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Int.Fin.Markets,Inst.and Money16(2006)180–197Empirical analysis of GARCH modelsin value at risk estimationMike K.P.So a,∗,Philip L.H.Yu ba Department of Information and Systems Management,School of Business and Management,The Hong Kong University of Science and Technology,Clear Water Bay,Hong Kongb Department of Statistics and Actuarial Science,The University of Hong Kong,Hong KongReceived26February2003;accepted2February2005Available online15August2005AbstractThis paper studies seven GARCH models,including RiskMetrics and two long memory GARCH models,in Value at Risk(VaR)estimation.Both long and short positions of investment were con-sidered.The seven models were applied to12market indices and four foreign exchange rates to assess each model in estimating VaR at various confidence levels.The results indicate that both sta-tionary and fractionally integrated GARCH models outperform RiskMetrics in estimating1%VaR. Although most return series show fat-tailed distribution and satisfy the long memory property,it is more important to consider a model with fat-tailed error in estimating VaR.Asymmetric behavior is also discovered in the stock market data that t-error models give better1%VaR estimates than normal-error models in long position,but not in short position.No such asymmetry is observed in the exchange rate data.©2005Elsevier B.V.All rights reserved.JEL classification:C53;G15Keywords:GARCH model;Long memory;Market risk∗Corresponding author.Tel.:+852********;fax:+852********.E-mail addresses:immkpso@ust.hk(M.K.P.So),plhyu@hku.hk(P.L.H.Yu).0021-9673/$–see front matter©2005Elsevier B.V.All rights reserved.doi:10.1016/j.intfin.2005.02.001M.K.P.So,P.L.H.Yu/Int.Fin.Markets,Inst.and Money16(2006)180–197181 1.IntroductionValue at Risk(VaR)is one of the most important measures of the market risk that has been widely used forfinancial risk management by institutions including banks,regulators and portfolio managers.Since the risk management group at J.P.Morgan developed the RiskMetrics model for measuring VaR in1994,RiskMetrics has become a benchmark for measuring market risk.Other methods,such as that based on extreme value theories,high frequency data and conditional moments of GARCH models can be found in Danielsson and de Vries(1997),Beltratti and Morana(1999),Ho et al.(2000)as well as Wong and So (2003).See Duffie and Pan(1997),Jorion(2001)for comprehensive overview of VaR.The common RiskMetrics model assumes that returns of afinancial asset follow a condi-tional normal distribution with zero mean and variance being expressed as an exponentially weighted moving average of historical squared returns.This model has two drawbacks. Firstly,it was well documented that a return distribution usually has a heavier tail than a normal distribution.Assuming conditional normality may generate substantial bias in VaR estimation which mainly concerns the tail properties of the return distribution.Secondly, recent empirical studies found that manyfinancial return series may exhibit long memory or long-term dependence on market volatility(Ding et al.,1993;So,2000).Such long term dependence was found to have significant impact on the pricing offinancial derivatives as well as forecasting market volatility.Besides the GARCH model of Bollerslev(1986)and its variants(Engle and(Bollerslev,1986;Nelson,1991))which can capture the time-varying volatility feature,several long memory GARCH models were proposed to incorporate the long memory volatility property infinancial time series;see for example Baillie et al.