empirical analysis of garch models in value at risk - e-journal

Int. Fin. Markets, Inst. and Money 16 (2006) 180–197

Empirical analysis of GARCH modelsin value at risk estimation

Mike K.P. Soa,∗, Philip L.H. Yub

a Department of Information and Systems Management, School of Business and Management,The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

b Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong

Received 26 February 2003; accepted 2 February 2005Available online 15 August 2005

Abstract

This paper studies seven GARCH models, including RiskMetrics and two long memory GARCHmodels, in Value at Risk (VaR) estimation. Both long and short positions of investment were con-sidered. The seven models were applied to 12 market indices and four foreign exchange rates toassess each model in estimating VaR at various confidence levels. The results indicate that both sta-tionary and fractionally integrated GARCH models outperform RiskMetrics in estimating 1% VaR.Although most return series show fat-tailed distribution and satisfy the long memory property, it ismore important to consider a model with fat-tailed error in estimating VaR. Asymmetric behavioris also discovered in the stock market data that t-error models give better 1% VaR estimates thannormal-error models in long position, but not in short position. No such asymmetry is observed in theexchange rate data.© 2005 Elsevier B.V. All rights reserved.

JEL classification: C53; G15

Keywords: GARCH model; Long memory; Market risk

∗ Corresponding author. Tel.: +852 23587726; fax: +852 23582421.E-mail addresses: [email protected] (M.K.P. So), [email protected] (P.L.H. Yu).

0021-9673/$ – see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.intfin.2005.02.001

M.K.P. So, P.L.H. Yu / Int. Fin. Markets, Inst. and Money 16 (2006) 180–197 181

1. Introduction

Value at Risk (VaR) is one of the most important measures of the market risk that hasbeen widely used for financial risk management by institutions including banks, regulatorsand portfolio managers. Since the risk management group at J.P. Morgan developed theRiskMetrics model for measuring VaR in 1994, RiskMetrics has become a benchmark formeasuring market risk. Other methods, such as that based on extreme value theories, highfrequency data and conditional moments of GARCH models can be found in Danielssonand 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 a financial asset follow a condi-tional normal distribution with zero mean and variance being expressed as an exponentiallyweighted 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 anormal distribution. Assuming conditional normality may generate substantial bias in VaRestimation which mainly concerns the tail properties of the return distribution. Secondly,recent empirical studies found that many financial return series may exhibit long memoryor long-term dependence on market volatility (Ding et al., 1993; So, 2000). Such long termdependence was found to have significant impact on the pricing of financial derivatives aswell as forecasting market volatility. Besides the GARCH model of Bollerslev (1986) and itsvariants (Engle and (Bollerslev, 1986; Nelson, 1991)) which can capture the time-varyingvolatility feature, several long memory GARCH models were proposed to incorporate thelong memory volatility property in financial time series; see for example Baillie et al. (1996),Bollerslev and Mikkelsen (1996). It is of interest to see whether long memory can affectthe measurement of market risk in the context of VaR.

In this paper, we compare the performance of seven GARCH-type models in estimatingVaR of market indices. Two of them are long memory GARCH models. Both conditionalnormal and conditional t-error distributions are considered. While most empirical studiesfocused only on holding a long position of a portfolio, we also consider a short position. Therest of this paper is organized as follows. Section 2 describes the basic concept of VaR andpresents various GARCH-type models for financial return series. In Section 3, we discussmaximum likelihood method of estimating the parameters of long memory GARCH models.We apply the seven models to 12 market indices in Section 4 to assess the performance inestimating VaR at various confidence levels. Applications to exchange data are presentedin Section 5. Section 6 gives some concluding remarks.

2. Value at risk and GARCH-type models

2.1. VaR

Value at Risk, or VaR, is a commonly used statistic for measuring potential risk ofeconomic losses in financial markets. With VaR, financial institutions can have a sense onthe minimum amount that is expected to lose with a small probability α over a given timehorizon k (usually 1-day or 10 days). For example, a α = 5% 1-day VaR of $10 million

182 M.K.P. So, P.L.H. Yu / Int. Fin. Markets, Inst. and Money 16 (2006) 180–197

can tell us that one out of 20 days, we could expect to realize a loss of at least $10 million.Alternatively, we could say that the maximum loss we would expect on 19 out of 20 days is$10 million. In other words, VaR is defined as the maximum loss over a given time horizonat a given confidence level. Mathematically, let Pt be the price of a financial asset on day t.A k-day VaR on day t is defined by

P(Pt−k − Pt ≤ VaR(t, k, α)) = 1 − α.

