VaR Estimation with Power EWMA Model ―

Conservativeness, Accuracy and Efficiency

Mei-Ying Liu*

Chi-Yeh Wu**

Hsien-Feng Lee***

EMF code: 450, 570

*Correspondent and attending author, Associate Professor, Department of Business Administration, Soochow University Address：156, Sec.1, Kwei-Yang St., Taipei 100, Taiwan Tel: 886-2-2311-1531 ext. 3602 Fax: 886-2-2382-2326 E-mail: meiying@scu.edu.tw ** Manager, Information Department, WK Technology Fund E-mail:kwu@wktech.com.tw *** Associate Professor, Department of Economics, National Taiwan University E-mail:leehsf@ntu.edu.tw

2

ABSTRACT

Financial asset returns are well-known to be non-normal and leptokurtic with the tails

fatter than normal distribution. The Standard EWMA estimator with the normality

assumption (used in JP Morgan's RiskMetrics® model) will be inefficient and lead to

understate the true value of risk if the asset returns are fat-tailed distributed. On the

basis of the power exponential distribution (also known as the generalized error

distribution) the family EWMA estimators, nesting Power EWMA, Standard EWMA

and Robust EWMA, are proposed by Guermat & Harris (2002). Using these newly

developed estimators, we first forecast the VaR of daily returns for TAIEX, FTSE 100,

and DJIA. Subsequently, the back-testing is performed to evaluate the VaR models.

Performance assessment is based on a range of measures that address the

conservativeness, accuracy and efficiency of each model. The results demonstrate that

the members of the family of EWMA estimators based on power exponential

distribution rather than normal distribution offer a superior coverage for the extreme

risk over the RiskMetrics®estimator, and show that Power EWMA performs excellent

accuracy in VaR estimation.

Key Words: VaR, Power EWMA, Standard EWMA, Robust EWMA,

Power Exponential Distribution, Fat-tailed

3

1. Introduction

Value-at-Risk (VaR) has emerged as the widely used tool for risk management of financial institutions. The successful implementation of VaR depends heavily on the accurate estimation of the conditional distribution of portfolio returns. Owing to the simple and rapid computations, the exponentially weighted moving average of past squared returns, or EWMA estimator, become the most common approach to forecast the conditional volatility of asset returns (JP Morgan, 1994; Dowd, 1998; Jorion, 2000). It has proved to be very effective at forecasting the volatility of returns over short horizons, and often outperforms the forecasts of more sophisticated models such as generalized autoregressive conditional heteroscedasticity (GARCH) when the underlying asset returns is normally distributed. (See Boudoukh, Richardson and Whitelaw, 1997; Alexander and Leigh, 1997)

However, a number of empirical studies show that asset returns are not normally distributed. In particular, the conditional distribution of short horizon asset returns is stylized found to be leptokurtic, with tails that are significantly fatter than those of the normal distribution (See Mandelbort, 1963；Fama, 1965； Baillie and DeGennaro, 1990; Jansen and de Vries, 1991; Bollerslev, Chou and Kroner, 1992; Koedijk and Kool, 1994; Loretan and Phillips 1994; Kearns and Pagan 1997). The Standard EWMA estimator with the normality assumption (used in JP Morgan's RiskMetrics® model) will be inefficient and lead to understate the true value of risk if the asset returns are fat-tail distributed.

To remedy this problem, two main directions are proceeded to characterize the tail behavior. The first is to set up the unconditional distribution as the mixture normal distribution such as normal-Poisson (Jorion, 1988), normal-lognormal (Hsieh, 1989) and Bernoulli-normal (Vlaar and Palm, 1993), still preserving the assumption of homoskedasticity. That is, the volatility of asset returns is time-independent. The second is to employ the non-normal distribution, for instance, Student-t distribution (Bollerslev, 1987 ; Baillie and Bollerslev, 1989; Kaiser, 1996; Beine, Laurent and Lecourt 2000), Laplace and double exponential distribution (Linden 2001) and exponential power distribution (Varma, 1999; Guermat & Harris, 2002) to substitute for normal distribution.

Lots of literatures have been addressed the weaknesses and strengths of various VaR models. However, no single consistent measure of VaR model performance has been developed. Of concern to supervisors, is whether the required minimum regulatory-capital calculated by the internal model of the bank can offer an appropriate coverage for its losses. Alternatively, it is the conservativeness of the model. We identify the relative conservative models as those that systematically

4

produce higher estimate of risk relative to other models. With respective to accuracy the risk manager should be concerned with whether the ex-post performance is compatible with the theoretical desired level. The regulatory capital-adequacy framework provides the incentive to develop efficient models, that is, models offer the enough coverage for the risk to meet the supervisors’ requirement with the minimum capital that must be held.

In this paper, therefore, we employ the family of nested power EWMA estimators based on exponential power distribution that are robust to fat-tailedness and leptokurtosis in the conditional distribution of returns to forecast the VaR. The data consists of daily aggregate equity portfolios returns for the TAIEX, FTSE 100, and DJIA. Next, we focus on three aspects- conservativeness, accuracy and efficiency of model and propose a range of statistics based on these criteria to compare the performance of the family models.