(1996), Bollerslev and Mikkelsen(1996).It is of interest to see whether long memory can affect the measurement of market risk in the context of VaR.In this paper,we compare the performance of seven GARCH-type models in estimating VaR of market indices.Two of them are long memory GARCH models.Both conditional normal and conditional t-error distributions are considered.While most empirical studies focused only on holding a long position of a portfolio,we also consider a short position.The rest of this paper is organized as follows.Section2describes the basic concept of VaR and presents various GARCH-type models forfinancial return series.In Section3,we discuss maximum likelihood method of estimating the parameters of long memory GARCH models. We apply the seven models to12market indices in Section4to assess the performance in estimating VaR at various confidence levels.Applications to exchange data are presented in Section5.Section6gives some concluding remarks.2.Value at risk and GARCH-type models2.1.VaRValue at Risk,or VaR,is a commonly used statistic for measuring potential risk of economic losses infinancial markets.With VaR,financial institutions can have a sense on the minimum amount that is expected to lose with a small probabilityαover a given time horizon k(usually1-day or10days).For example,aα=5%1-day VaR of$10million182M.K.P.So,P.L.H.Yu/Int.Fin.Markets,Inst.and Money16(2006)180–197can tell us that one out of20days,we could expect to realize a loss of at least$10million. Alternatively,we could say that the maximum loss we would expect on19out of20days is $10million.In other words,VaR is defined as the maximum loss over a given time horizon at a given confidence level.Mathematically,let P t be the price of afinancial asset on day t.A k-day VaR on day t is defined byP(P t−k−P t≤VaR(t,k,α))=1−α.Given a distribution of return,VaR can be determined and expressed in terms of a percentile of the return distribution(Dowd,1998;Jorion,2001).Specifically,if qαis theαth percentile of the continuously compounded return log(P t)−log(P t−k),then VaR can be written as VaR(t,k,α)=(1−e qα)P t−k.(1) The above expression implies that good VaR estimates can only be produced with accurate forecasts of the percentiles qα,which realizes on appropriate volatility modeling.To incor-porate the time varying feature of the market volatility,we adopt various heteroskedastic models.2.2.GARCH-type modelsWe define the1-day logarithmic return on day t as r t=log(P t)−log(P t−1)and denote the information up to time t by t.In this paper,we investigate the performance of the following GARCH-type models in estimating VaR.2.2.1.RiskMetrics modelThe RiskMetrics model assumes that returns are generated as followsr t= t, t| t−1∼N(0,σ2t)σ2t=λσ2t−1+(1−λ) 2t−1where0≤λ≤1is the smoothing parameter.The formulation in the mean equation implies that the conditional distribution of returns is normal with mean zero.One main feature of the RiskMetrics model is that the conditional variance can be written as an exponentially weighted moving average(EWMA)of the past squared innovations or returns,that is,σ2t=(1−λ)(r2t−1+λr2t−2+λ2r2t−3+...).The smaller the smoothing parameter,the greater the weight is given to recent return data. RiskMetrics(1996)advised that we can useλ=0.94for daily data andλ=0.97for monthly data.It was also shown in the literature thatλ=0.94produces very good forecasts for1-day volatility(RiskMetrics,1996;Fleming et al.,2001).Under the RiskMetrics model, the1-day VaR on day t in(1)is reduced to(1−e qα)P t−1≈−σt zαP t−1,where zαis the 100αth percentile of N(0,1).M.K.P.So,P.L.H.Yu/Int.Fin.Markets,Inst.and Money16(2006)180–1971832.2.2.GARCH(p,q)modelThe GARCH(p,q)model(p>0and q≥0are integers)is defined asr t=µ+ t, t| t−1∼D(0,σ2t)(2)σ2t=ω+β(B)σ2t+α(B) 2twhereω>0,α(B)=α1B+...