Given a distribution of return, VaR can be determined and expressed in terms of a percentileof the return distribution (Dowd, 1998; Jorion, 2001). Specifically, if qα is the αth percentileof the continuously compounded return log(Pt) − log(Pt−k), then VaR can be written as

VaR(t, k, α) = (1 − eqα )Pt−k. (1)

The above expression implies that good VaR estimates can only be produced with accurateforecasts of the percentiles qα, which realizes on appropriate volatility modeling. To incor-porate the time varying feature of the market volatility, we adopt various heteroskedasticmodels.

2.2. GARCH-type models

We define the 1-day logarithmic return on day t as rt = log(Pt) − log(Pt−1) and denotethe information up to time t by �t . In this paper, we investigate the performance of thefollowing GARCH-type models in estimating VaR.

2.2.1. RiskMetrics modelThe RiskMetrics model assumes that returns are generated as follows

rt = εt, εt|�t−1 ∼ N(0, σ2t )

σ2t = λσ2

t−1 + (1 − λ)ε2t−1

where 0 ≤ λ ≤ 1 is the smoothing parameter. The formulation in the mean equation impliesthat the conditional distribution of returns is normal with mean zero. One main feature ofthe RiskMetrics model is that the conditional variance can be written as an exponentiallyweighted moving average (EWMA) of the past squared innovations or returns, that is,

σ2t = (1 − λ)(r2

t−1 + λr2t−2 + λ2r2

t−3 + . . .).

The smaller the smoothing parameter, the greater the weight is given to recent return data.RiskMetrics (1996) advised that we can use λ = 0.94 for daily data and λ = 0.97 formonthly data. It was also shown in the literature that λ = 0.94 produces very good forecastsfor 1-day volatility (RiskMetrics, 1996; Fleming et al., 2001). Under the RiskMetrics model,the 1-day VaR on day t in (1) is reduced to (1 − eqα )Pt−1 ≈ −σtzαPt−1, where zα is the100αth percentile of N(0, 1).


2.2.2. GARCH(p,q) modelThe GARCH(p, q) model (p > 0 and q ≥ 0 are integers) is defined as

rt = µ + εt, εt|�t−1 ∼ D(0, σ2t )

σ2t = ω + β(B)σ2

t + α(B)ε2t

(2)

where ω > 0, α(B) = α1B + . . . + αqBq and β(B) = β1B + . . . + βpBp, with αi ≥ 0 for

i = 1, . . . , q and βj ≥ 0 for j = 1, . . . , p, and D(0, σ2t ) represents a conditional distribution

with zero mean and variance σ2t . Bollerslev (1986) showed that the GARCH process of {rt}

is covariance stationary if and only if α(1) + β(1) < 1. Since the variance equation of theGARCH(p, q) process can be expressed as

σ2t = ω(1 − β(B))−1 + α(B)(1 − β(B))−1ε2

t ,

the above stationary condition implies that the effect of the past squared innovations onthe current conditional variance decays exponentially with the lag length. Note that theRiskMetrics model can be viewed as a special case of GARCH(1,1) model with µ = 0, ω =0, and λ = β1 = 1 − α1. Since GARCH(1,1) model was found to be adequate to manyfinancial time series (Bollerslev et al., 1992), we focused on this model in our empiricalanalysis.

2.2.3. IGARCH(p, q) modelNote that the variance equation of the GARCH model can be written as

(1 − α(B) − β(B))ε2t = ω + (1 − β(B))νt, νt = ε2

t − σ2t .

