extreme value copula methodsrdquo EconomicModelling vol 38 pp 470ndash485 2014
[2] Z-R Wang X-H Chen Y-B Jin and Y-J Zhou ldquoEstimatingrisk of foreign exchange portfolio using VaR and CVaR basedonGARCH-EVT-CopulamodelrdquoPhysicaA StatisticalMechan-ics and Its Applications vol 389 no 21 pp 4918ndash4928 2010
[3] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics amp Economics vol 45 no 3 pp 315ndash3242009
[4] R K Y Low J Alcock R Faff and T Brailsford ldquoCanonicalvine copulas in the context of modern portfolio managementare they worth itrdquo Journal of Banking and Finance vol 37 no8 pp 3085ndash3099 2013
[5] R Aloui M S Aıssa and D K Nguyen ldquoConditional depen-dence structure between oil prices and exchange rates a copula-GARCHapproachrdquo Journal of InternationalMoney and Financevol 32 no 1 pp 719ndash738 2013
[6] YH Chen andAH Tu ldquoEstimating hedged portfolio value-at-risk using the conditional copula an illustration of model riskrdquoInternational Review of Economics and Finance vol 27 pp 514ndash528 2013
[7] G N F Weiszlig and H Supper ldquoForecasting liquidity-adjustedintraday value-at-riskwith vine copulasrdquo Journal of Banking andFinance vol 37 no 9 pp 3334ndash3350 2013
[8] T Berger ldquoForecasting value-at-risk using time varying copulasandEVT return distributionsrdquo International Economics vol 133pp 93ndash106 2013
[9] R Aloui M S B Aıssa S Hammoudeh and D K NguyenldquoDependence and extreme dependence of crude oil and naturalgas prices with applications to risk managementrdquo Energy Eco-nomics vol 42 pp 332ndash342 2014
[10] D A Dickey and W A Fuller ldquoDistribution of the estimatorsfor autoregressive time series with a unit rootrdquo Journal of theAmerican Statistical Association vol 74 no 366 pp 427ndash4311979
[11] D Kwiatkowski P C B Phillips P Schmidt and Y Shin ldquoTest-ing the null hypothesis of stationarity against the alternative ofa unit root How sure are we that economic time series have aunit rootrdquo Journal of Econometrics vol 54 no 1ndash3 pp 159ndash1781992
[12] T Bollerslev ldquoGeneralized autoregressive conditional het-eroskedasticityrdquo Journal of Econometrics vol 31 no 3 pp 307ndash327 1986
[13] W H DuMouchel ldquoEstimating the stable index 120572 in order tomeasure tail thickness a critiquerdquo The Annals of Statistics vol11 no 4 pp 1019ndash1031 1983
[14] P Embrechts F Lindskog and A McNeil ldquoModelling depen-dence with copulas and applications to risk managementrdquo inHandbook of Heavy Tailed Distributions in Finance S RachevEd pp 25ndash26 Elsevier Amsterdam The Netherlands 2003
[15] A J Patton ldquoModelling asymmetric exchange rate depen-dencerdquo International Economic Review vol 47 no 2 pp 527ndash556 2006
[16] J C Reboredo ldquoHow do crude oil prices co-move A copulaapproachrdquo Energy Economics vol 33 no 5 pp 948ndash955 2011
is the cumulative distribution function of thestandardized error from the marginal models Φ
sum120588denotes
the standardized multivariate normal distribution with thecorrelation matrix sum120588 Φminus1 is the inverse of the univariatestandard normal distribution and 120588 isin (minus1 1) measures thedependence
The Gaussian copula cannot capture the dependence of afat tail In order to capture the fat tail we also introduced themultivariate Student 119905-copula in this study The multivariateStudent 119905-copula [14] is defined as
where 119905sum120588V is the standardized multivariate Student 119905-
distribution with correlation and degrees of freedom matrixsum120588 V correlation 120588 isin (minus1 1) 119905minus1V denotes the inverse of theunivariate Student 119905-distribution The parameters of V and 120588determine the extent of symmetric extreme dependence bythe upper and lower tail dependences 120582
331 VaR and CVaR In order to analyze risk managementthis paper used VaR and CVaR to measure the risk Weestimate the VaR and the CVaR by taking the 120572-quantilesof the portfolio return forecasts as given in the followingequation
We assume that the weight in each asset is the same thatis 120596 = [(15) (15) (15) (15) (15)]
1015840 As a result wecan compute the VaR and the CVaR of the equally weightedportfolio
332 Optimal Portfolio with Minimum Risk We estimatethe VaR and the CVaR of an equally weighted portfoliobut the major concern for commercial banks and individual
investors is to minimize the risk of the investment portfolioTo address this concern in this paper we also compute theoptimal weights of each asset thatminimize the portfolio riskAccording to Wang et al [2] the algorithm is as follows Weassume that the weight in individual asset within a portfoliois 120596 = [120596
1 1205962 1205963 1205964 1205965]1015840 where 0 le 120596
119904le 1 (119904 = 1 2 5)
and 1205961+ 1205962+ 1205963+ 1205964+ 1205965= 1 Many weight vectors
can satisfy these conditions Given a weight vector 120596 119873returns of the portfolio are obtained this means that returns= 119903119905+1
times 120596 Subsequently the VaR of these portfolios at agiven confidence level can be computed After getting allthe VaR values corresponding to all the weight vectors theminimumVaR can be calculatedThus we can get the weightof each asset that minimizes the VaR that is minVaR
120572=
risk(1198871 1198872 1198873 1198874 1198875)
34 Estimation Procedure We follow a seven-step estimationprocedure in this paper
(1) We first complete the parameter estimation andobtain the standardized residuals by fitting the uni-variate ARMA(119901 119902)-GARCH(1 1) model for eachnatural gas return series
(2) For EVT we chose the exceedances to be the 10thpercentile of the sample and used the sample MEFplot and the Hill plot to determine an appropriatethreshold for the upper and lower tails of the dis-tribution of the residuals We assume that excessresiduals over this threshold follow the generalizedPareto distribution (GPD) and estimate for the upperand lower tails and use theGaussian kernel estimationfor the remaining part
(3) We transform the standardized residuals 119911119904119905from the
GARCHmodel into the variates 119906119904119905 using the ECDFs
(empirical cumulative distribution functions)There-after we perform the goodness-of-fit test for each 119906
119904119905
(4) We fit a copula model and estimate its parameter(5) We simulate 119873 times from the estimated copula
model to convert to 119873 standardized residuals usingthe inverse of the corresponding distribution functionfor each model
(6) We forecast 119873 portfolio returns using the ARMA(119901119902)-GARCH(1 1)model
(7) We estimate the VaR and the CVaR of the equallyweighted portfolio Additionally we compute theoptimal portfolio with the minimum risk
4 Empirical Results
41 Results for Marginal Model Firstly estimate theARMA(119901 119902)-GARCH(1 1) model for each natural gasreturn The AIC BIC and Loglik statistics are adopted toselect the most suitable models The estimated coefficientsand standard errors for the marginal model are presentedin Table 2 Table 2 shows that the AR(1) estimates 120601
1199041are
significant in all the series except for TTFFM and TTFFQwhich implies that the current returns have spillover effects
The Scientific World Journal 5
Table 2 Estimated parameters of agricultural commodity returnsfor marginal model
TTFFD TTFFM TTFFQ TTFFS TTFFY
119888119904119900
minus00002(00004)
minus00007lowastlowast(00003)
minus00008(00003)
minus00005lowast(00002)
minus00003(00002)
1206011199041
minus01350lowastlowast(00234)
00363(00206)
00206(00190)
00668lowastlowast(00218)
00588lowastlowast(00223)
120596119904
00000lowastlowast(00000)
00000(00000)
00000lowastlowast(00000)
00000lowastlowast(00000)