+αq B q andβ(B)=β1B+...+βp B p,withαi≥0for i=1,...,q andβj≥0for j=1,...,p,and D(0,σ2t)represents a conditional distribution with zero mean and varianceσ2t.Bollerslev(1986)showed that the GARCH process of{r t} is covariance stationary if and only ifα(1)+β(1)<1.Since the variance equation of the GARCH(p,q)process can be expressed asσ2t=ω(1−β(B))−1+α(B)(1−β(B))−1 2t,the above stationary condition implies that the effect of the past squared innovations on the current conditional variance decays exponentially with the lag length.Note that the RiskMetrics model can be viewed as a special case of GARCH(1,1)model withµ=0,ω= 0,andλ=β1=1−α1.Since GARCH(1,1)model was found to be adequate to many financial time series(Bollerslev et al.,1992),we focused on this model in our empirical analysis.2.2.3.IGARCH(p,q)modelNote that the variance equation of the GARCH model can be written as(1−α(B)−β(B)) 2t=ω+(1−β(B))νt,νt= 2t−σ2t.According to the empirical studies in Engle and Bollerslev(1986),Chou(1988),the estimated lag polynomial(1−α(B)−β(B))is found to have a significant unit root in some applications of GARCH models.Factoring this polynomial as(1−α(B)−β(B))= (1−B)φ(B),whereφ(B)has all the roots outside the unit circle,Engle and Bollerslev (1986)proposed the following integrated GARCH,or IGARCH(p,q)model:φ(B)(1−B) 2t=ω+(1−β(B))νt,νt= 2t−σ2t,(3) whereφ(B)=1−φ1B−···−φq B q.As many empirical studies using GARCH(1,1)mod-els giveα1+β1very close to1implying high persistent volatility,the impact of past in-formation on future volatility forecasts decays very slowly.Therefore,we believe that the IGARCH(1,1)model given byr t=µ+ t,σ2t=ω+β1σ2t−1+(1−β1) 2t−1is a good alternative to GARCH(1,1)model.Whenµ=0,the IGARCH(1,1)model reduces to RiskMetrics model withλ=β1.From the good performance of RiskMetrics for some αsuch as5%or10%documented in the literature,it is anticipated that IGARCH(1,1)can also be a good model for VaR estimation.In our empirical studies,we estimatedβ1from the data rather than taking it to be0.94in RiskMetrics.184M.K.P .So,P .L.H.Yu /Int.Fin.Markets,Inst.and Money 16(2006)180–1972.2.4.FIGARCH(p,d,q)modelThere has been significant empirical evidence of long memory volatility in financial markets (Ding et al.,1993;So,2000).In order to capture long memory property in financial market volatility,Baillie et al.(1996)extended the IGARCH model by replacing the first difference operator (1−B )in (3)by the fractional differencing operator (1−B )d with 0<d <1and developed the following FIGARCH(p,d,q )model:φ(B )(1−B )d 2t =ω+(1−β(B ))νt ,νt = 2t −σ2t .Clearly,the above FIGARCH model covers GARCH and IGARCH as special cases whend =0or 1.To better understand the properties of the models,we rewrite the variance equation of the FIGARCH(p,d,q )model asσ2t =ω(1−β(B ))−1+(1−β(B ))−1(1−φ(B )(1−B )d ) 2t ,(4)where (1−B )d can be expressed by the Maclaurin series expansion(1−B )d=∞ k =0Γ(k −d )Γ(k +1)Γ(−d )B k=1−dB +(1−d )(−d )2B 2+(2−d )(1−d )(−d )3!B 3+ (5)As Γ(k −d )/Γ(k +1)≈k −d −1if k is large,the coefficients in the above infinite polyno-mial decay hyperbolically.Therefore,in the FIGARCH model (0<d <1),the effect of thepast innovations on the current conditional variance dies out at a hyperbolic rate with the lag length.This makes a clear difference from GARCH and IGARCH models (d =1)that the effect of the past squared innovations on the current conditional variance dies out ex-ponentially in GARCH and remains important for all lags in IGARCH.Hence,FIGARCH models can be good compromise of IGARCH and GARCH in capturing volatility dynamic structure.Since in some stock markets the parameter µis likely to be significantly positive,assum-ing µ=0in RiskMetrics may generate biased VaR estimates.Therefore,we allow µin the underlying GARCH,IGARCH and FIGARCH models and estimate it using returns data.In the empirical investigation,the mean specification in (2)was adopted for the three GARCH models.Two conditional distributions D (0,σ2t )for the error term t were considered:(a)a normal distribution N (0,σ2t ),and (b)a standardized t distribution with νd.f.and variance σ2t (i.e., t ∼σt t (ν)/√ν/(ν−2)).It then follows that the 1-day VaR on day t in (1)is(1−e µ+cσt )P t −1,(6)where c =z αand t α(ν)/√ν/(ν−2)for normal and t -distributed error,respectively,and t α(ν)is the 100αth percentile of the standardized t distribution with degrees of freedom ν.Once estimates of µand GARCH parameters are available,estimates of the above VaR are readily obtained.M.K.P .So,P .L.H.Yu /Int.Fin.Markets,Inst.and Money 16(2006)180–1971853.Estimation of GARCH-type modelsThe maximum likelihood method is used to estimate GARCH-type models.The pa-rameters under GARCH and IGARCH models with normal and t -distributed errors can be estimated by standard statistical software.Here,we only discuss the estimation of param-eters under the FIGARCH models.We first consider the case of t -distributed errors.Given a time series of returns over a period of T days r t ,t =1,...,T ,the maximum likelihood estimate of µunder any GARCH-type model is the sample mean ˆµ= T t =1r t /T .With µ=ˆµ,the log-likelihood function reduces tot =T ln Γ((ν+1)/2)√(ν−2)πΓ(ν/2)−12T t =1 ln σ2t +(ν+1)ln 1+ˆ 2t(ν−2)σ2t where σ2t is given in (4)and ˆ t =r t −ˆµ.Similarly,the log-likelihood function under normalerrors is given byn =−12T ln(2π)−12T t =1ln σ2t +ˆ 2t σ2t.Calculating the log-likelihood function ( t or n )is nontrivial in FIGARCH models because the log-likelihood involves σ2t ,and hence the term (1−B )d .Since d is a real number in (0,1),computing (1−B )d amounts to getting the infinite sum shown in (5).In practice,the maximum likelihood techniques for FIGARCH models require the truncation of the infinite sum at a certain lag.Since the fractional differencing operator d is designed to capture the long memory feature of the process,truncating at too low lag may distort some important long-term dependencies.To mitigate these effects,the truncation lag was fixed at 1000.In other words,we used the following approximation(1−B )d≈1000 k =0Γ(k −d )Γ(k +1)Γ(−d )B kwhen evaluating σ2t in the log-likelihood.Once the maximum likelihood estimates of the model parameters in FIGARCH model are obtained,we can forecast the 1-day ahead vari-ance ˆσ2t by substituting the estimates into (4).Finally,putting ˆµand ˆσ2t into (6)gives an estimate of 1-day VaR on day t .4.Applications