According to the empirical studies in Engle and Bollerslev (1986), Chou (1988), theestimated lag polynomial (1 − α(B) − β(B)) is found to have a significant unit root insome applications of GARCH models. Factoring this polynomial as (1 − α(B) − β(B)) =(1 − B)φ(B), where φ(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)ε2t = ω + (1 − β(B))νt, νt = ε2

t − σ2t , (3)

where φ(B) = 1 − φ1B − · · · − φqBq. As many empirical studies using GARCH(1,1) mod-

els give α1 + β1 very close to 1 implying high persistent volatility, the impact of past in-formation on future volatility forecasts decays very slowly. Therefore, we believe that theIGARCH(1,1) model given by

rt = µ + εt, σ2t = ω + β1σ

2t−1 + (1 − β1)ε2

t−1

is a good alternative to GARCH(1,1) model. When µ = 0, the IGARCH(1,1) model reducesto RiskMetrics model with λ = β1. From the good performance of RiskMetrics for someα such as 5% or 10% documented in the literature, it is anticipated that IGARCH(1,1) canalso be a good model for VaR estimation. In our empirical studies, we estimated β1 fromthe data rather than taking it to be 0.94 in RiskMetrics.


2.2.4. 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 financialmarket volatility, Baillie et al. (1996) extended the IGARCH model by replacing the firstdifference operator (1 − B) in (3) by the fractional differencing operator (1 − B)d with0 < d < 1 and developed the following FIGARCH(p, d, q) model:

φ(B)(1 − B)dε2t = ω + (1 − β(B))νt, νt = ε2

t − σ2t .

Clearly, the above FIGARCH model covers GARCH and IGARCH as special cases whend = 0 or 1. To better understand the properties of the models, we rewrite the varianceequation of the FIGARCH(p, d, q) model as

σ2t = ω(1 − β(B))−1 + (1 − β(B))−1(1 − φ(B)(1 − B)d)ε2

t , (4)

where (1 − B)d can be expressed by the Maclaurin series expansion

(1 − B)d =∞∑

k=0

Γ (k − d)

Γ (k + 1)Γ (−d)Bk

= 1 − dB + (1 − d)(−d)

2B2 + (2 − d)(1 − d)(−d)

3!B3 + . . . (5)

As Γ (k − d)/Γ (k + 1) ≈ k−d−1 if 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 thelag length. This makes a clear difference from GARCH and IGARCH models (d = 1) thatthe 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, FIGARCHmodels can be good compromise of IGARCH and GARCH in capturing volatility dynamicstructure.

Since in some stock markets the parameter µ is likely to be significantly positive, assum-ing µ = 0 in RiskMetrics may generate biased VaR estimates. Therefore, we allow µ in theunderlying GARCH, IGARCH and FIGARCH models and estimate it using returns data. Inthe empirical investigation, the mean specification in (2) was adopted for the three GARCHmodels. Two conditional distributions D(0, σ2

t ) for the error term εt were considered: (a) anormal distribution N(0, σ2

t ), and (b) a standardized t distribution with ν d.f. and varianceσ2

t (i.e., εt ∼ σtt(ν)/√

ν/(ν − 2)). It then follows that the 1-day VaR on day t in (1) is

(1 − eµ+cσt )Pt−1, (6)

where c = zα and tα(ν)/√

ν/(ν − 2) for normal and t-distributed error, respectively, andtα(ν) is the 100αth percentile of the standardized t distribution with degrees of freedom ν.Once estimates of µ and GARCH parameters are available, estimates of the above VaR arereadily obtained.


3. Estimation of GARCH-type models

The 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 beestimated 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. Givena time series of returns over a period of T days rt, t = 1, . . . , T , the maximum likelihoodestimate of µ under any GARCH-type model is the sample mean µ̂ = ∑T

t=1 rt/T . Withµ = µ̂, the log-likelihood function reduces to

�t = T lnΓ ((ν + 1)/2)√(ν − 2)πΓ (ν/2)

− 1

2

T∑t=1

[ln σ2

t + (ν + 1) ln

(1 + ε̂2

t

(ν − 2)σ2t

)]

where σ2t is given in (4) and ε̂t = rt − µ̂. Similarly, the log-likelihood function under normal

errors is given by

�n = −1

2T ln(2π) − 1

2

T∑t=1

[ln σ2

t + ε̂2t

σ2t

].

Calculating the log-likelihood function (�t or �n) is nontrivial in FIGARCH models becausethe log-likelihood involves σ2

t , 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, themaximum likelihood techniques for FIGARCH models require the truncation of the infinitesum at a certain lag. Since the fractional differencing operator d is designed to capture thelong memory feature of the process, truncating at too low lag may distort some importantlong-term dependencies. To mitigate these effects, the truncation lag was fixed at 1000. Inother words, we used the following approximation

(1 − B)d ≈1000∑k=0

Γ (k − d)

Γ (k + 1)Γ (−d)Bk

when 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 σ̂2

t by substituting the estimates into (4). Finally, putting µ̂ and σ̂2t into (6) gives an

estimate of 1-day VaR on day t.