00000lowastlowast(00000)
120572119904
02045lowastlowast(00263)
00560lowastlowast(00169)
00120lowastlowast(00035)
01865lowastlowast(00341)
01661lowastlowast(00243)
120573119904
07945lowastlowast(00238)
09430lowastlowast(00170)
09848lowastlowast(00030)
08125lowastlowast(00305)
08329lowastlowast(00213)
120582119904
43963lowastlowast(03857)
32386lowastlowast(02172)
28659lowastlowast(01965)
31684lowastlowast(01866)
36138lowastlowast(02616)
LL 39100000 49437790 51541850 54091680 57945190AIC minus40372 minus51063 minus53239 minus55876 minus59861ARCH(10) 07074 08258 09988 10000 09996NoteThe table shows the estimates and the standard errors of the parametersfor the marginal distribution model defined in (3) and (4) lowastlowast and lowastdenote rejection of the null hypothesis at the 1 and 5 significance levelsrespectively ARCH(10) is the P value of Englersquos LM test for the ARCH effectin the residuals up to the 10th order
from their returns in one of the previous periods Secondit can be observed that the ARCH parameter 120572
that all the returns experience ARCH and GARCH effectsThe 119875 values of ARCH(10) are more than 10 which showsthat there is no heteroskedasticity effect in the residualsAdditionally the degrees of freedom 120582
119904of the Student
119905-distribution for all the returns are significant suggestingthat the error terms are not normal distribution and thatthe Student 119905-distribution works reasonably well for all thereturn series
As described in Section 31 we apply EVT to thoseresiduals GPD specially is for tail estimation and we choosethe exceedances to be the 10th percentile of the sampleThe Gaussian kernel estimate is for the remaining partThe parameter estimation for each natural gasrsquos residuals isdemonstrated in Table 3
Given the availability of the estimates ofmarginalmodelswe proceed to estimate the copula functions For that wetransform the standardized residuals 119911
119904119905from the GARCH
model into the variates 119906119904119905 using the ECDF Each variate 119906
119904119905
should be uniform (0 1) otherwise the copula model couldbe misspecifiedThis paper follows Patton [15] and Reboredo[16] to carry out the goodness-of-fit test which are Ljung-Box(LB) test and Kolmogorov-Smirnov (KS) test LB test is usedto examine the serial correlation under the null hypothesisof serial independence To do so (119906
119904119905minus 119906119904)119896 is regressed on
the first 10 lags of the variables And KS test is used to testthe null hypothesis that the 119906
119894119905are uniform (0 1) Table 4
reports the 119875 values of these two tests At the 5 significancelevel all the null hypotheses are rejected which implies thatthe marginal distribution model demonstrates that these are
Table 3 Parameter estimation for each natural gasrsquos residuals
KS test 09997 10000 09436 09997 08967Note This table reports the P values from the Ljung-Box (LB) test for serialindependence of the first four moments of the variable 119906
119904119905 We regress
(119906119904119905minus 119906119904)119896 on the first five lags of the variables for 119896 = 1 2 3 4 In
addition we present the P values of the Kolmogorov-Smirnov (KS) test forthe adequacy of the distribution model
not misspecified Hence it can be concluded that the copulamodel can correctly capture the dependence
42 Results for Copula After estimating the parameters ofthe marginal distribution we proceed further to estimatethe copula parameter The multivariate Gaussian copula andStudent 119905-copula are applied in this study The dependencematrix is estimated by the maximum-likelihood estimation(MLE) method The results of the dependence matrix forthe multivariate Gaussian copula model and the Student 119905-copula model are reported in Tables 5 and 6 respectivelyThedependence parameters are positive and strongly significantat 10 level for theGaussian and the Student 119905-copulamodelsThese findings point out the presence of significantly strongpositive relationships between the TTF natural gas portfoliosSecond the degree of freedom for the multivariate Student119905-copula V is reasonably low (49837) which suggests theexistence of substantially extreme comovements and taildependence for all the pairs of the TTF natural gas portfoliosThis finding indicates that extreme events or crises are likelyto spread from one series to another during the bull and bearperiods
43 Portfolio Risk Analysis After modeling the dependencestructure we simulate one-month natural gas portfolioreturns based on the dependence structure specified in boththe multivariate Gaussian copula and the Student 119905-copulaand estimate the risk of a five-dimensional natural gasportfolio With an equally weighted portfolio of five natural
6 The Scientific World Journal
Table 5 Estimated parameters of natural gas portfolio for multi-variate Gaussian copula
Note lowastlowast denotes rejection of the null hypothesis at the 1 and 5 signifi-cance levels respectively
gas returns (TTFFD TTFFM TTFFQ TTFFS and TTFFY)we can calculate the VaR and the CVaR of the portfolio Theresults of the VaR and the CVaR calculations are presentedin Table 7 From Table 7 we can perceive that the VaR andthe CVaR calculated from the Student 119905-copula are more thanthose from the Gaussian copula under the same confidencelevel The reason is that the Student 119905-copula considers theextreme dependence whereas the Gaussian copula does not
Likewise in this paper we also compute the optimalweights of each asset that minimize the portfolio risk Theoptimal portfolio weight under the minimum portfolio riskand the VaR (CVaR) under different copulas and confidencelevels are presented in Table 8 From Table 8 we can draw thefollowing interesting conclusions
(1) The optimal portfolio weights are similar acrossthe multivariate Gaussian copula and Student 119905-copula and different confidence levels The optimalinvestment tends to concentrate on TTFFM TTFFQTTFFS and TTFFY Moreover the optimal portfolioweights of TTFFM TTFFQ TTFFS and TTFFY arealmost equal However the optimal portfolio weightof TTFFD is nearly zero This is because TTFFD isthe spot (one-day-ahead) price and TTFFM TTFFQTTFFS and TTFFY are the futures prices Futures
Table 7 Portfolio risk under equally weighted natural gas portfolio
Confidence level Risk value Normal copula 119905-copula
090 VaR 00746 00755CVaR 01094 01112
095 VaR 00977 00991CVaR 01344 01365
099 VaR 01565 01624CVaR 01989 01994
markets have risk aversion functionUnderminimumportfolio risk investors prefer the futures market Onthe other hand with higher confidence levels theweight of TTFFD investment becomes larger Thereason is that with higher confidence levels investorswill be more risk tolerant and therefore willing totake more risk to achieve higher expected returns
(2) Comparing the risk value calculated by the Gaussiancopula and Student 119905-copula we observe that mostof the VaR and all the CVaR from the Gaussiancopula are less than those from the Student 119905-copulaThis result is consistent with the conclusions whencomputing the VaR and the CVaR of the equallyweighted foreign exchange portfolio This is becausethe Student 119905-copula on account of its heavy tail isable to capture the extreme dependence The extremeevents of the asset returns have higher dependenceand hence larger VaR and CVaR
5 Conclusion