The paper is organized as follows. The following section introduces the methodology, including the data description, the illustration of the nested power EWMA family, the tail-index estimation and the measures of model evaluation while Section 3 presents the results of the empirical evaluation. Some concluding remarks are offered in Section 4.

2. Methodology 2.1 Data The empirical evaluation uses aggregate daily equity returns for the US, UK and Taiwan. The raw data used are daily price observations for the DJIA, FTSE100 and TAIEX indices obtained from Dow Jones Indexes' Web Site, Datastream and Taiwan Economic Journal (TEJ) databank respectively, for the period 01/01/84 to 31/12/02. Continuously compounded returns were then calculated as the first difference of the

natural logarithm of each series, 1lnln −−= ttt IIr , where It is the price index value

for date t. Moreover, to investigate the diversifiable effects among different equity returns, we compose the equally weighted portfolio of TAIEX, FTSE 100 and DJIA.

2.2 Tail Index Estimation

Financial asset returns are well-known to exhibit fatter tail than normal distribution.

To account for tail behavior, we directly estimate the tail index. The tail index is a

measure of the degree of tail fatness of underlying distribution and estimates with

extreme value theory (EVT) which addresses the characteristic of tail behavior of

distribution. The most famous and often applied estimator for tail index, due to its

5

easy implementation and asymptotic unbiasedness is proposed by Hill(1975) as

following:

( ) ( )mn

m

1i1in xlnxln

m1)m( −

=+− −

= ∑ξ ， 2m ≥ (1)

=

−

+−

=∑ )

xx

(lnm1

mn

1inm

1i

where m is the pre-specified number of tail observations to be included. The

selection of m is crucial to obtain unbiased estimator of the tail index. n is the sample

size. ix is the ith increasing order statistic ( ni ....2,1= .), while the tail index α is equal

to ξ1 . Equation (1) means that Hill estimator measures the average of the ratios of

the each observed value relative to the threshold value in the predetermined tail area.

The larger the average, the smaller the tail index and the greater magnitude of the

fat-tailedness is.

However, there is considerable empirical evidence shows that it is biased in relatively small samples and limited to the cases in which a larger sample is available. To improve the Hill estimator, a recently developed alternative approach proposed by Huisman et al. (2001) is especially useful for small samples. Their regression-based approach is based on an approximation of the asymptotic expected value of m

cmmE −≈α

ξ 1))(( (2)

where c is a constant depending on parameters of the distribution and the sample size. If m becomes small, the bias goes down and the expectation goes to the true value

αξ 1

= . The variance of the estimator increases with small m.

2

1))((α

ξm

mVar ≈ (3)

The idea of Huisman et al. (2001) is to use equation (2) in a regression analysis and regress the values )(mξ (computed with conventional Hill estimator) against m as follows:

)()( 10 mmm εββξ ++= , m = 1…κ (4)

6

The estimated ∧

0β is an estimator ofα

ξ 1= . The authors propose to choose threshold

value κ to equal half the sample (2n ). Although the parameters in equation (4) can be

estimated by ordinary least squares (OLS). Equation (3) indicates that the variance of Hill estimates is not constant for different m. The error term )(mε is heteroscedastic. Accordingly, they propose a weighted least squares (WLS) approach to correct the form of heteroscedasticity and improve the efficiency of the estimator. We apply the modified Hill estimator to obtain the tail index estimates by use of both OLS and WLS.

2.3 The Power EWMA Variance Estimator We introduce a general power EWMA estimator proposed by Guermat and Harris (2002), nesting EWMA models that are more robust to the leptokurtosis of returns, and which would therefore be expected to be more efficient when the conditional distribution of returns is fat tailed. The power EWMA estimator is based on the maximum likelihood estimator of the variance of the power exponential distribution (also known as the generalized error distribution, or Box-Tiao distribution). The probability density function of the power exponential distribution is given by

δ

ϕ

δδ

δϕ

δδ σr

21

1

e)σΓ(1/2

)σ,f(r,−

+

= (5)

where

21

2

)/3()/1(2

ΓΓ

=−

δδϕ

δ

(6)

and ( )•Γ is the gamma function. The power exponential distribution has variance equal to 2σ , zero skewness and a kurtosis coefficient that depends on the value of power parameterδ . When δ = 2, the power exponential distribution reduces to the normal distribution. Whenδ > 2, the power exponential distribution is thin tailed and platykurtic, and whenδ < 2, the power exponential distribution is fat-tailed and leptokurtic. When δ = 1, the power exponential distribution reduces to the Laplace distribution. The power exponential distribution with different power parameterδ is shown in Figure 1. From Figure 1 we can find that the larger the power parameterδ , the more fat-tailed and leptokurtic the distribution is.