to stock market indices 4.1.Data descriptionIn this section,we apply various GARCH models and RiskMetrics to stock market data for VaR estimation.Twelve market indices namely,All Ordinaries Index (AOI)of Australia,FTSE100of United Kingdom,Jakarta Composite (JSX)of Indonesia,Hang Seng Index186M.K.P.So,P.L.H.Yu/Int.Fin.Markets,Inst.and Money16(2006)180–197Table1Summary statistics of stock market and exchange rate returnsSkewness Kurtosis Index Period n Mean StandarddeviationAOI06/08/84–31/12/9836400.0377 1.0221−6.76177.68 FTSE10014/02/84–31/12/9837580.04660.9580−1.2017.67 HSI03/01/75–31/12/9859260.0691 1.8375−2.3951.27 JSX03/01/85–31/12/9834520.0513 1.69804.40120.77 KLSE04/01/77–31/12/9854140.0334 1.5966−0.2932.84 KOSPI05/01/77–31/12/9864420.0274 1.3174−0.3510.15 NASDAQ12/10/84–31/12/9835940.06100.9800−1.5017.61 NIKKEI05/01/84–31/12/9837580.0088 1.3610−0.1111.66 SET02/05/75–31/12/9858300.0218 1.43970.079.04 SP50004/01/50–31/12/98124160.03460.8504−1.8148.67 STII03/01/80–31/12/9847480.0245 1.3363−1.6536.06 WEIGHT06/01/75–31/12/9869050.0509 1.6438−0.262.64 Exchange rate Period n Mean StandardSkewness KurtosisdeviationGBP/US01/01/80–31/12/9847730.00600.66470.062.94 YEN/US01/01/80–31/12/984773−0.01580.6916−0.543.91 AUD/US01/01/80–31/12/9847730.01240.61431.8932.12 CAD/US01/01/80–31/12/9847730.00580.27480.103.58(HSI)of Hong Kong,Kuala Lumpur Composite Price Index(KLSE)of Malaysia,KOSPI of South Korea,NASDAQ of US,Nikkei225Index(NIKKEI)of Japan,Stock Exchange of Thailand Daily Index(SET)of Thailand,Standard&Poor500Index(SP500)of US, Straits Times Industrial Index(STII)of Singapore and Taiwan Stock Exchange Weighted Stock Index(WEIGHT)of Taiwan were selected for illustration.The above list includes indices of global markets and major markets in Asia.To allow enough data forfitting the FIGARCH models,the starting year of most data series ranges from1975to1985 so that we can have at least3000observations in the sample.The time span of the data sets is presented in Table1.All data sets end at the last trading day of1998.The whole data range was divided into two parts;the estimation period and the validation period.We estimated VaR in the validation period to assess the performance of different methods. For the sake of comparisons,we chose the validation period to be1995–1998for all data sets.In this section,the return r t is expressed in percentages,i.e.r t=100×(log P t−log P t−1).Summary statistics of returns are given in Table1.Highest returns are recorded in HSI and NASDAQ where investing in the two markets generates more than15%annual return on the average in the time period of investigation.The lowest returns is recorded in NIKKEI which has about2.2%average annual return from1984to1998.According to the standard deviations in the table,investing in SP500and FTSE100are the least risky.All the market returns show negative skewness except JSX and SET.Typical phenomenon of excess kurtosis is also revealed in the indices.To summarize the mean-variance relationship of the12markets,we also plot the mean daily returns against daily standard deviation in Fig.1.The four non-Asian market indices,AOI,FTSE100,NASDAQ and SP500form one cluster and all the eight Asian market indices form another cluster.It is clear that the twoM.K.P.So,P.L.H.Yu/Int.Fin.Markets,Inst.and Money16(2006)180–197187Fig.1.Sample mean and standard deviation of the market index returns.clusters differ mainly in the standard n market returns