We employed the GARCH-EVT-copula model to estimatethe risk of a multidimensional natural gas portfolio Thisstudy focuses on the natural gas portfolio at the Title TransferFacility (TTF) Hub The data consist of daily series ofone-day-ahead gas prices (TTFFD) one-month-ahead cash-settled futures natural gas prices (TTFFM) one-quarter-ahead cash-settled futures natural gas prices (TTFFQ) one-season-ahead cash-settled futures natural gas prices (TTFFS)and one-year-ahead cash-settled futures natural gas prices(denoted by TTFFY) Our dataset is from January 1 2007 toMay 30 2014 The multivariate Gaussian copula and Student119905-copula have been employed to model the dependencestructure
The empirical results show that for an equally weightedportfolio of five natural gases theVaR and theCVaR obtainedfrom the Student 119905-copula are larger than those obtained fromthe Gaussian copula The reason is that the Gaussian copulacannot capture the extreme dependence but the Student119905-copula can Second the optimal portfolio weights arefound to be similar across the multivariate Gaussian copulaand Student 119905-copula and different confidence levels uponminimizing the portfolio risk The optimal investment tendsto concentrate on TTFFM TTFFQ TTFFS and TTFFY Inaddition the optimal portfolio weights of TTFFM TTFFQTTFFS and TTFFY are almost equal At the same timethe optimal portfolio weight of TTFFD is nearly zero With
The Scientific World Journal 7
Table 8 Optimal investment proportion of natural gas portfolio with minimum risk
higher confidence levels theweight of the TTFFD investmentbecomes larger
Further studies on estimating the risk of natural gasportfolios should concentrate on modeling dependenceSpecifically we should employ more copulas multivariateClaytonmultivariateGumbel and vine copula Furthermorethe risk of natural gas portfolio in different Hubs should beconsidered Third we should work on backtesting to checkthe performance of the approach compared to other popularapproaches of VaR estimation We intend to address thesematters in our future research
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A Ghorbel and A Trabelsi ldquoEnergy portfolio risk managementusing time-varying extreme value copula methodsrdquo EconomicModelling vol 38 pp 470ndash485 2014
[2] Z-R Wang X-H Chen Y-B Jin and Y-J Zhou ldquoEstimatingrisk of foreign exchange portfolio using VaR and CVaR basedonGARCH-EVT-CopulamodelrdquoPhysicaA StatisticalMechan-ics and Its Applications vol 389 no 21 pp 4918ndash4928 2010
[3] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics amp Economics vol 45 no 3 pp 315ndash3242009
[4] R K Y Low J Alcock R Faff and T Brailsford ldquoCanonicalvine copulas in the context of modern portfolio managementare they worth itrdquo Journal of Banking and Finance vol 37 no8 pp 3085ndash3099 2013
[5] R Aloui M S Aıssa and D K Nguyen ldquoConditional depen-dence structure between oil prices and exchange rates a copula-GARCHapproachrdquo Journal of InternationalMoney and Financevol 32 no 1 pp 719ndash738 2013
[6] YH Chen andAH Tu ldquoEstimating hedged portfolio value-at-risk using the conditional copula an illustration of model riskrdquoInternational Review of Economics and Finance vol 27 pp 514ndash528 2013
[7] G N F Weiszlig and H Supper ldquoForecasting liquidity-adjustedintraday value-at-riskwith vine copulasrdquo Journal of Banking andFinance vol 37 no 9 pp 3334ndash3350 2013
[8] T Berger ldquoForecasting value-at-risk using time varying copulasandEVT return distributionsrdquo International Economics vol 133pp 93ndash106 2013
[9] R Aloui M S B Aıssa S Hammoudeh and D K NguyenldquoDependence and extreme dependence of crude oil and naturalgas prices with applications to risk managementrdquo Energy Eco-nomics vol 42 pp 332ndash342 2014
[10] D A Dickey and W A Fuller ldquoDistribution of the estimatorsfor autoregressive time series with a unit rootrdquo Journal of theAmerican Statistical Association vol 74 no 366 pp 427ndash4311979
[11] D Kwiatkowski P C B Phillips P Schmidt and Y Shin ldquoTest-ing the null hypothesis of stationarity against the alternative ofa unit root How sure are we that economic time series have aunit rootrdquo Journal of Econometrics vol 54 no 1ndash3 pp 159ndash1781992
[12] T Bollerslev ldquoGeneralized autoregressive conditional het-eroskedasticityrdquo Journal of Econometrics vol 31 no 3 pp 307ndash327 1986
[13] W H DuMouchel ldquoEstimating the stable index 120572 in order tomeasure tail thickness a critiquerdquo The Annals of Statistics vol11 no 4 pp 1019ndash1031 1983
[14] P Embrechts F Lindskog and A McNeil ldquoModelling depen-dence with copulas and applications to risk managementrdquo inHandbook of Heavy Tailed Distributions in Finance S RachevEd pp 25ndash26 Elsevier Amsterdam The Netherlands 2003
[15] A J Patton ldquoModelling asymmetric exchange rate depen-dencerdquo International Economic Review vol 47 no 2 pp 527ndash556 2006
[16] J C Reboredo ldquoHow do crude oil prices co-move A copulaapproachrdquo Energy Economics vol 33 no 5 pp 948ndash955 2011
is the cumulative distribution function of thestandardized error from the marginal models Φ
sum120588denotes
the standardized multivariate normal distribution with thecorrelation matrix sum120588 Φminus1 is the inverse of the univariatestandard normal distribution and 120588 isin (minus1 1) measures thedependence
The Gaussian copula cannot capture the dependence of afat tail In order to capture the fat tail we also introduced themultivariate Student 119905-copula in this study The multivariateStudent 119905-copula [14] is defined as
where 119905sum120588V is the standardized multivariate Student 119905-
distribution with correlation and degrees of freedom matrixsum120588 V correlation 120588 isin (minus1 1) 119905minus1V denotes the inverse of theunivariate Student 119905-distribution The parameters of V and 120588determine the extent of symmetric extreme dependence bythe upper and lower tail dependences 120582
331 VaR and CVaR In order to analyze risk managementthis paper used VaR and CVaR to measure the risk Weestimate the VaR and the CVaR by taking the 120572-quantilesof the portfolio return forecasts as given in the followingequation
We assume that the weight in each asset is the same thatis 120596 = [(15) (15) (15) (15) (15)]
1015840 As a result wecan compute the VaR and the CVaR of the equally weightedportfolio
332 Optimal Portfolio with Minimum Risk We estimatethe VaR and the CVaR of an equally weighted portfoliobut the major concern for commercial banks and individual
investors is to minimize the risk of the investment portfolioTo address this concern in this paper we also compute theoptimal weights of each asset thatminimize the portfolio riskAccording to Wang et al [2] the algorithm is as follows Weassume that the weight in individual asset within a portfoliois 120596 = [120596
1 1205962 1205963 1205964 1205965]1015840 where 0 le 120596
119904le 1 (119904 = 1 2 5)
and 1205961+ 1205962+ 1205963+ 1205964+ 1205965= 1 Many weight vectors
can satisfy these conditions Given a weight vector 120596 119873returns of the portfolio are obtained this means that returns= 119903119905+1
times 120596 Subsequently the VaR of these portfolios at agiven confidence level can be computed After getting allthe VaR values corresponding to all the weight vectors theminimumVaR can be calculatedThus we can get the weightof each asset that minimizes the VaR that is minVaR
120572=
risk(1198871 1198872 1198873 1198874 1198875)
34 Estimation Procedure We follow a seven-step estimationprocedure in this paper
(1) We first complete the parameter estimation andobtain the standardized residuals by fitting the uni-variate ARMA(119901 119902)-GARCH(1 1) model for eachnatural gas return series