7

Conditional on the power parameterδ , the maximum likelihood estimator of the

standard deviation of the power exponential distribution is given by

δδ δσ t

T

tr

Tg

1

1)(ˆ=Σ= (7)

where

2

)/1()/3()(

δ

δδδδ

ΓΓ

=g (8)

Equation (7) is the unconditional variance estimator independent of past information.

Guermat and Harris (2002) transformed it into the conditional variance estimator and

replaced the unweighted average in (7) by an exponentially weighted average to yield

the power EWMA estimator

k

iti

i

kt rkg −

∞

=+ ∑−= λλσ

01 )()1( (9)

By recursive substitution, the Power EWMA estimator can be rewritten as

k

tkt

kt rkg )()1(1 λλσσ −+=+ (10)

and so the Power EWMA estimator can be seen as an infinite weighted average of past squared returns, incorporating information from all past shocks to the power parameter k of returns, but with exponentially declining weights. Alternatively, by

Figure 1 Power Exponential Distribution with Different Power Parameter δ

8

using the fact that 21

21

21 +++ += tttr εσ where 2

1+tε is a zero mean random shock that is

orthogonal to the time t information set, the power EWMA estimator can also be interpreted as an infinite order autoregressive model for the kth powered return. When k = 2, the power EWMA estimator coincides with the standard EWMA estimator given by

2221 )1( ttt rλλσσ −+=+ (11)1

The standard EWMA estimator is a special case of the generalized autoregressive conditional heteroscedasticity, or GARCH model (Engle, 1982; Bollerslev, 1986). The GARCH(1,1) model for the conditional variance of returns is given by

2

12

102

1 ttt rβσαασ ++=+ (12)

where 0α , 1α and 1β are parameters to be estimated. When 0α =0 and 1β = 1− 1α ,

the GARCH model reduces to the standard EWMA estimator, and is alternatively known as Integrated GARCH or IGARCH. For k < 2, the power EWMA estimator is less sensitive to extreme observations and thus we may expect it to be more efficient when the conditional distribution of returns is leptokurtic. The role of the function g (k) is to preserve the integrated nature of the volatility, in keeping with the standard EWMA model.

When k = 1, the power exponential reduces to the Laplace distribution, and the power EWMA estimator reduces to

tt

iti

0i1t

r2)1(

r2)1(

λλσ

λλσ

−+=

∑−= −

∞

=+

(13) 2

The Laplace distribution is commonly used in the context of robust estimation, and so the EWMA estimator given by (13) might therefore be thought of as a ‘robust’

1 when 2k = , 1)2/1(

)2/1(*21

*2)

21(

)121(

*2)2/1()2/3(*2)(

22

=

Γ

Γ=

Γ

+Γ=

ΓΓ

=kg

2 when 1k = , 22)1()2(2

)1()12(

)1/1()1/3(*1)k(g 2

121

21

21

==

=

+=

=

ΓΓ

ΓΓ

ΓΓ

9

EWMA estimator. The power EWMA estimator therefore nests the standard EWMA estimator, the robust EWMA estimator, and a continuum of estimators that lie between the two, as well as estimators that are even more sensitive to outlying observations than the standard EWMA estimator, and those that are even less sensitive to them than the robust EWMA estimator. The power EWMA estimator described above is a special case of the NGARCH model of Higgins and Bera (1992), given by

kt

kt

kt r1101 βσαασ ++=+ (14)

When 00 =α and ( ) ( )kg11 1 αβ −= , the NGARCH model reduces to the power EWMA estimator. As with the standard GARCH model, the parameters of the NGARCH model can be estimated by maximum likelihood. In contrast, members of the family of power EWMA estimators have only a single parameter – the decay factor – and consequently their implementation is very much more straightforward than that of the more sophisticated NGARCH model. The relationship among these nested models can be summarized as Figure 2. The parameters of the models for the four equity returns are firstly estimated by maximum likelihood based on power exponential distribution, using BHHH algorithm (Berndt et al, 1974) with a convergence criterion of 0.00001 applied to the function value. Next, we apply the likelihood ratio statistics3 to test the restrictions on the NGARCH model that are implied by the power EWMA estimator, and introduce the standard EWMA based on normal distribution (well known as the RiskMetrics) as the benchmark to compare the out-of-sample performance of various power EWMA estimators.

3Under regularity, the large sample of likelihood ratio test statistic λχ ln22 −= is chi-squared, with

degrees of freedom equal to the number of restrictions imposed, where UR L/L=λ ， UL denotes

the likelihood function evaluated without regard to constraints and RL is the constrained likelihood function estimate. Since both likelihoods are positive, and a restricted optimum is never superior to an unrestricted one. Such that0＜λ ＜1. If λ is too small, then the doubt is cast on the restriction, otherwise when λ is close to 1, the null hypothesis is accepted.（see Greene (2000).