4. Applications to stock market indices

4.1. Data description

In this section, we apply various GARCH models and RiskMetrics to stock market datafor VaR estimation. Twelve market indices namely, All Ordinaries Index (AOI) of Australia,FTSE100 of United Kingdom, Jakarta Composite (JSX) of Indonesia, Hang Seng Index


Table 1Summary statistics of stock market and exchange rate returns

Index Period n Mean Standarddeviation

Skewness Kurtosis

AOI 06/08/84–31/12/98 3640 0.0377 1.0221 −6.76 177.68FTSE100 14/02/84–31/12/98 3758 0.0466 0.9580 −1.20 17.67HSI 03/01/75–31/12/98 5926 0.0691 1.8375 −2.39 51.27JSX 03/01/85–31/12/98 3452 0.0513 1.6980 4.40 120.77KLSE 04/01/77–31/12/98 5414 0.0334 1.5966 −0.29 32.84KOSPI 05/01/77–31/12/98 6442 0.0274 1.3174 −0.35 10.15NASDAQ 12/10/84–31/12/98 3594 0.0610 0.9800 −1.50 17.61NIKKEI 05/01/84–31/12/98 3758 0.0088 1.3610 −0.11 11.66SET 02/05/75–31/12/98 5830 0.0218 1.4397 0.07 9.04SP500 04/01/50–31/12/98 12416 0.0346 0.8504 −1.81 48.67STII 03/01/80–31/12/98 4748 0.0245 1.3363 −1.65 36.06WEIGHT 06/01/75–31/12/98 6905 0.0509 1.6438 −0.26 2.64

Exchange rate Period n Mean Standarddeviation

Skewness Kurtosis

GBP/US 01/01/80–31/12/98 4773 0.0060 0.6647 0.06 2.94YEN/US 01/01/80–31/12/98 4773 −0.0158 0.6916 −0.54 3.91AUD/US 01/01/80–31/12/98 4773 0.0124 0.6143 1.89 32.12CAD/US 01/01/80–31/12/98 4773 0.0058 0.2748 0.10 3.58

(HSI) of Hong Kong, Kuala Lumpur Composite Price Index (KLSE) of Malaysia, KOSPIof South Korea, NASDAQ of US, Nikkei 225 Index (NIKKEI) of Japan, Stock Exchangeof Thailand Daily Index (SET) of Thailand, Standard & Poor 500 Index (SP500) of US,Straits Times Industrial Index (STII) of Singapore and Taiwan Stock Exchange WeightedStock Index (WEIGHT) of Taiwan were selected for illustration. The above list includesindices of global markets and major markets in Asia. To allow enough data for fittingthe FIGARCH models, the starting year of most data series ranges from 1975 to 1985so that we can have at least 3000 observations in the sample. The time span of the datasets is presented in Table 1. All data sets end at the last trading day of 1998. The wholedata range was divided into two parts; the estimation period and the validation period. Weestimated VaR in the validation period to assess the performance of different methods.For the sake of comparisons, we chose the validation period to be 1995–1998 for all datasets.

In this section, the return rt is expressed in percentages, i.e. rt = 100 × (log Pt −log Pt−1). Summary statistics of returns are given in Table 1. Highest returns are recordedin HSI and NASDAQ where investing in the two markets generates more than 15% annualreturn on the average in the time period of investigation. The lowest returns is recorded inNIKKEI which has about 2.2% average annual return from 1984 to 1998. According to thestandard deviations in the table, investing in SP500 and FTSE100 are the least risky. Allthe market returns show negative skewness except JSX and SET. Typical phenomenon ofexcess kurtosis is also revealed in the indices. To summarize the mean-variance relationshipof the 12 markets, we also plot the mean daily returns against daily standard deviation inFig. 1. The four non-Asian market indices, AOI, FTSE100, NASDAQ and SP500 form onecluster and all the eight Asian market indices form another cluster. It is clear that the two


Fig. 1. Sample mean and standard deviation of the market index returns.

clusters differ mainly in the standard deviation. Asian market returns generally are morevolatile than the non-Asian market returns. Typical pattern of ‘high risk high return’ forinvestment can also be seen in the Asian indices.