(2) For EVT we chose the exceedances to be the 10thpercentile of the sample and used the sample MEFplot and the Hill plot to determine an appropriatethreshold for the upper and lower tails of the dis-tribution of the residuals We assume that excessresiduals over this threshold follow the generalizedPareto distribution (GPD) and estimate for the upperand lower tails and use theGaussian kernel estimationfor the remaining part
(3) We transform the standardized residuals 119911119904119905from the
GARCHmodel into the variates 119906119904119905 using the ECDFs
(empirical cumulative distribution functions)There-after we perform the goodness-of-fit test for each 119906
119904119905
(4) We fit a copula model and estimate its parameter(5) We simulate 119873 times from the estimated copula
model to convert to 119873 standardized residuals usingthe inverse of the corresponding distribution functionfor each model
(6) We forecast 119873 portfolio returns using the ARMA(119901119902)-GARCH(1 1)model
(7) We estimate the VaR and the CVaR of the equallyweighted portfolio Additionally we compute theoptimal portfolio with the minimum risk
4 Empirical Results
41 Results for Marginal Model Firstly estimate theARMA(119901 119902)-GARCH(1 1) model for each natural gasreturn The AIC BIC and Loglik statistics are adopted toselect the most suitable models The estimated coefficientsand standard errors for the marginal model are presentedin Table 2 Table 2 shows that the AR(1) estimates 120601
1199041are
significant in all the series except for TTFFM and TTFFQwhich implies that the current returns have spillover effects
The Scientific World Journal 5
Table 2 Estimated parameters of agricultural commodity returnsfor marginal model
TTFFD TTFFM TTFFQ TTFFS TTFFY
119888119904119900
minus00002(00004)
minus00007lowastlowast(00003)
minus00008(00003)
minus00005lowast(00002)
minus00003(00002)
1206011199041
minus01350lowastlowast(00234)
00363(00206)
00206(00190)
00668lowastlowast(00218)
00588lowastlowast(00223)
120596119904
00000lowastlowast(00000)
00000(00000)
00000lowastlowast(00000)
00000lowastlowast(00000)
00000lowastlowast(00000)
120572119904
02045lowastlowast(00263)
00560lowastlowast(00169)
00120lowastlowast(00035)
01865lowastlowast(00341)
01661lowastlowast(00243)
120573119904
07945lowastlowast(00238)
09430lowastlowast(00170)
09848lowastlowast(00030)
08125lowastlowast(00305)
08329lowastlowast(00213)
120582119904
43963lowastlowast(03857)
32386lowastlowast(02172)
28659lowastlowast(01965)
31684lowastlowast(01866)
36138lowastlowast(02616)
LL 39100000 49437790 51541850 54091680 57945190AIC minus40372 minus51063 minus53239 minus55876 minus59861ARCH(10) 07074 08258 09988 10000 09996NoteThe table shows the estimates and the standard errors of the parametersfor the marginal distribution model defined in (3) and (4) lowastlowast and lowastdenote rejection of the null hypothesis at the 1 and 5 significance levelsrespectively ARCH(10) is the P value of Englersquos LM test for the ARCH effectin the residuals up to the 10th order
from their returns in one of the previous periods Secondit can be observed that the ARCH parameter 120572
that all the returns experience ARCH and GARCH effectsThe 119875 values of ARCH(10) are more than 10 which showsthat there is no heteroskedasticity effect in the residualsAdditionally the degrees of freedom 120582
119904of the Student
119905-distribution for all the returns are significant suggestingthat the error terms are not normal distribution and thatthe Student 119905-distribution works reasonably well for all thereturn series
As described in Section 31 we apply EVT to thoseresiduals GPD specially is for tail estimation and we choosethe exceedances to be the 10th percentile of the sampleThe Gaussian kernel estimate is for the remaining partThe parameter estimation for each natural gasrsquos residuals isdemonstrated in Table 3
Given the availability of the estimates ofmarginalmodelswe proceed to estimate the copula functions For that wetransform the standardized residuals 119911
119904119905from the GARCH
model into the variates 119906119904119905 using the ECDF Each variate 119906
119904119905
should be uniform (0 1) otherwise the copula model couldbe misspecifiedThis paper follows Patton [15] and Reboredo[16] to carry out the goodness-of-fit test which are Ljung-Box(LB) test and Kolmogorov-Smirnov (KS) test LB test is usedto examine the serial correlation under the null hypothesisof serial independence To do so (119906
119904119905minus 119906119904)119896 is regressed on
the first 10 lags of the variables And KS test is used to testthe null hypothesis that the 119906
119894119905are uniform (0 1) Table 4
reports the 119875 values of these two tests At the 5 significancelevel all the null hypotheses are rejected which implies thatthe marginal distribution model demonstrates that these are
Table 3 Parameter estimation for each natural gasrsquos residuals
KS test 09997 10000 09436 09997 08967Note This table reports the P values from the Ljung-Box (LB) test for serialindependence of the first four moments of the variable 119906
119904119905 We regress
(119906119904119905minus 119906119904)119896 on the first five lags of the variables for 119896 = 1 2 3 4 In
addition we present the P values of the Kolmogorov-Smirnov (KS) test forthe adequacy of the distribution model
not misspecified Hence it can be concluded that the copulamodel can correctly capture the dependence
42 Results for Copula After estimating the parameters ofthe marginal distribution we proceed further to estimatethe copula parameter The multivariate Gaussian copula andStudent 119905-copula are applied in this study The dependencematrix is estimated by the maximum-likelihood estimation(MLE) method The results of the dependence matrix forthe multivariate Gaussian copula model and the Student 119905-copula model are reported in Tables 5 and 6 respectivelyThedependence parameters are positive and strongly significantat 10 level for theGaussian and the Student 119905-copulamodelsThese findings point out the presence of significantly strongpositive relationships between the TTF natural gas portfoliosSecond the degree of freedom for the multivariate Student119905-copula V is reasonably low (49837) which suggests theexistence of substantially extreme comovements and taildependence for all the pairs of the TTF natural gas portfoliosThis finding indicates that extreme events or crises are likelyto spread from one series to another during the bull and bearperiods
43 Portfolio Risk Analysis After modeling the dependencestructure we simulate one-month natural gas portfolioreturns based on the dependence structure specified in boththe multivariate Gaussian copula and the Student 119905-copulaand estimate the risk of a five-dimensional natural gasportfolio With an equally weighted portfolio of five natural
6 The Scientific World Journal
Table 5 Estimated parameters of natural gas portfolio for multi-variate Gaussian copula
Note lowastlowast denotes rejection of the null hypothesis at the 1 and 5 signifi-cance levels respectively
gas returns (TTFFD TTFFM TTFFQ TTFFS and TTFFY)we can calculate the VaR and the CVaR of the portfolio Theresults of the VaR and the CVaR calculations are presentedin Table 7 From Table 7 we can perceive that the VaR andthe CVaR calculated from the Student 119905-copula are more thanthose from the Gaussian copula under the same confidencelevel The reason is that the Student 119905-copula considers theextreme dependence whereas the Gaussian copula does not
Likewise in this paper we also compute the optimalweights of each asset that minimize the portfolio risk Theoptimal portfolio weight under the minimum portfolio riskand the VaR (CVaR) under different copulas and confidencelevels are presented in Table 8 From Table 8 we can draw thefollowing interesting conclusions