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GARCH2

t12t10

21t rβσαασ ++=+

GARCH2

t12t10

21t rβσαασ ++=+

Standard EWMA2

t2t

21t r)1( λλσσ −+=+

Standard EWMA2

t2t

21t r)1( λλσσ −+=+

Power EWMAk

tkt

k1t r)k(g)1( λλσσ −+=+

Power EWMAk

tkt

k1t r)k(g)1( λλσσ −+=+

Robust EWMAtt1t r2)1( λλσσ −+=+

Robust EWMAtt1t r2)1( λλσσ −+=+

NGARCHk

t1kt10

k1t rβσαασ ++=+

NGARCHk

t1kt10

k1t rβσαασ ++=+

When ( ) ( )kg1 0α 110 αβ −== 、

When k = 1

When k = 2

When 110 1 0、α αβ −==

2.4 The Estimation of VaR We compute out-of-sample one-day VaR forecasts for the four portfolios, using each of the EWMA estimators. A rolling window is used for the estimation of each model. Each estimated model is then used to forecast the VaR of the portfolios with window length of 10, 50, 100, 250, 500, 10004 observations respectively. Moreover, VaR is computed for the 99% and 95% confidence levels. The VaR of each portfolio in each period t is forecast by the formula

1t1t )(VaR ++ −= σαδ (15)

where 1+tσ is the standard deviation of the portfolio’s return, r 1+t , conditional on the

time t information set, andδ (α ) is theα - quantile of the standardized (i.e. zero mean, unit variance) empirical power exponential distribution and α is one minus the VaR confidence level. The standardized empirical distribution is defined as the return series over the window, scaled by the estimated standard deviation for each of those days. The standard deviation estimate used to standardize the return is obtained from 4 According to the guideline of BIS the window lengths must be at least 250 business days. Our results

of VaR estimated show that when the window lengths greater than 100 days, the discrepancy among estimated VaRs of different window lengths are such small that can be neglected. Consequently, window length of 250 observations is enough. It is not necessary to extend window length longer.

Figure 2 The Nested Power EWMA Models

11

the EWMA model (see Hull and White, 1998). 2.5 Model Evaluation The evaluation of VaR forecasts is not straightforward. As with the evaluation of volatility forecasting models, a direct comparison between the forecast VaR and the actual VaR can not be made, since the latter is unobservable. A variety of evaluation methods have been proposed (see, for instance, Kupiec, 1995; Christofferson, 1998; Lopez; 1999). Up to now, no single definitive of VaR model performance has been developed. To evaluate the performance of the family models we propose a range of statistics that address different aspect of the usefulness of VaR models to risk manager and supervisory authorities. We focus on three aspects of model: conservativeness, accuracy and efficiency (Engel and Gizycki, 1999).

2.5.1 Conservativeness

Mean Relative Bias (MRB) Engel and Gizycki (1999) defined the conservativeness of models in terms of the relative size of VaR for the risk assessment. The larger the VaR value is, the more conservative the model. To measure relative size of VaR among different models, they apply the mean relative bias developed by Hendricks (1996). The mean relative bias statistic captures the degree of the average bias of the VaR of specific model from the all-model average. Given T time period and N VaR models, the MRB of model i can be calculated as:

∑=

−=

T

1t t

titi VaR

VaRVaRT1MRB (16)

where, ∑=

=N

1iitt VaR

N1VaR

2.5.2 Accuracy

Different user of the VaR model will focus on different types of inaccuracies. It may be expected that supervisors will pay more attention to the underestimate of losses while the financial institutions will be concerned more about over-predictions of losses due to the capital adequacy requirement. In this study, we define the accuracy as what extent is the rate of failure (or exception) of specific model close to the preset significant level. The three accuracy measures：binary loss function, LR test of unconditional coverage (Kupiec ,1995) and the scaling multiple to obtain coverage are presented as follows. a. Binary Loss Function (BLF)

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The binary loss function is based on whether the actual loss is larger or smaller than the VaR estimate. Here we are simply concerned with the number of the failure rather than the magnitude of the exception. If the actual loss is larger than the VaR then it is termed as an“exception”(or failure) and has the equal value of 1 , all others are 0. That is

≥∆

<∆=

+

++

titi

tititi

VaRPif

VaRPifL

,1,

,1,1,

0

1 (17)

Aggregate the number of failure across the all dates and divide it by the sample size. The BLF is obtained as the rate of failure. The closer the BLF value is to the confidence level of the model, the more accuracy the model. b. LR Test of Unconditional Coverage (LRuc) The BLF provides a point estimate of the probability of failure. In other words, the accuracy of the VaR model requires that the BLF on average be equal to one minus the prescribed confidential level of VaR model. The model should provide the correct unconditional coverage of loss. Kupiec (1995) proposed a likelihood ratio test based on the binomial process which can be applied to determine if the rate of failure is statistically compatible with the expected level of confidence. Given the sample size T and the frequency of failure N governed by a binomial probability, the likelihood ratio statistic of the unconditional coverage hypothesis α=p:H0 can be stated as

[ ] 2,1~1ln2)1(ln2 αχ

−+−−=

−−

NNTNNT

uc TN

TNppLR (18)

Under the null correct hypothesis of correct unconditional coverage, the LRuc has a chi-squared distribution with one degree of freedom. c. Multiple to Obtain Coverage (MOC) To highlight the magnitude of the deviation that the losses from VaR estimate, we compare the multiple to obtain coverage proposed by Engel and Gizycki (1999). The multiple equivalent, X i , of risk measure for model i is calculate so that

αii TF = ，

≥∆

<∆Σ=

+

+

=tiiti

tiitiT

tiVaRXPif

VaRXPifF

i

,1,

,1,

1 0

1 (19)

where the F i is equivalent to the total number of failures. T i is the sample size and α

is the significant level of the model. 1, +∆ tiP denotes the realized profit or loss on t +1

day .