generally are more volatile than the non-Asian market returns.Typical pattern of‘high risk high return’for investment can also be seen in the Asian indices.We observe that the sample autocorrelations of the absolute centered returns,|r t−¯r|, of the12market indices decay slowly with lag.This is especially true for JSX,KOSPI, SET and WEIGHT where autocorrelations are found to be highly significant even at lag 200.Therefore,some long memory features are observed in the absolute centered returns of the four indices.For AOI and FTSE100,it is hard to judge whether market volatil-ity exhibits long range dependence because the autocorrelations are declared to be in-significant after lag50.To provide better insights about the time series properties of the stock market returns,we also examine the partial correlations(PACF).All PACFs are found to decay quite quickly.It is not surprising even though we believe on the existence of long memory in|r t−¯r|because PACFs of ordinary long memory ARIMA processes also decay quickly;see for example(Beran,1994,p.66).Although empirical evidence of long memory is observed in most of the indices,we need to confirm the existence of long memory by the model selection using AIC and SBC.In the next sections,we also study whether fractionally integrated GARCH models can produce superior VaR esti-mates.4.2.GARCHfittingIn our empirical study,the whole time span was divided into two parts.Data in thefirst part were used for estimating unknown parameters in GARCH models.VaR estimates of the last four years were then computed based on the parameter estimates obtained in the first part of the data.These estimates were used to assess the out-sample performance of various GARCH models in forecasting VaR.188M.K.P.So,P.L.H.Yu/Int.Fin.Markets,Inst.and Money16(2006)180–197 The following four conditional variance specifications were adopted:RiskMetrics:σ2t=λσ2t−1+(1−λ) 2t−1GARCH(1,1):σ2t=ω+β1σ2t−1+α1 2t−1IGARCH(1,1):σ2t=ω+β1σ2t−1+(1−β1) 2t−1FIGARCH(1,d,0):σ2t=ω+β1σ2t−1+(1−β1B−(1−B)d) 2tBoth standardized normal and t assumptions on t were assumed for GARCH,IGARCHand FIGARCH models.Together with RiskMetrics,we considered seven VaR estimationmethods based on seven models.As suggested in RiskMetrics(1996),the parameterλwastaken as0.94.Maximum likelihood estimation was performed for the above three GARCHmodels.Table2gives an extract of the parameter estimates.1A‘(t)’is added to indicate thata model with t errors wasfitted.From the GARCH(1,1)modelfitting,the typicalfinding thatφ1=α1+β1is close to one is observed.Without restricting thefitted GARCH(1,1)model to be stationary,wefindfiveφ>1.An increase inφis also noted when a t error model isfittedinstead of normal error.In the IGARCH estimation results with normal errors,we observelargeβ1of0.90and0.91for SP500and WEIGHT,paring withλ=0.94in RiskMetrics,including the interceptωin the IGARCH(1,1)model substantially lowersthe parameter value associated withσ2t−1,implying that the previous innovation 2t−1hasa greater impact on the current variance.For FIGARCH modelfitting,all d which rangesfrom0.09to0.69,are found to be significantly different from zero.The lowest two areassociated with FIGARCH(t)for AOI and FTSE100.In summary,the estimates of d agreewith the argument that market volatility exhibits long range dependence.To compare thequality offit among the six GARCH models,we also report the ranking based on AIC andSBC in Table3.Better models