(1) The optimal portfolio weights are similar acrossthe multivariate Gaussian copula and Student 119905-copula and different confidence levels The optimalinvestment tends to concentrate on TTFFM TTFFQTTFFS and TTFFY Moreover the optimal portfolioweights of TTFFM TTFFQ TTFFS and TTFFY arealmost equal However the optimal portfolio weightof TTFFD is nearly zero This is because TTFFD isthe spot (one-day-ahead) price and TTFFM TTFFQTTFFS and TTFFY are the futures prices Futures
Table 7 Portfolio risk under equally weighted natural gas portfolio
Confidence level Risk value Normal copula 119905-copula
090 VaR 00746 00755CVaR 01094 01112
095 VaR 00977 00991CVaR 01344 01365
099 VaR 01565 01624CVaR 01989 01994
markets have risk aversion functionUnderminimumportfolio risk investors prefer the futures market Onthe other hand with higher confidence levels theweight of TTFFD investment becomes larger Thereason is that with higher confidence levels investorswill be more risk tolerant and therefore willing totake more risk to achieve higher expected returns
(2) Comparing the risk value calculated by the Gaussiancopula and Student 119905-copula we observe that mostof the VaR and all the CVaR from the Gaussiancopula are less than those from the Student 119905-copulaThis result is consistent with the conclusions whencomputing the VaR and the CVaR of the equallyweighted foreign exchange portfolio This is becausethe Student 119905-copula on account of its heavy tail isable to capture the extreme dependence The extremeevents of the asset returns have higher dependenceand hence larger VaR and CVaR
5 Conclusion
We employed the GARCH-EVT-copula model to estimatethe risk of a multidimensional natural gas portfolio Thisstudy focuses on the natural gas portfolio at the Title TransferFacility (TTF) Hub The data consist of daily series ofone-day-ahead gas prices (TTFFD) one-month-ahead cash-settled futures natural gas prices (TTFFM) one-quarter-ahead cash-settled futures natural gas prices (TTFFQ) one-season-ahead cash-settled futures natural gas prices (TTFFS)and one-year-ahead cash-settled futures natural gas prices(denoted by TTFFY) Our dataset is from January 1 2007 toMay 30 2014 The multivariate Gaussian copula and Student119905-copula have been employed to model the dependencestructure
The empirical results show that for an equally weightedportfolio of five natural gases theVaR and theCVaR obtainedfrom the Student 119905-copula are larger than those obtained fromthe Gaussian copula The reason is that the Gaussian copulacannot capture the extreme dependence but the Student119905-copula can Second the optimal portfolio weights arefound to be similar across the multivariate Gaussian copulaand Student 119905-copula and different confidence levels uponminimizing the portfolio risk The optimal investment tendsto concentrate on TTFFM TTFFQ TTFFS and TTFFY Inaddition the optimal portfolio weights of TTFFM TTFFQTTFFS and TTFFY are almost equal At the same timethe optimal portfolio weight of TTFFD is nearly zero With
The Scientific World Journal 7
Table 8 Optimal investment proportion of natural gas portfolio with minimum risk
higher confidence levels theweight of the TTFFD investmentbecomes larger
Further studies on estimating the risk of natural gasportfolios should concentrate on modeling dependenceSpecifically we should employ more copulas multivariateClaytonmultivariateGumbel and vine copula Furthermorethe risk of natural gas portfolio in different Hubs should beconsidered Third we should work on backtesting to checkthe performance of the approach compared to other popularapproaches of VaR estimation We intend to address thesematters in our future research
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A Ghorbel and A Trabelsi ldquoEnergy portfolio risk managementusing time-varying extreme value copula methodsrdquo EconomicModelling vol 38 pp 470ndash485 2014
[2] Z-R Wang X-H Chen Y-B Jin and Y-J Zhou ldquoEstimatingrisk of foreign exchange portfolio using VaR and CVaR basedonGARCH-EVT-CopulamodelrdquoPhysicaA StatisticalMechan-ics and Its Applications vol 389 no 21 pp 4918ndash4928 2010
[3] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics amp Economics vol 45 no 3 pp 315ndash3242009
[4] R K Y Low J Alcock R Faff and T Brailsford ldquoCanonicalvine copulas in the context of modern portfolio managementare they worth itrdquo Journal of Banking and Finance vol 37 no8 pp 3085ndash3099 2013
[5] R Aloui M S Aıssa and D K Nguyen ldquoConditional depen-dence structure between oil prices and exchange rates a copula-GARCHapproachrdquo Journal of InternationalMoney and Financevol 32 no 1 pp 719ndash738 2013
[6] YH Chen andAH Tu ldquoEstimating hedged portfolio value-at-risk using the conditional copula an illustration of model riskrdquoInternational Review of Economics and Finance vol 27 pp 514ndash528 2013
[7] G N F Weiszlig and H Supper ldquoForecasting liquidity-adjustedintraday value-at-riskwith vine copulasrdquo Journal of Banking andFinance vol 37 no 9 pp 3334ndash3350 2013
[8] T Berger ldquoForecasting value-at-risk using time varying copulasandEVT return distributionsrdquo International Economics vol 133pp 93ndash106 2013
[9] R Aloui M S B Aıssa S Hammoudeh and D K NguyenldquoDependence and extreme dependence of crude oil and naturalgas prices with applications to risk managementrdquo Energy Eco-nomics vol 42 pp 332ndash342 2014
[10] D A Dickey and W A Fuller ldquoDistribution of the estimatorsfor autoregressive time series with a unit rootrdquo Journal of theAmerican Statistical Association vol 74 no 366 pp 427ndash4311979
[11] D Kwiatkowski P C B Phillips P Schmidt and Y Shin ldquoTest-ing the null hypothesis of stationarity against the alternative ofa unit root How sure are we that economic time series have aunit rootrdquo Journal of Econometrics vol 54 no 1ndash3 pp 159ndash1781992
[12] T Bollerslev ldquoGeneralized autoregressive conditional het-eroskedasticityrdquo Journal of Econometrics vol 31 no 3 pp 307ndash327 1986
[13] W H DuMouchel ldquoEstimating the stable index 120572 in order tomeasure tail thickness a critiquerdquo The Annals of Statistics vol11 no 4 pp 1019ndash1031 1983
[14] P Embrechts F Lindskog and A McNeil ldquoModelling depen-dence with copulas and applications to risk managementrdquo inHandbook of Heavy Tailed Distributions in Finance S RachevEd pp 25ndash26 Elsevier Amsterdam The Netherlands 2003
[15] A J Patton ldquoModelling asymmetric exchange rate depen-dencerdquo International Economic Review vol 47 no 2 pp 527ndash556 2006
[16] J C Reboredo ldquoHow do crude oil prices co-move A copulaapproachrdquo Energy Economics vol 33 no 5 pp 948ndash955 2011
Table 2 Estimated parameters of agricultural commodity returnsfor marginal model
TTFFD TTFFM TTFFQ TTFFS TTFFY
119888119904119900
minus00002(00004)
minus00007lowastlowast(00003)
minus00008(00003)
minus00005lowast(00002)
minus00003(00002)
1206011199041
minus01350lowastlowast(00234)
00363(00206)
00206(00190)
00668lowastlowast(00218)
00588lowastlowast(00223)
120596119904
00000lowastlowast(00000)
00000(00000)
00000lowastlowast(00000)
00000lowastlowast(00000)
00000lowastlowast(00000)
120572119904
02045lowastlowast(00263)
00560lowastlowast(00169)
00120lowastlowast(00035)
01865lowastlowast(00341)
01661lowastlowast(00243)
120573119904
07945lowastlowast(00238)
09430lowastlowast(00170)
09848lowastlowast(00030)
08125lowastlowast(00305)
08329lowastlowast(00213)
120582119904
43963lowastlowast(03857)
32386lowastlowast(02172)
28659lowastlowast(01965)
31684lowastlowast(01866)
36138lowastlowast(02616)