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2.5.3 Efficiency

Mean Relative Scaled Bias (MRSB)

Efficiency is important since VaR measures are used by supervisor and the internal management of financial institutions to influence investors’ incentives. A more efficient VaR model provides more precise resource allocation signals to the financial institutions. Hence we address the aspect of efficiency on the capacity for a model to provide adequate risk coverage with minimum capital. The Mean Relative Scaled Bias (MRSB) (Engel and Gizycki (1999)) is aimed to evaluate which model, once suitably obtain the desired risk coverage level, produces the smallest VaR measure. There are two steps in calculating the MRSB measure. First, the scaling should be calculated by multiplying the VaR for each model by the multiple needed to obtain the 95% or 99% coverage as described in MOC measure. Subsequently, we compare the scaled VaR measures with the average relative size to all-model average. The MRSB measure is given as following.

∑= ⋅

⋅−⋅=

T

t t

ttiii VaRX

VaRXVaRXT

MRSB1

,1 (20)

where, ∑=

⋅=⋅N

itiit VaRX

NVaRX

1,

1

3. Empirical Results 3.1 Descriptive Statistics

The preliminary statistics for all four equity returns are summarized in Table1. As is commonly found daily financial asset returns are not normally distributed. In all cases the Jarque-Bera test for normality are highly significant with excess kurtosis5 and negatively skewed. Moreover, DJIA exhibits the most leptokurtic distributed with the kurtosis of 66.52. The variance of TAIEX is the largest of all whereas that of portfolio is the smallest due to the diversification effects.

Table1 Descriptive Statistics

5 Excess kurtosis means the distribution has kurtosis greater than that of normal distribution whose kurtosis equal to 3.

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TAIEX FTSE 100 DJIA Portfolio Mean 0.00034 0.00030 0.00041 0.00031 Standard deviation 0.01841 0.01060 0.01116 0.00836 Maximum 6.58% 7.60% 9.67% 5.04% Minimum -7.05% -13.03% -25.63% -11.32%

Skewness -0.21543 (0.00000)

-0.813053 (0.00000)

-2.75528 (0.00000)

-0.737642 (0.00000)

Kurtosis 4.87460

(0.00000) 13.52859 (0.00000)

66.52633 (0.00000)

11.43150 (0.00000)

Jarque-Bera6 815.3398 (0.00000)

22547.77 (0.00000)

805907.4 (0.00000)

17156.62 (0.00000)

Sample size 5289 4768 4757 5620 Note: P-values are reported in parentheses for the skewness, kurtosis and the

Jarque-Bera statistic. .

3.2 Estimation of Tail Index The most important information in terms of characterizing the limiting extreme

distribution of the tail is the tail index. The index of right-tail, left-tail and both

tails7 for the four equity return distributions are estimated by OLS and WLS.

The modified Hill estimate results are shown in Table 2. From Table 2 we can

find that the estimates of normal distribution, as a benchmark, are all around 8.5.

All tail index estimates, varying between 3.24 and 5.88, are greater than 8.5

indicates that all the equity return distributions exhibit fatter tails than the normal

distribution, as commonly found in literatures. Further, the DJIA has the highest

degree of fat-tailedness, also consistent with the findings of preliminary statistics.

If these estimates differ significantly over both tails, it is inappropriate to use the

estimate obtained from the combined information in both. However, our results

show that the left-tail estimates display little fatter tail than the right-tail do for

the majority of equity return.

Table2 Tail- Index Estimates

6 The Jarque-Bera statistic has a chi-squared distribution with two degrees of freedom under the null

hypothesis of normality. 7 All observations are taken in excess of their sample mean. The left tail is examined by using the

absolute value of all negative returns, the right tail is examined by using the absolute value of all positive returns and to both tails simultaneously we use the absolute values of all returns.