according to AIC and SBC are rankedfirst.Mean ranks arecomputed to aggregate the information from the12indices.It is obvious that t-error modelsperform better than the normal error models.In addition,FIGARCH(t),as a compromise ofGARCH(t)and IGARCH(t),gives the bestfit in terms of the mean ranks.It is interesting tosee whether long memory GARCH models can produce more accurate VaR estimates thanother models.4.3.VaR estimation resultsIn the validation exercise,VaR estimates for the period1995to1998were producedby using the maximum likelihood estimates from thefirst part of the data.Three commonvalues ofαwere chosen for illustration;they are1%,2.5%and5%.In this empiricalstudy,we computed for each index the sample coverageˆαwhich is the proportion of losses(P t−1−P t)greater than the VaR estimates.Table4gives the sample coverages for different αof the12indices.According to the definition in(1),we expect thatˆαis close toαfor a good VaR estimation method.Therefore,the smaller the discrepancy betweenˆαandα, the better performance is the estimation method.To assess the overall performance of the seven methods,we ranked the methods according to|α−ˆα|for each index.A smaller rank1Complete tables of parameter estimates are available upon request from the authors.M.K.P.So,P.L.H.Yu/Int.Fin.Markets,Inst.and Money16(2006)180–197189 Table2An extract of parameter estimates of the12market returnsφ1β1dGARCH GARCH(t)IGARCH IGARCH(t)FIGARCH FIGARCH(t) AOI0.84250.90990.61910.86510.45150.0898 FTSE1000.92550.94270.85540.90840.27620.1915HSI0.97950.98170.78780.83900.60270.6215JSX 1.0386 3.32220.03980.70350.50230.6403 KLSE0.95730.97930.74970.78970.36150.4737 KOSPI 1.0006 1.00770.79600.81340.45360.5152 NASDAQ0.93460.94700.77760.82430.28920.3230 NIKKEI0.95530.98980.69290.81210.39860.5043SET 1.0261 1.06040.80640.77280.41950.4609SP5000.98510.99010.90320.91610.26740.2902STII0.87970.90810.63710.74600.69250.3762 WEIGHT0.99260.99460.91360.90600.36160.4256was assigned to a smaller|α−ˆα|.Then,the average of all the ranks for each method were calculated and displayed as mean rank in Table4.A smaller mean rank indicates an overall better match betweenαandˆαwhich is a sign of superior performance.Similarly,we studied the cases for holding a short position in the investment.In these cases,the VaR is defined by P(P t−P t−k<V aR(t,k,α))=1−αwhich implies thatV aR(t,k,α)=(e q1−α−1)P t−k,where q1−α=µ−cσt,with c=zαor tα(ν)/√ν/(ν−2),is the upperαth percentilesof returns.Sample coverages and mean ranks for short position are presented in Table3Model selection:ranking of AIC and SBC(in parantheses)Index GARCH IGARCH FIGARCH GARCH(t)IGARCH(t)FIGARCH(t) AOI5(5)6(6)4(4)1(1)3(3)2(2)FTSE1004(4)6(6)5(5)1(1)3(3)2(2)HSI4(5)5(4)6(6)2(3)3(1)1(2)JSX5(5)6(6)4(4)1(1)3(3)2(2)KLSE5(5)6(6)4(4)2(3)3(2)1(1) KOSPI6(6)5(5)4(4)3(3)2(2)1(1) NASDAQ5(5)6(6)4(4)1(1)3(3)2(2) NIKKEI5(5)6(6)4(4)3(3)2(2)1(1)SET5(5)6(6)4(4)2(2)3(3)1(1)SP5005(5)6(6)4(4)1(1)2(2)3(3)STII4(4)6(5)5(6)1(1)3(3)2(2) WEIGHT5(6)6(5)4(4)2(3)3(2)1(1)Mean rank 4.8(5.0) 5.8(5.6) 4.3(4.4) 1.7(1.9) 2.8(2.4) 1.6(1.7)Table4Sample coverage for the long position of the12market indicesRM GARCH IGARCH FIGARCH GARCH(t)IGARCH(t)FIGARCH(t)α=1%AOI 1.88 1.18 1.09 1.580.890.99 1.19FTSE100 1.88 1.19 1.29 1.59 1.190.99 1.39HSI 2.02 2.23 2.23 2.13 1.82 1.62 1.42JSX 3.34 1.82 4.25 2.230.71 2.63 1.82KLSE 2.03 2.33 1.93 1.52 1.11 1.01 1.12KOSPI 1.28 1.28 1.28 1.37 1.03 1.11 1.20NASDAQ 2.08 3.07 2.57 2.97 2.08 1.19 