We observe that the sample autocorrelations of the absolute centered returns, |rt − r̄|,of the 12 market 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 lag200. Therefore, some long memory features are observed in the absolute centered returnsof 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 lag 50. To provide better insights about the time series properties of thestock market returns, we also examine the partial correlations (PACF). All PACFs arefound to decay quite quickly. It is not surprising even though we believe on the existenceof long memory in |rt − r̄| because PACFs of ordinary long memory ARIMA processesalso decay quickly; see for example (Beran, 1994, p. 66). Although empirical evidenceof long memory is observed in most of the indices, we need to confirm the existenceof long memory by the model selection using AIC and SBC. In the next sections, wealso study whether fractionally integrated GARCH models can produce superior VaR esti-mates.

4.2. GARCH fitting

In our empirical study, the whole time span was divided into two parts. Data in the firstpart were used for estimating unknown parameters in GARCH models. VaR estimates ofthe last four years were then computed based on the parameter estimates obtained in thefirst part of the data. These estimates were used to assess the out-sample performance ofvarious GARCH models in forecasting VaR.


The following four conditional variance specifications were adopted:

RiskMetrics : σ2t = λσ2

t−1 + (1 − λ)ε2t−1

GARCH(1, 1) : σ2t = ω + β1σ

2t−1 + α1ε

2t−1

IGARCH(1, 1) : σ2t = ω + β1σ

2t−1 + (1 − β1)ε2

t−1

FIGARCH(1, d, 0) : σ2t = ω + β1σ

2t−1 + (1 − β1B − (1 − B)d)ε2

t

Both standardized normal and t assumptions on εt 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 parameter λ wastaken as 0.94. Maximum likelihood estimation was performed for the above three GARCHmodels. Table 2 gives an extract of the parameter estimates. 1 A ‘(t)’ is added to indicate thata model with t errors was fitted. From the GARCH(1,1) model fitting, the typical finding thatφ1 = α1 + β1 is close to one is observed. Without restricting the fitted GARCH(1,1) modelto be stationary, we find fiveφ > 1. An increase inφ is also noted when a t error model is fittedinstead of normal error. In the IGARCH estimation results with normal errors, we observelarge β1 of 0.90 and 0.91 for SP500 and WEIGHT, respectively. Comparing with λ = 0.94in RiskMetrics, including the intercept ω in the IGARCH(1, 1) model substantially lowersthe parameter value associated with σ2

t−1, implying that the previous innovation ε2t−1 has

a greater impact on the current variance. For FIGARCH model fitting, all d which rangesfrom 0.09 to 0.69, 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 of fit among the six GARCH models, we also report the ranking based on AIC andSBC in Table 3. Better models according to AIC and SBC are ranked first. Mean ranks arecomputed to aggregate the information from the 12 indices. 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 best fit 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 results

In the validation exercise, VaR estimates for the period 1995 to 1998 were producedby using the maximum likelihood estimates from the first part of the data. Three commonvalues of α were chosen for illustration; they are 1%, 2.5% and 5%. In this empiricalstudy, we computed for each index the sample coverage α̂ which is the proportion of losses(Pt−1 − Pt) greater than the VaR estimates. Table 4 gives the sample coverages for differentα of the 12 indices. According to the definition in (1), we expect that α̂ is close to α fora good VaR estimation method. Therefore, the smaller the discrepancy between α̂ and α,the better performance is the estimation method. To assess the overall performance of theseven methods, we ranked the methods according to |α − α̂| for each index. A smaller rank

1 Complete tables of parameter estimates are available upon request from the authors.


Table 2An extract of parameter estimates of the 12 market returns

φ1 β1 d

GARCH GARCH(t) IGARCH IGARCH (t) FIGARCH FIGARCH (t)