15

TAIEX FTSE

100 DJIA Portfolio Normal

Both tailOLS 5.56616

(0.00000) 3.59839

(0.00000)3.59066

(0.00000)4.50852

(0.00000)8.41893

(0.00000)

Both tailWLS4.93622

(0.00000) 3.8069

(0.00000)3.87973

(0.00000)4.68356

(0.00000)─

Sample size 2,644 2,383 2,378 2,808 24,999

Left tailOLS 5.46890

(0.00000) 3.38424

(0.00000)3.24098

(0.00000)4.33279

(0.00000)8.49762

(0.00000)

Left tailWLS 4.80790

(0.00000) 3.63574

(0.00000)3.52230

(0.00000)4.62216

(0.00000)─

Sample size 1,310 1,155 1,171 1,380 12,527

Right tailOLS 5.87510

(0.00000) 3.87090

(0.00000)4.00489

(0.00000)4.93776

(0.00000)8.74279

(0.00000)

Right tailWLS 5.19921

(0.00000) 3.98977

(0.00000)4.22635

(0.00000)4.95584

(0.00000)─

Sample size 1,333 1,228 1,206 1,428 12,472 Note: P-values are reported in parentheses for the estimate of the tail

indexξ

α 1=

3.2 Estimation of Power EWMA

The NGARCH model, and the power EWMA estimator that it nests, are based on the maximum likelihood estimators of the variance of the power exponential distribution. Table 3 reports the parameter estimates of each model for each of the three return series. The first column of each panel in Table 3 gives the results for the unrestricted NGARCH model. The sum of the parameters 1α and 2α is 0.99168 for the TAIEX, 0.97781 for the FTSE 100, 0.99096 for the DJIA, 0.98757 for the portfolio. They are all very close but not equal to 1.The second column reports the results for the power EWMA estimator which imposes the restrictions that 0α =0 and 2α = 1- 1α .The estimated power parameter of the conditional distribution,δ , in the power EWMA model is 1.52346 for the TAIEX ,1.57267 for the FTSE 100, 1.24451for the DJIA and 1.50543 for the portfolio. The estimated power parameter of the conditional distribution of the power EWMA model in each case is very close to the estimated power parameter, k, of the conditional variance model, which is consistent with the results of the previous studies (Nelson and Foster ,1994; Guermat and Harris, 2002).The power parameter, k, varying between 1.10735 for the DJIA and 1.34356 for

16

the FTSE 100, for all four series is closer to unity than to two suggest that the robust EWMA estimator might be expected to perform better than the standard EWMA estimator. The estimated decay factors of the power EWMA model are 0.95535 for the DJIA, 0.94212 for the portfolio, 0.93792 for the FTSE 100 and 0.92986 for the TAIEX. These are all very close to the value of 0.94 that is suggested by JP Morgan. The estimated power parameters of the conditional distribution,δ , varying between 1.22393 and 1.60166, are all smaller than 2 which confirms again our previous findings that all the equity returns exhibit fat tailed and leptokurtic distribution.

Table 3 Estimates of EWMAs

NGARCH Power EWMA

Standard

EWMA

Robust

EWMA

TAIEX

0α 0.00001

(0.23825)

[0] [0] [0]

1α 0.90227

(0.00000)

0.92967

(0.00000)

0.92986

(0.00000)

0.92670

(0.00000)

2α 0.08941

(0.00000)

[ 11 α− ] [ 11 α− ] [ 11 α− ]

k 1.64705

(0.00000)

1.34356

(0.00000)

[2] [1]

δ 1.56534

(0.00000)

1.52346

(0.00000)

1.55402

(0.00000)

1.43966

(0.00000)

LOGL 18216.77374 18194.93186 18190.44856 18174.73734

FTSE 100

0α 0.00001

(0.27084)

[0] [0] [0]

1α 0.89057

(0.00000)

0.93125

(0.00000)

0.93792

(0.00000)

0.92505

(0.00000)

2α 0.08724

(0.00000)

[ 11 α− ] [ 11 α− ] [ 11 α− ]

k 1.79268

(0.00000)

1.32207

(0.00000)

[2] [1]

δ 1.60166

(0.00000)

1.57267

(0.00000)

1.57728

(0.00000)

1.51066

(0.00000)

LOGL 18859.79677 18834.25421 18824.40090 18811.12709

DJIA

17

0α 0.00002

(0.31736)

[0] [0] [0]

1α 0.93652

(0.00000)

0.95328

(0.00000)

0.95535

(0.00000)

0.95236

(0.00000)

2α 0.05444

(0.00000)

[ 11 α− ] [ 11 α− ] [ 11 α− ]

k 1.33747

(0.00000)

1.10735

(0.00000)

[2] [1]

δ 1.25417

(0.00000)

1.24451

(0.00000)

1.25896

(0.00000)

1.22393

(0.00000)

LOGL 18856.36568 18846.36007 18828.91324 18843.71799

Portfolio

0α 0.00003

(0.35462)

[0] [0] [0]

1α 0.92574

(0.00000)

0.94236

(0.00000)

0.94212

(0.00000)

0.94124

(0.00000)

2α 0.06183

(0.00000)

[ 11 α− ] [ 11 α− ] [ 11 α− ]

k 1.24067

(0.00000)

1.24380

(0.00000)

[2] [1]

δ 1.52604

(0.00000)

1.50543

(0.00000)

1.51714

(0.00000)

1.44407

(0.00000)

LOGL 23680.06706 23665.59646 23653.39508 23650.26249 Note: 1. The restricted parameter values imposed on NGARCH are

reported in square brackets [ ]. 2. P-values are reported in parentheses ( ) for the parameter

estimates. 3. LOGL is the maximum value of the log likelihood function.