LL 39100000 49437790 51541850 54091680 57945190AIC minus40372 minus51063 minus53239 minus55876 minus59861ARCH(10) 07074 08258 09988 10000 09996NoteThe table shows the estimates and the standard errors of the parametersfor the marginal distribution model defined in (3) and (4) lowastlowast and lowastdenote rejection of the null hypothesis at the 1 and 5 significance levelsrespectively ARCH(10) is the P value of Englersquos LM test for the ARCH effectin the residuals up to the 10th order
from their returns in one of the previous periods Secondit can be observed that the ARCH parameter 120572
that all the returns experience ARCH and GARCH effectsThe 119875 values of ARCH(10) are more than 10 which showsthat there is no heteroskedasticity effect in the residualsAdditionally the degrees of freedom 120582
119904of the Student
119905-distribution for all the returns are significant suggestingthat the error terms are not normal distribution and thatthe Student 119905-distribution works reasonably well for all thereturn series
As described in Section 31 we apply EVT to thoseresiduals GPD specially is for tail estimation and we choosethe exceedances to be the 10th percentile of the sampleThe Gaussian kernel estimate is for the remaining partThe parameter estimation for each natural gasrsquos residuals isdemonstrated in Table 3
Given the availability of the estimates ofmarginalmodelswe proceed to estimate the copula functions For that wetransform the standardized residuals 119911
119904119905from the GARCH
model into the variates 119906119904119905 using the ECDF Each variate 119906
119904119905
should be uniform (0 1) otherwise the copula model couldbe misspecifiedThis paper follows Patton [15] and Reboredo[16] to carry out the goodness-of-fit test which are Ljung-Box(LB) test and Kolmogorov-Smirnov (KS) test LB test is usedto examine the serial correlation under the null hypothesisof serial independence To do so (119906
119904119905minus 119906119904)119896 is regressed on
the first 10 lags of the variables And KS test is used to testthe null hypothesis that the 119906
119894119905are uniform (0 1) Table 4
reports the 119875 values of these two tests At the 5 significancelevel all the null hypotheses are rejected which implies thatthe marginal distribution model demonstrates that these are
Table 3 Parameter estimation for each natural gasrsquos residuals
KS test 09997 10000 09436 09997 08967Note This table reports the P values from the Ljung-Box (LB) test for serialindependence of the first four moments of the variable 119906
119904119905 We regress
(119906119904119905minus 119906119904)119896 on the first five lags of the variables for 119896 = 1 2 3 4 In
addition we present the P values of the Kolmogorov-Smirnov (KS) test forthe adequacy of the distribution model
not misspecified Hence it can be concluded that the copulamodel can correctly capture the dependence
42 Results for Copula After estimating the parameters ofthe marginal distribution we proceed further to estimatethe copula parameter The multivariate Gaussian copula andStudent 119905-copula are applied in this study The dependencematrix is estimated by the maximum-likelihood estimation(MLE) method The results of the dependence matrix forthe multivariate Gaussian copula model and the Student 119905-copula model are reported in Tables 5 and 6 respectivelyThedependence parameters are positive and strongly significantat 10 level for theGaussian and the Student 119905-copulamodelsThese findings point out the presence of significantly strongpositive relationships between the TTF natural gas portfoliosSecond the degree of freedom for the multivariate Student119905-copula V is reasonably low (49837) which suggests theexistence of substantially extreme comovements and taildependence for all the pairs of the TTF natural gas portfoliosThis finding indicates that extreme events or crises are likelyto spread from one series to another during the bull and bearperiods
43 Portfolio Risk Analysis After modeling the dependencestructure we simulate one-month natural gas portfolioreturns based on the dependence structure specified in boththe multivariate Gaussian copula and the Student 119905-copulaand estimate the risk of a five-dimensional natural gasportfolio With an equally weighted portfolio of five natural
6 The Scientific World Journal
Table 5 Estimated parameters of natural gas portfolio for multi-variate Gaussian copula
Note lowastlowast denotes rejection of the null hypothesis at the 1 and 5 signifi-cance levels respectively
gas returns (TTFFD TTFFM TTFFQ TTFFS and TTFFY)we can calculate the VaR and the CVaR of the portfolio Theresults of the VaR and the CVaR calculations are presentedin Table 7 From Table 7 we can perceive that the VaR andthe CVaR calculated from the Student 119905-copula are more thanthose from the Gaussian copula under the same confidencelevel The reason is that the Student 119905-copula considers theextreme dependence whereas the Gaussian copula does not
Likewise in this paper we also compute the optimalweights of each asset that minimize the portfolio risk Theoptimal portfolio weight under the minimum portfolio riskand the VaR (CVaR) under different copulas and confidencelevels are presented in Table 8 From Table 8 we can draw thefollowing interesting conclusions
(1) The optimal portfolio weights are similar acrossthe multivariate Gaussian copula and Student 119905-copula and different confidence levels The optimalinvestment tends to concentrate on TTFFM TTFFQTTFFS and TTFFY Moreover the optimal portfolioweights of TTFFM TTFFQ TTFFS and TTFFY arealmost equal However the optimal portfolio weightof TTFFD is nearly zero This is because TTFFD isthe spot (one-day-ahead) price and TTFFM TTFFQTTFFS and TTFFY are the futures prices Futures
Table 7 Portfolio risk under equally weighted natural gas portfolio
Confidence level Risk value Normal copula 119905-copula
090 VaR 00746 00755CVaR 01094 01112
095 VaR 00977 00991CVaR 01344 01365
099 VaR 01565 01624CVaR 01989 01994
markets have risk aversion functionUnderminimumportfolio risk investors prefer the futures market Onthe other hand with higher confidence levels theweight of TTFFD investment becomes larger Thereason is that with higher confidence levels investorswill be more risk tolerant and therefore willing totake more risk to achieve higher expected returns
(2) Comparing the risk value calculated by the Gaussiancopula and Student 119905-copula we observe that mostof the VaR and all the CVaR from the Gaussiancopula are less than those from the Student 119905-copulaThis result is consistent with the conclusions whencomputing the VaR and the CVaR of the equallyweighted foreign exchange portfolio This is becausethe Student 119905-copula on account of its heavy tail isable to capture the extreme dependence The extremeevents of the asset returns have higher dependenceand hence larger VaR and CVaR
5 Conclusion
We employed the GARCH-EVT-copula model to estimatethe risk of a multidimensional natural gas portfolio Thisstudy focuses on the natural gas portfolio at the Title TransferFacility (TTF) Hub The data consist of daily series ofone-day-ahead gas prices (TTFFD) one-month-ahead cash-settled futures natural gas prices (TTFFM) one-quarter-ahead cash-settled futures natural gas prices (TTFFQ) one-season-ahead cash-settled futures natural gas prices (TTFFS)and one-year-ahead cash-settled futures natural gas prices(denoted by TTFFY) Our dataset is from January 1 2007 toMay 30 2014 The multivariate Gaussian copula and Student119905-copula have been employed to model the dependencestructure