AOI 0.8425 0.9099 0.6191 0.8651 0.4515 0.0898FTSE100 0.9255 0.9427 0.8554 0.9084 0.2762 0.1915HSI 0.9795 0.9817 0.7878 0.8390 0.6027 0.6215JSX 1.0386 3.3222 0.0398 0.7035 0.5023 0.6403KLSE 0.9573 0.9793 0.7497 0.7897 0.3615 0.4737KOSPI 1.0006 1.0077 0.7960 0.8134 0.4536 0.5152NASDAQ 0.9346 0.9470 0.7776 0.8243 0.2892 0.3230NIKKEI 0.9553 0.9898 0.6929 0.8121 0.3986 0.5043SET 1.0261 1.0604 0.8064 0.7728 0.4195 0.4609SP500 0.9851 0.9901 0.9032 0.9161 0.2674 0.2902STII 0.8797 0.9081 0.6371 0.7460 0.6925 0.3762WEIGHT 0.9926 0.9946 0.9136 0.9060 0.3616 0.4256

was assigned to a smaller |α − α̂|. Then, the average of all the ranks for each method werecalculated and displayed as mean rank in Table 4. A smaller mean rank indicates an overallbetter match between α and α̂ which is a sign of superior performance. Similarly, we studiedthe cases for holding a short position in the investment. In these cases, the VaR is defined by

P(Pt − Pt−k < VaR(t, k, α)) = 1 − α

which implies that

VaR(t, k, α) = (eq1−α − 1)Pt−k,

where q1−α = µ − cσt , with c = zα or tα(ν)/√

ν/(ν − 2), is the upper αth percentilesof returns. Sample coverages and mean ranks for short position are presented in

Table 3Model selection: ranking of AIC and SBC (in parantheses)

Index GARCH IGARCH FIGARCH GARCH (t) IGARCH(t) FIGARCH(t)

AOI 5 (5) 6 (6) 4 (4) 1 (1) 3 (3) 2 (2)FTSE100 4 (4) 6 (6) 5 (5) 1 (1) 3 (3) 2 (2)HSI 4 (5) 5 (4) 6 (6) 2 (3) 3 (1) 1 (2)JSX 5 (5) 6 (6) 4 (4) 1 (1) 3 (3) 2 (2)KLSE 5 (5) 6 (6) 4 (4) 2 (3) 3 (2) 1 (1)KOSPI 6 (6) 5 (5) 4 (4) 3 (3) 2 (2) 1 (1)NASDAQ 5 (5) 6 (6) 4 (4) 1 (1) 3 (3) 2 (2)NIKKEI 5 (5) 6 (6) 4 (4) 3 (3) 2 (2) 1 (1)SET 5 (5) 6 (6) 4 (4) 2 (2) 3 (3) 1 (1)SP500 5 (5) 6 (6) 4 (4) 1 (1) 2 (2) 3 (3)STII 4 (4) 6 (5) 5 (6) 1 (1) 3 (3) 2 (2)WEIGHT 5 (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)


Table 4Sample coverage for the long position of the 12 market indices

RM GARCH IGARCH FIGARCH GARCH (t) IGARCH (t) FIGARCH (t)

α = 1%AOI 1.88 1.18 1.09 1.58 0.89 0.99 1.19FTSE100 1.88 1.19 1.29 1.59 1.19 0.99 1.39HSI 2.02 2.23 2.23 2.13 1.82 1.62 1.42JSX 3.34 1.82 4.25 2.23 0.71 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 2.03 0.91 0.71 1.11SET 1.53 1.63 2.15 2.46 0.82 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 1.30 0.80 0.90WEIGHT 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 2.5

α = 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 4.3


Table 5. The best coverages with the smallest |α − α̂| in each index and the smallestmean ranks for each α are bold faced to highlight the best performance. We con-struct boxplots in Fig. 2 for describing the distribution of the sample coverage for allmodels.


Table 5Sample coverage for the short position of the 12 market indices



α = 2.5%AOI 2.67 1.68 1.97 1.58 1.97 0.99 2.08FTSE100 2.48 1.58 1.09 1.68 1.48 0.99 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 4.3


For the long position with α = 1%, RiskMetrics, GARCH, IGARCH and FIGARCHperform poorly that all sample coverages are greater than 1%, indicating that biased VaRestimates are generated by the three methods. The biasedness is also observed from Fig. 2that all the four boxplots for the four models lie above the reference line of 1%. In terms


Fig. 2. Boxplots showing the distribution of the sample coverage α̂ in various scenarios. Dotted reference lineslocate the position of α. The top, medium and bottom panels correspond to the 1st, 2.5th and 5th percentiles,respectively. The left and right columns refer to the long and short positions.

of the mean ranks, IGARCH(t), GARCH(t) and FIGARCH(t) show superior results in thisparticular scenario. For α = 2.5%, while RiskMetrics suffers from the same problem ofgenerating coverages greater than 2.5%, IGARCH and IGARCH(t) have mean ranks sub-stantially smaller than the others. In the relatively less extreme case of α = 5%, RiskMetricsperforms the best. We can also see from the boxplots in Fig. 2 that the sample coverages ofRiskMetrics have the smallest dispersion, probably indicating that this method is robust tothe characteristics of different financial markets.