3.3 Restrictions Test on the Nested Model

The power EWMA estimator is nested by the NGARCH model, and therefore imposes certain restrictions on the NGARCH model. In this section, we test whether those restrictions are supported by the data. Table 4 reports likelihood ratio tests of the various restrictions. Table 4 shows that, owing to the precision with which the NGARCH parameters are estimated, the null hypothesis that the sum of the NGARCH parameters is unity can be rejected for all four series at significant level of 1%. These results suggest that while the true data generating process is not quite integrated, the sum of the estimated parameters is very close to unity and so, over short horizons, their dynamic properties should be reasonably well described by an

18

integrated NGARCH, or power EWMA process. The third and fourth rows report results for the standard EWMA estimator and the

robust EWMA estimator. These models impose the restrictions on the power EWMA estimator that k = 2 and k = 1, respectively. On the basis of likelihood ratio tests reported in Table 4, both models can be rejected for the TAIEX, FTSE 100 and portfolio at significant level of 1% with the except of DJIA, the robust EWMA estimator can be rejected only at significant level of 5%. A more inspection of the results, we can find that, due to the different degree of the fat-tailedness, the extent of rejection of Standard EWMA for the DJIA is stronger than the others, whereas the rejections of Robust EWMA for the TAIEX, FTSE 100 and portfolio are stronger than the DJIA. In sum, all the restrictions imposed on the EWMA models can not be supported by the sample asset returns. The unconstrained model, NGARCH, is the most suitable one for our data.

Table 4 L R Test of Restrictions on the Nested Model

H 0 TAIEX FTSE 100 DJIA Portfolio

Power EWMA vs.

NGARCH

00 =α 121 =+αα

43.68376** (0.00000)

51.08512** (0.00000)

20.01122** (0.00005)

28.94120** (0.00000)

Standard EWMA vs.

Power EWMA 2k =

8.96660** (0.00275)

19.70662** (0.00001)

34.89366** (0.00000)

24.40276** (0.00000)

Robust EWMA vs.

Power EWMA 1k =

40.38904** (0.00000)

46.25424** (0.00000)

5.28416* (0.02152)

30.66794** (0.00000)

Note: 1.This table reports the likelihood ratio test statistics to test the respective restrictions. The LR statistics are chi-square distributed with degrees of freedom equal to the number of restrictions imposed.

2.*denotes the significant level at 5% and ** denotes the significant level at 1%. 3. P-values are reported in parentheses.

3.4 Model Evaluation

For each model, the average value of the performance criteria across sample assets

for the 95 th , 99 th percentile and the average of the two are summarized in Table 5.

The mean relative bias tends to fall between 7.5% and -7% for average, indicating that

there is little difference in the magnitude of risk measure across the models. The most

conservative model is the robust EWMA which produces the largest average VaR

estimate, while the RiskMetrics® is the least conservative model which produces the

lowest average VaR estimate.

19

With the exceptions of the RiskMetrics® in 99 th VaR estimate and the Robust

EWMA in 95 th VaR estimate, the VaR measures produce the rates of failure (i.e. BLF)

close to the benchmarks of 0.01 and 0.05 respectively. The RiskMetrics® based on

normal distribution appears to be much more accurate when forecasting the 95 th VaR

than the 99 th VaR, which suggests that it is less sensitive to the outlying observations

than other models based exponential power distribution. The most conservative model

Robust EWMA thereby produce the least BLF either in 99 th VaR estimate or in

99 th VaR estimate. The average BLF is 2.36%, for the Robust EWMA, 2.72% for the

NGARCH and 2.85% for the Power EWMA, all lower than the benchmarks of 3%,

indicating that these models understate risk while others 3.20% for the Standard

EWMA and 3.46% for the RiskMetrics®, higher than the benchmark of 3%, indicating

that these models overstate risk. The Power EWMA provides the BLF which most

close to the benchmark, suggesting that Power EWMA is the most accurate model.

The results of likelihood ratio test show that the numbers of the NGARCH, Power

EWMA, Standard EWMA, Robust EWMA, RiskMetrics models that reject the null

hypothesis, for the 99 th VaR estimates are 2, 0, 1, 2, 4 and for the 95 th VaR estimates

are 2, 1, 0, 4, respectively. The Power EWMA, rejecting the null hypothesis only once

and having the least average LR statistics, achieve the best accurate results. Next to

the Power EWMA, the sequence are Standard EWMA, NGARCH, Robust EWMA

and RiskMetrics®。.

For the 99 th VaR estimate the multiples needed to obtain coverage are all larger

than one except for the Robust EWMA model. Put it another way, the Robust EWMA

model overestimates risk while the other models under-predict risk. Worthy to note

that for the 95 th VaR estimate the two models, the Standard EWMA and the

RiskMetrics®, with power parameter of 2 both require the multiples very close to one.