1.68NIKKEI 1.92 2.43 2.22 2.030.910.71 1.11SET 1.53 1.63 2.15 2.460.82 1.33 1.33SP500 2.37 2.18 2.08 2.48 1.98 1.58 1.88STII 1.90 1.70 1.30 1.40 1.300.800.90WEIGHT 2.56 2.12 2.21 2.04 1.94 1.94 1.86Mean rank 5.5 5.2 5.0 5.5 2.3 2.0 2.5α=2.5%AOI 2.96 2.57 2.17 2.57 2.67 2.07 3.26FTSE100 2.87 2.77 2.28 2.67 2.77 2.27 3.07HSI 3.14 3.64 3.44 3.14 3.24 3.14 3.24JSX 4.65 3.34 5.36 4.86 3.34 5.66 6.58KLSE 3.44 3.65 3.34 3.55 3.85 3.55 3.25KOSPI 3.42 2.48 2.48 3.34 2.48 2.56 3.17NASDAQ 3.46 5.24 4.06 5.35 5.14 3.86 5.05NIKKEI 3.54 4.55 3.94 4.36 4.75 4.25 4.76SET 2.96 3.98 4.39 4.30 3.78 4.49 4.09SP500 3.66 3.66 3.36 3.67 3.66 3.46 3.56STII 3.30 3.00 2.30 2.10 2.90 2.40 2.40WEIGHT 4.24 4.24 3.98 3.63 3.89 3.80 3.54Mean rank 4.0 4.5 3.4 4.0 4.1 3.6 4.3α=5%AOI 4.84 3.95 3.46 3.86 5.13 4.54 6.23FTSE100 4.85 4.85 4.06 5.15 5.04 4.35 5.84HSI 5.16 5.97 5.36 5.67 6.28 5.97 6.59JSX 6.37 4.757.287.19 6.9810.3111.23 KLSE 5.47 5.88 5.27 5.08 6.59 6.38 6.49KOSPI 6.07 5.98 5.98 6.857.097.187.87NASDAQ 5.938.017.328.428.908.319.01NIKKEI 6.278.097.487.607.897.588.40SET 5.52 6.337.157.477.568.898.79SP500 4.55 4.85 4.45 5.05 5.24 4.95 5.94STII 5.80 4.70 3.90 4.01 6.10 5.30 6.51WEIGHT 6.18 6.01 5.74 6.19 6.36 6.27 6.28Mean rank 2.3 3.0 3.7 3.5 4.6 4.3 6.6Table5.The best coverages with the smallest|α−ˆα|in each index and the smallest mean ranks for eachαare bold faced to highlight the best performance.We con-struct boxplots in Fig.2for describing the distribution of the sample coverage for all models.Table5Sample coverage for the short position of the12market indicesRM GARCH IGARCH FIGARCH GARCH(t)IGARCH(t)FIGARCH(t)α=1%AOI 1.18 1.090.69 1.090.490.200.79FTSE100 1.190.790.690.790.590.490.79HSI 2.53 1.010.91 1.010.710.510.71JSX 1.42 1.11 3.03 1.930.30 1.42 1.52KLSE 1.62 1.82 1.42 1.420.910.81 1.12KOSPI 1.71 1.37 1.37 1.54 1.11 1.11 1.03NASDAQ0.790.890.590.690.400.300.20NIKKEI 1.52 1.52 1.31 1.82 1.21 1.11 1.11SET 2.66 2.45 2.66 2.66 1.43 2.15 2.15SP500 1.38 1.090.89 1.290.400.200.59STII 1.60 1.100.800.900.700.400.80WEIGHT 1.330.970.970.800.710.620.71Mean rank 4.7 2.8 4.2 3.8 4.1 4.9 3.5α=2.5%AOI 2.67 1.68 1.97 1.58 1.970.99 2.08FTSE100 2.48 1.58 1.09 1.68 1.480.99 1.68HSI 3.74 2.53 2.13 2.13 2.43 2.13 2.43JSX 3.03 1.92 4.04 3.34 2.43 4.15 4.15KLSE 3.14 2.43 2.43 2.23 2.53 2.43 2.13KOSPI 3.42 2.56 2.56 3.00 2.65 2.82 2.91NASDAQ 2.47 2.47 1.48 2.08 1.98 1.19 1.68NIKKEI 2.83 2.73 2.43 2.43 2.12 2.02 2.43SET 4.49 3.47 3.98 4.20 2.86 3.68 4.09SP500 3.36 2.37 2.08 3.27 2.27 1.98 2.97STII 2.90 2.00 1.60 1.90 2.30 1.60 1.80WEIGHT 2.83 2.39 2.12 2.04 1.94 1.86 1.86Mean rank 4.1 2.5 4.0 4.5 2.9 5.6 4.3α=5%AOI 5.53 3.75 3.65 4.06 3.85 3.26 4.25FTSE100 5.64 3.66 2.97 3.87 4.35 3.26 4.95HSI 5.47 4.05 3.85 4.25 5.06 4.66 5.17JSX 4.55 3.34 6.37 4.76 5.668.099.21KLSE 4.46 4.46 3.95 3.65 5.17 4.76 4.77KOSPI 6.32 5.38 5.38 5.91 6.15 6.32 6.33NASDAQ 5.14 4.95 4.06 4.56 5.74 4.75 5.64NIKKEI 4.55 4.45 4.35 4.76 4.45 4.35 5.16SET 5.72 4.90 5.92 6.04 5.41 6.54 6.65SP500 6.63 5.24 4.35 3.77 5.64 5.24 6.44STII 5.10 3.40 2.90 2.81 4.00 3.40 4.30WEIGHT 5.48 4.59 4.06 4.16 4.51 4.24 4.16Mean rank 3.1 3.4 5.5 4.3 3.1 4.7 4.0For the long position withα=1%,RiskMetrics,GARCH,IGARCH and FIGARCH perform poorly that all sample coverages are greater than1%,indicating that biased VaR estimates are generated by the three methods.The biasedness is also observed from Fig.2 that all the four boxplots for the four models lie above the reference line of1%.In terms。