The empirical results show that for an equally weightedportfolio of five natural gases theVaR and theCVaR obtainedfrom the Student 119905-copula are larger than those obtained fromthe Gaussian copula The reason is that the Gaussian copulacannot capture the extreme dependence but the Student119905-copula can Second the optimal portfolio weights arefound to be similar across the multivariate Gaussian copulaand Student 119905-copula and different confidence levels uponminimizing the portfolio risk The optimal investment tendsto concentrate on TTFFM TTFFQ TTFFS and TTFFY Inaddition the optimal portfolio weights of TTFFM TTFFQTTFFS and TTFFY are almost equal At the same timethe optimal portfolio weight of TTFFD is nearly zero With
The Scientific World Journal 7
Table 8 Optimal investment proportion of natural gas portfolio with minimum risk
higher confidence levels theweight of the TTFFD investmentbecomes larger
Further studies on estimating the risk of natural gasportfolios should concentrate on modeling dependenceSpecifically we should employ more copulas multivariateClaytonmultivariateGumbel and vine copula Furthermorethe risk of natural gas portfolio in different Hubs should beconsidered Third we should work on backtesting to checkthe performance of the approach compared to other popularapproaches of VaR estimation We intend to address thesematters in our future research
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A Ghorbel and A Trabelsi ldquoEnergy portfolio risk managementusing time-varying extreme value copula methodsrdquo EconomicModelling vol 38 pp 470ndash485 2014
[2] Z-R Wang X-H Chen Y-B Jin and Y-J Zhou ldquoEstimatingrisk of foreign exchange portfolio using VaR and CVaR basedonGARCH-EVT-CopulamodelrdquoPhysicaA StatisticalMechan-ics and Its Applications vol 389 no 21 pp 4918ndash4928 2010
[3] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics amp Economics vol 45 no 3 pp 315ndash3242009
[4] R K Y Low J Alcock R Faff and T Brailsford ldquoCanonicalvine copulas in the context of modern portfolio managementare they worth itrdquo Journal of Banking and Finance vol 37 no8 pp 3085ndash3099 2013
[5] R Aloui M S Aıssa and D K Nguyen ldquoConditional depen-dence structure between oil prices and exchange rates a copula-GARCHapproachrdquo Journal of InternationalMoney and Financevol 32 no 1 pp 719ndash738 2013
[6] YH Chen andAH Tu ldquoEstimating hedged portfolio value-at-risk using the conditional copula an illustration of model riskrdquoInternational Review of Economics and Finance vol 27 pp 514ndash528 2013
[7] G N F Weiszlig and H Supper ldquoForecasting liquidity-adjustedintraday value-at-riskwith vine copulasrdquo Journal of Banking andFinance vol 37 no 9 pp 3334ndash3350 2013
[8] T Berger ldquoForecasting value-at-risk using time varying copulasandEVT return distributionsrdquo International Economics vol 133pp 93ndash106 2013
[9] R Aloui M S B Aıssa S Hammoudeh and D K NguyenldquoDependence and extreme dependence of crude oil and naturalgas prices with applications to risk managementrdquo Energy Eco-nomics vol 42 pp 332ndash342 2014
[10] D A Dickey and W A Fuller ldquoDistribution of the estimatorsfor autoregressive time series with a unit rootrdquo Journal of theAmerican Statistical Association vol 74 no 366 pp 427ndash4311979
[11] D Kwiatkowski P C B Phillips P Schmidt and Y Shin ldquoTest-ing the null hypothesis of stationarity against the alternative ofa unit root How sure are we that economic time series have aunit rootrdquo Journal of Econometrics vol 54 no 1ndash3 pp 159ndash1781992
[12] T Bollerslev ldquoGeneralized autoregressive conditional het-eroskedasticityrdquo Journal of Econometrics vol 31 no 3 pp 307ndash327 1986
[13] W H DuMouchel ldquoEstimating the stable index 120572 in order tomeasure tail thickness a critiquerdquo The Annals of Statistics vol11 no 4 pp 1019ndash1031 1983
[14] P Embrechts F Lindskog and A McNeil ldquoModelling depen-dence with copulas and applications to risk managementrdquo inHandbook of Heavy Tailed Distributions in Finance S RachevEd pp 25ndash26 Elsevier Amsterdam The Netherlands 2003
[15] A J Patton ldquoModelling asymmetric exchange rate depen-dencerdquo International Economic Review vol 47 no 2 pp 527ndash556 2006
[16] J C Reboredo ldquoHow do crude oil prices co-move A copulaapproachrdquo Energy Economics vol 33 no 5 pp 948ndash955 2011
Note lowastlowast denotes rejection of the null hypothesis at the 1 and 5 signifi-cance levels respectively
gas returns (TTFFD TTFFM TTFFQ TTFFS and TTFFY)we can calculate the VaR and the CVaR of the portfolio Theresults of the VaR and the CVaR calculations are presentedin Table 7 From Table 7 we can perceive that the VaR andthe CVaR calculated from the Student 119905-copula are more thanthose from the Gaussian copula under the same confidencelevel The reason is that the Student 119905-copula considers theextreme dependence whereas the Gaussian copula does not
Likewise in this paper we also compute the optimalweights of each asset that minimize the portfolio risk Theoptimal portfolio weight under the minimum portfolio riskand the VaR (CVaR) under different copulas and confidencelevels are presented in Table 8 From Table 8 we can draw thefollowing interesting conclusions
(1) The optimal portfolio weights are similar acrossthe multivariate Gaussian copula and Student 119905-copula and different confidence levels The optimalinvestment tends to concentrate on TTFFM TTFFQTTFFS and TTFFY Moreover the optimal portfolioweights of TTFFM TTFFQ TTFFS and TTFFY arealmost equal However the optimal portfolio weightof TTFFD is nearly zero This is because TTFFD isthe spot (one-day-ahead) price and TTFFM TTFFQTTFFS and TTFFY are the futures prices Futures
Table 7 Portfolio risk under equally weighted natural gas portfolio
Confidence level Risk value Normal copula 119905-copula
090 VaR 00746 00755CVaR 01094 01112
095 VaR 00977 00991CVaR 01344 01365
099 VaR 01565 01624CVaR 01989 01994
markets have risk aversion functionUnderminimumportfolio risk investors prefer the futures market Onthe other hand with higher confidence levels theweight of TTFFD investment becomes larger Thereason is that with higher confidence levels investorswill be more risk tolerant and therefore willing totake more risk to achieve higher expected returns
(2) Comparing the risk value calculated by the Gaussiancopula and Student 119905-copula we observe that mostof the VaR and all the CVaR from the Gaussiancopula are less than those from the Student 119905-copulaThis result is consistent with the conclusions whencomputing the VaR and the CVaR of the equallyweighted foreign exchange portfolio This is becausethe Student 119905-copula on account of its heavy tail isable to capture the extreme dependence The extremeevents of the asset returns have higher dependenceand hence larger VaR and CVaR
5 Conclusion
We employed the GARCH-EVT-copula model to estimatethe risk of a multidimensional natural gas portfolio Thisstudy focuses on the natural gas portfolio at the Title TransferFacility (TTF) Hub The data consist of daily series ofone-day-ahead gas prices (TTFFD) one-month-ahead cash-settled futures natural gas prices (TTFFM) one-quarter-ahead cash-settled futures natural gas prices (TTFFQ) one-season-ahead cash-settled futures natural gas prices (TTFFS)and one-year-ahead cash-settled futures natural gas prices(denoted by TTFFY) Our dataset is from January 1 2007 toMay 30 2014 The multivariate Gaussian copula and Student119905-copula have been employed to model the dependencestructure