This indicates that they are both accurate models at 95% confidence level. For

average the Power EWMA need the multiple most close to one consistent with the

results of previous accuracy measures, achieving the best accuracy, The followings

are Standard EWMA, NGARCH, Robust EWMA, RiskMetrics® in sequence.

Comparing across all models, we find that the NGARCH exhibit greatly different

performance of efficiency. It provides the most efficient at 99 th measure whereas the

poorest efficient at 95 th measure. The RiskMetrics® dominates the other models for

20

95 th VaR estimate.

Table 5 The Performance Measures of Models

NGARCH Power EWMA

Standard EWMA Robust EWMA RiskMetrics®

99 VaR

ConservativenessMRB 0.02537 0.01204 -0.02654 0.08799 -0.09886

Accuracy BLF 0.98% 1.04% 1.25% 0.80% 1.72%

MOC 1.00286 1.02196 1.06662 0.95036 1.14936

LRuc 2.26346(2) 0.75996(0) 2.74555(1) 3.16126(2) 17.16379(4)

Effiency MRSB -0.00650 -0.00005 0.00448 0.00008 0.00199

95 VaR

ConservativenessMRB 0.01729 -0.00128 -0.03687 0.06299 -0.04213

Accuracy BLF 4.46% 4.66% 5.15% 3.92% 5.19%

MOC 0.96373 0.97798 1.01458 0.91985 1.01823

LRuc 6.40036(2) 1.64764(1) 1.01450(0) 11.83509(4) 0.89814(0)

Effiency MRSB 0.00214 -0.00065 -0.00021 0.00072 -0.00201

Average

ConservativenessMRB 0.02133 0.00538 -0.03170 0.07549 -0.07049

Accuracy BLF 2.72% 2.85% 3.20% 2.36% 3.46%

MOC 0.98330 0.99997 1.04060 0.93511 1.08380

LRuc 4.33191(4) 1.20380(1) 1.88002(1) 7.49817(6) 9.03097(5)

Effiency MRSB -0.00218 -0.00035 0.00213 0.00040 -0.00001 Note: The numbers of the model that reject the null hypothesis are reported in

parentheses.

4. Conclusion

21

The RiskMetrics®, standard EWMA based on normal distribution, is widely used to

forecast the variance of the conditional distribution of asset returns. It is appropriate

when the asset returns are drown from a normal distribution. However, there is

considerable evidence that the distribution of most financial returns is not well

approximated by normal distribution, even conditionally. The conditional distribution

of assets returns is typically found to be leptokurtic, and have fatter tail than normal

distribution. To improve the inefficiency of EWMA estimators, we introduce the

power exponential distribution to construct a serial of EWMA family estimators such

as Power EWMA, Standard EWMA, and Robust EWMA proposed by Guerma &

Harris (2002). The daily returns of TAIEX, FTSE 100 and DJIA are used to forecast

the VaR. Considering the different aspects of the usefulness of VaR models to risk

manager and supervisory authorities, we focus on three aspects- conservativeness,

accuracy and efficiency of model and propose a range of statistics based on these

criteria to evaluate the performance of the family models. The concluding remarks are

provided from the empirical findings as follows.

From the results of descriptive statistics, estimated tail-index and the estimated

power parameter of power exponential distribution, we obtain the consistent findings

that all equity returns have significant fat-tailed and leptokurtic distribution. Among

of them, the DJIA exhibits the highest degree of the fat-tailedness with leptokurtosis.

Next are FTSE 100, portfolio and the least is TAIEX. The estimated decay factors of

the power EWMA model are 0.95535 for the DJIA, 0.94212 for the portfolio, 0.93792

for the FTSE 100 and 0.92986 for the TAIEX. These are all very close to the value of

0.94 that is suggested by JP Morgan.

Comparing across all models, we find that the most conservative model is the

robust EWMA which produces the largest average VaR estimate, while the

RiskMetrics® is the least conservative model which produces the lowest average VaR

estimate. The Power EWMA providing the BLF and MOC which are both most close

to the benchmark, rejecting the null hypothesis of LR test only once and having the

least average LR statistics, achieve the best accurate results. A closer inspection of the

results reveal that for the 95 th VaR estimate, the Standard EWMA and the

RiskMetrics®, with power parameter of 2 both require the multiples very close to one..

They are more accurate at 95% confidence level (less extreme risk) than at 99%

confidence level (more extreme risk). It is worthy to note that the NGARCH exhibit

22

greatly different performance of efficiency, which provides the most efficient at

99 th measure whereas the poorest efficient at 95 th measure. Another finding is that the

RiskMetrics® dominates the other models for 95 th VaR estimate.

The back-testing results demonstrate that the members of the family of EWMA

estimators with the power exponential distribution have improved the inefficiency of

the RiskMetrics®- the traditional Standard EWMA estimator based on normal

distribution, and offer an appropriate coverage of the extreme risk. Due to the

flexibility of power parameter of the model, the Power EWMA performs a superior

accuracy in VaR estimation over the other EWMA estimators.

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