In the short position with α = 1%, we do not observe the same problem with the normalerror models that all sample coverages are greater than 1% as in the long position. Con-versely, GARCH performs exceptionally well, and then followed by FIGARCH(t). This


asymmetry between the long and short positions may be due to skewed distribution of re-turns and/or the well-known volatility asymmetry in response to good and bad news. Fromthe table, we can see that the t-error models produce sample coverages substantially smallerthan 1%. Similarly, the overperformance of GARCH and GARCH(t) can be identified inα = 2.5%. In addition, RiskMetrics and GARCH(t) give the least mean rank in α = 5%. Inthis case, both IGARCH and IGARCH(t) are not good in estimating VaR, having most sam-ple coverages greater than 5%. As in the long position, the sample coverages of RiskMetricsappear to be less volatile than other methods.

To check the performance of the seven models in each stock market, we present theboxplots showing the distribution of sample coverage in Fig. 3. Among the 12 indices, AOIand FTSE100 have the least variation, and JSX, NASDAQ and NIKKEI show the largest

Fig. 3. Boxplots showing the distribution of the sample coverage α̂ for the 12 market indices. Dotted referencelines locate the position of α. The top, medium and bottom panels correspond to the 1st, 2.5th and 5th percentiles,respectively.


dispersion. The above tells us that VaR estimation results of the latter three index returns arelargely dependent on the models adopted. It is interesting to note that excluding HSI, JSXis the most volatile market and the lowest and highest returns are recorded, respectively inNIKKEI and NASDAQ.

5. Applications to exchange rate data

As the trading of currency constitutes an important part of the financial investment, weare also interested to see the performance of the seven models in estimating VaR. It appearsin the literature some empirical results on exchange rate volatility forecasting. For example,Vilauso (2002) argued that FIGARCH outperforms GARCH and IGARCH in terms ofout-sample mean squared error and mean absolute error. Balaban (2004) concluded thatthe standard GARCH(1, 1) provides relatively better forecasts of monthly exchange ratevolatility. We collected Pound/Dollar (GBP/US), Yen/Dollar (YEN/US), Australian/Dollar(AUD/US) and Canadian/Dollar (CAN/US) from 1980 to 1998. As in the stock marketanalysis, we reserve the last four year data for validating the VaR estimation methods.In Table 1, we observe close-to-zero sample mean returns and uniformly smaller standarddeviations than that of the market index returns. Furthermore, the skewness are close to zeroand the kurtosis are generally not as large as that of the market index returns. Concerningthe autocorrelation and partial correlation patterns, they behave similarly as those in marketindex returns that the former exhibits very slow decay whereas the latter decays quitequickly. Apparently, time series properties of the exchange rate returns are coherent to themarket index returns though the two type of returns have different moment structures.

Table 6Sample coverage for the long position of the four exchange rates


α = 1%GBP/US 1.89 1.29 1.49 1.29 0.89 0.89 0.90YEN/US 2.68 2.29 1.89 2.49 1.49 1.19 1.59AUD/US 1.89 1.69 1.49 1.39 1.19 1.39 1.59CAD/US 2.09 1.79 1.89 1.99 1.59 1.59 1.49Mean rank 7.0 4.9 4.8 4.8 2.0 2.1 2.5

α = 2.5%GBP/US 3.48 2.29 2.39 2.79 2.29 1.99 2.39YEN/US 4.37 3.68 3.58 3.88 3.78 3.58 3.78AUD/US 3.48 2.19 2.09 1.99 2.68 2.78 2.99CAD/US 3.28 3.28 2.98 3.39 2.78 2.68 3.18Mean rank 6.6 3.8 2.5 6.0 2.8 2.6 3.8