The empirical results show that for an equally weightedportfolio of five natural gases theVaR and theCVaR obtainedfrom the Student 119905-copula are larger than those obtained fromthe Gaussian copula The reason is that the Gaussian copulacannot capture the extreme dependence but the Student119905-copula can Second the optimal portfolio weights arefound to be similar across the multivariate Gaussian copulaand Student 119905-copula and different confidence levels uponminimizing the portfolio risk The optimal investment tendsto concentrate on TTFFM TTFFQ TTFFS and TTFFY Inaddition the optimal portfolio weights of TTFFM TTFFQTTFFS and TTFFY are almost equal At the same timethe optimal portfolio weight of TTFFD is nearly zero With
The Scientific World Journal 7
Table 8 Optimal investment proportion of natural gas portfolio with minimum risk
higher confidence levels theweight of the TTFFD investmentbecomes larger
Further studies on estimating the risk of natural gasportfolios should concentrate on modeling dependenceSpecifically we should employ more copulas multivariateClaytonmultivariateGumbel and vine copula Furthermorethe risk of natural gas portfolio in different Hubs should beconsidered Third we should work on backtesting to checkthe performance of the approach compared to other popularapproaches of VaR estimation We intend to address thesematters in our future research
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A Ghorbel and A Trabelsi ldquoEnergy portfolio risk managementusing time-varying extreme value copula methodsrdquo EconomicModelling vol 38 pp 470ndash485 2014
[2] Z-R Wang X-H Chen Y-B Jin and Y-J Zhou ldquoEstimatingrisk of foreign exchange portfolio using VaR and CVaR basedonGARCH-EVT-CopulamodelrdquoPhysicaA StatisticalMechan-ics and Its Applications vol 389 no 21 pp 4918ndash4928 2010
[3] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics amp Economics vol 45 no 3 pp 315ndash3242009
[4] R K Y Low J Alcock R Faff and T Brailsford ldquoCanonicalvine copulas in the context of modern portfolio managementare they worth itrdquo Journal of Banking and Finance vol 37 no8 pp 3085ndash3099 2013
[5] R Aloui M S Aıssa and D K Nguyen ldquoConditional depen-dence structure between oil prices and exchange rates a copula-GARCHapproachrdquo Journal of InternationalMoney and Financevol 32 no 1 pp 719ndash738 2013
[6] YH Chen andAH Tu ldquoEstimating hedged portfolio value-at-risk using the conditional copula an illustration of model riskrdquoInternational Review of Economics and Finance vol 27 pp 514ndash528 2013
[7] G N F Weiszlig and H Supper ldquoForecasting liquidity-adjustedintraday value-at-riskwith vine copulasrdquo Journal of Banking andFinance vol 37 no 9 pp 3334ndash3350 2013
[8] T Berger ldquoForecasting value-at-risk using time varying copulasandEVT return distributionsrdquo International Economics vol 133pp 93ndash106 2013
[9] R Aloui M S B Aıssa S Hammoudeh and D K NguyenldquoDependence and extreme dependence of crude oil and naturalgas prices with applications to risk managementrdquo Energy Eco-nomics vol 42 pp 332ndash342 2014
[10] D A Dickey and W A Fuller ldquoDistribution of the estimatorsfor autoregressive time series with a unit rootrdquo Journal of theAmerican Statistical Association vol 74 no 366 pp 427ndash4311979
[11] D Kwiatkowski P C B Phillips P Schmidt and Y Shin ldquoTest-ing the null hypothesis of stationarity against the alternative ofa unit root How sure are we that economic time series have aunit rootrdquo Journal of Econometrics vol 54 no 1ndash3 pp 159ndash1781992
[12] T Bollerslev ldquoGeneralized autoregressive conditional het-eroskedasticityrdquo Journal of Econometrics vol 31 no 3 pp 307ndash327 1986
[13] W H DuMouchel ldquoEstimating the stable index 120572 in order tomeasure tail thickness a critiquerdquo The Annals of Statistics vol11 no 4 pp 1019ndash1031 1983
[14] P Embrechts F Lindskog and A McNeil ldquoModelling depen-dence with copulas and applications to risk managementrdquo inHandbook of Heavy Tailed Distributions in Finance S RachevEd pp 25ndash26 Elsevier Amsterdam The Netherlands 2003
[15] A J Patton ldquoModelling asymmetric exchange rate depen-dencerdquo International Economic Review vol 47 no 2 pp 527ndash556 2006
[16] J C Reboredo ldquoHow do crude oil prices co-move A copulaapproachrdquo Energy Economics vol 33 no 5 pp 948ndash955 2011
higher confidence levels theweight of the TTFFD investmentbecomes larger
Further studies on estimating the risk of natural gasportfolios should concentrate on modeling dependenceSpecifically we should employ more copulas multivariateClaytonmultivariateGumbel and vine copula Furthermorethe risk of natural gas portfolio in different Hubs should beconsidered Third we should work on backtesting to checkthe performance of the approach compared to other popularapproaches of VaR estimation We intend to address thesematters in our future research
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A Ghorbel and A Trabelsi ldquoEnergy portfolio risk managementusing time-varying extreme value copula methodsrdquo EconomicModelling vol 38 pp 470ndash485 2014
[2] Z-R Wang X-H Chen Y-B Jin and Y-J Zhou ldquoEstimatingrisk of foreign exchange portfolio using VaR and CVaR basedonGARCH-EVT-CopulamodelrdquoPhysicaA StatisticalMechan-ics and Its Applications vol 389 no 21 pp 4918ndash4928 2010
[3] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics amp Economics vol 45 no 3 pp 315ndash3242009
[4] R K Y Low J Alcock R Faff and T Brailsford ldquoCanonicalvine copulas in the context of modern portfolio managementare they worth itrdquo Journal of Banking and Finance vol 37 no8 pp 3085ndash3099 2013
[5] R Aloui M S Aıssa and D K Nguyen ldquoConditional depen-dence structure between oil prices and exchange rates a copula-GARCHapproachrdquo Journal of InternationalMoney and Financevol 32 no 1 pp 719ndash738 2013
[6] YH Chen andAH Tu ldquoEstimating hedged portfolio value-at-risk using the conditional copula an illustration of model riskrdquoInternational Review of Economics and Finance vol 27 pp 514ndash528 2013
[7] G N F Weiszlig and H Supper ldquoForecasting liquidity-adjustedintraday value-at-riskwith vine copulasrdquo Journal of Banking andFinance vol 37 no 9 pp 3334ndash3350 2013
[8] T Berger ldquoForecasting value-at-risk using time varying copulasandEVT return distributionsrdquo International Economics vol 133pp 93ndash106 2013
[9] R Aloui M S B Aıssa S Hammoudeh and D K NguyenldquoDependence and extreme dependence of crude oil and naturalgas prices with applications to risk managementrdquo Energy Eco-nomics vol 42 pp 332ndash342 2014
[10] D A Dickey and W A Fuller ldquoDistribution of the estimatorsfor autoregressive time series with a unit rootrdquo Journal of theAmerican Statistical Association vol 74 no 366 pp 427ndash4311979
[11] D Kwiatkowski P C B Phillips P Schmidt and Y Shin ldquoTest-ing the null hypothesis of stationarity against the alternative ofa unit root How sure are we that economic time series have aunit rootrdquo Journal of Econometrics vol 54 no 1ndash3 pp 159ndash1781992
[12] T Bollerslev ldquoGeneralized autoregressive conditional het-eroskedasticityrdquo Journal of Econometrics vol 31 no 3 pp 307ndash327 1986
[13] W H DuMouchel ldquoEstimating the stable index 120572 in order tomeasure tail thickness a critiquerdquo The Annals of Statistics vol11 no 4 pp 1019ndash1031 1983
[14] P Embrechts F Lindskog and A McNeil ldquoModelling depen-dence with copulas and applications to risk managementrdquo inHandbook of Heavy Tailed Distributions in Finance S RachevEd pp 25ndash26 Elsevier Amsterdam The Netherlands 2003
[15] A J Patton ldquoModelling asymmetric exchange rate depen-dencerdquo International Economic Review vol 47 no 2 pp 527ndash556 2006
[16] J C Reboredo ldquoHow do crude oil prices co-move A copulaapproachrdquo Energy Economics vol 33 no 5 pp 948ndash955 2011