Table 7Sample coverage for the short position of the four exchange rates



α = 2.5%GBP/US 3.28 2.78 2.49 2.29 2.49 2.49 2.19YEN/US 3.18 2.98 2.39 3.09 2.49 2.19 2.39AUD/US 3.28 2.88 3.38 2.89 3.38 3.38 3.58CAD/US 3.28 3.58 3.18 4.08 3.28 2.88 3.68Mean rank 5.1 4.0 2.9 4.8 2.9 3.0 5.4


We fitted the seven models to the exchange rate returns and computed estimates of VaRin 1995–1998. Comparing the summary results in Tables 6 and 7, we find very consistentperformance results in long and short positions. For α = 1%, all the t-error models out-perform the normal-error models. GARCH(t) and IGARCH(t) are the best two models inestimating VaR for both positions. IGARCH appears to be the best choice for α = 2.5%though GARCH(t) performs equally well in terms of the mean rank in short position. Inthe cases of the above two α values, RiskMetrics performs the worst in all cases except inthe short position with α = 2.5% where FIGARCH(t) shows even higher mean rank. Thissuggests that RiskMetrics should not be recommended to estimate small percentage VaRfor exchange rates. On the other hand, for α = 5%, RiskMetrics is recorded as the best andthe second best in long and short positions, respectively. Fig. 4 shows the distribution of

Fig. 4. Boxplots showing the distribution of the sample coverage α̂ for the four exchange rates. Dotted referencelines locate the position of α. The top, medium and bottom panels correspond to the 1st, 2.5th and 5th percentiles,respectively.


Table 8Best two models based on mean ranks

α Market indices Exchange rates

Long position Short position Long position Short position

1% IGARCH(t) GARCH GARCH(t) GARCH(t)GARCH(t) FIGARCH(t) IGARCH(t) IGARCH(t)

2.5% IGARCH GARCH IGARCH IGARCHIGARCH(t) GARCH(t) IGARCH(t) GARCH(t)

5% RM RM RM IGARCHGARCH GARCH(t) FIGARCH RM

the sample coverage for the four exchange rates. Comparing this distribution with that forthe 12 market indices in Fig. 3, the sample coverage is less dispersed in the exchange rates.This indicates the VaR estimation for exchange rates is less dependent on the methods usedthan stock market index data. It is also noteworthy that unlike models for stock marketindices, models with fat-tailed errors for exchange rates demonstrate superior performancefor estimating extreme VaR, such as α = 1%, in both long and short positions.

6. Concluding remarks

In this paper, we explore the performance of seven methods using various GARCHmodels on VaR estimation of market returns and exchange rate returns. We focus on threeextreme percentiles α = 1%, 2.5% and 5% in the empirical study. Referring to the best twomodels derived based on mean ranks in Table 8, we observe that t-error models give better1% VaR estimates than normal-error models in long position, but not in short position, ofinvesting in stock markets. While this striking result of asymmetric behavior in long andshort positions should be highlighted for market indices, we do not observe any asymmetricproperty in exchange rate data where t-error models beat normal-error models in bothpositions, particularly for α = 1%. From Table 8, IGARCH seems to work best for α =2.5%. While RiskMetrics performs very well in comparison to the other models at recoveringthe lower and upper 5th percentiles, it predominantly underestimates 1% and 2.5% VaR, thuscausing most of the sample coverages greater than the respective nominal values. Amongthe seven models we have considered, RiskMetrics seems to be more robust of having lessvariation in the sample coverage. We also discovered that VaR estimation for exchange ratesis less relied on the volatility models than stock market data.

FIGARCH(t) cannot outperform GARCH(t) though their autocorrelation plots displaysome indication of long memory volatility. This probably tells us that the long memoryvolatility feature is not very crucial in determining a proper value of VaR. The best fittedmodel according to AIC and SBC does not necessarily lead to better VaR estimates.

Although we only look into market indices and exchange rates, methods found to havesuperior performance in this paper may also be applied to other assets such as bonds.For example, Reilly et al. (2000), Jones and Wilson (2004) pointed out the increasingcorrelation between bond returns and stock returns. The results of Werner and Upper(2004) demonstrate the fat-tailed behavior of bond futures returns. The above literature


makes us query that the same methods can work well in both stock and bond returns. Thisis a direction for future research.

Acknowledgements

The authors thank Professor Ike Mathur and an annoymous referee for their valuablesuggestions and comments.

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