Fat--tails and VaR estimation using power EWMA models.

ABSTRACT

It is well known that financial asset returns have a non-normal distribution and are leptokurtic, with the tails fatter than those of the normal distribution. The Standard EWMA estimator with a normality assumption (used in JP Morgan's RiskMetrics[R] model) is inefficient and leads to understating the true value of risk if the asset returns are fat-tail distributed. On the basis of the power exponential distribution (also known as the generalized error distribution) the family of EWMA estimators, nesting Power EWMA, Standard EWMA and Robust EWMA, were proposed by Guermat & Harris (2002). Using these newly developed estimators, we first forecast the VaR of daily returns for the TAIEX, FTSE 100, and DJIA. Subsequently, back-testing is performed to evaluate the VaR models. The performance assessment is based on a range of measures that address the conservativeness, accuracy and efficiency of each model The results demonstrate that the family of EWMA estimators, based on a power exponential distribution, rather than normal distribution, offer superior coverage for extreme risk over the RiskMetrics[R] estimator, and show that the Power EWMA performs with the highest accuracy in VaR estimation.

1. INTRODUCTION

Value-at-Risk (VaR) has emerged as a widely used tool of financial institutions for risk management. The successful implementation of VaR depends heavily on the accurate estimation of the conditional distribution of portfolio returns. Owing to simple and rapid computations, the exponentially weighted moving average of past squared returns, or the EWMA estimator, has become the most common approach to forecast the conditional volatility of asset returns (JP Morgan, 1994; Dowd, 1998; Jorion, 2000). It has proved very effective for 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 are 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 has been found to be stylized as leptokurtic, with tails that are significantly fatter than those of the normal distribution (see Mandelbrot, 1963; Fama, 1965; Baillie and de Gennaro, 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[R] model) is inefficient and can lead to understating the true value of risk if the asset returns are fat-tail distributed.

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

Lots of literatures have addressed the weaknesses and strengths of various VaR models. However, no single consistent measurement of VaR model performance has been developed. Of concern to supervisors, is whether the required minimum regulatory-capital, calculated by the bank's internal model, can offer an appropriate coverage for its losses. We identify conservative models as those that systematically produce higher estimates of risk, relative to other models. With respect to accuracy, the risk manager should be concerned with whether ex-post performance is compatible with theoretical desired level. The regulatory capital-adequacy framework provides the incentive to develop efficient models, that is, models offering enough risk coverage to meet the supervisors' requirement for minimum holdings of capital.

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

The paper is organized as follows. The second section introduces methodology, including data descriptions, illustrations of the nested Power EWMA family, tail-index estimation and model evaluation measures. Section 3 presents the results of the empirical evaluation and finally, 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 the Dow Jones Indexes' Web Site, Datastream and the 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, [r.sub.t] = ln[I.sub.t] - ln[I.sub.t-1], where lt is the price index value for date t. Moreover, to investigate the diversifiable effects among different equity returns, we compose an equally weighted portfolio from the TAIEX, FTSE 100 and DJIA.

2.2 Tail Index Estimation

An important feature of the normal distribution is that the tail decay as the square of an exponential and thus faster than an exponential toward zero, implying that the large positive or negative returns are rare event. However, this seems to contradict the empirical evidence that many financial asset returns exhibit fatter tails than those of the normal distribution. Consequently, VaR under the normality assumption tends to underestimate the actual losses. Fat-tailed distributions are characterized by having higher probability of extreme events than in the case of normal distribution. The tail index is a measure of the degree of tail fatness of underlying distributions and is estimated using the extreme value theory (EVT), which addresses tail distribution behavior characteristics. The most famous and commonly applied estimator for tail index, due to its easy implementation and asymptotic unbiasedness, was proposed by Hill (1975) as follows:

(1) [xi] (m) = [1/m [m.summation over i=1] ln ([x.sub.n-i+1])]-ln([x.sub.n-m]), m [greater than or equal to] 2

= 1/m [[m.summation over.i=1]ln([x.sub.n-i+1]/[x.sub.n-m])]

where [xi] is the Hill's estimator which is the inverse of tail index a. m is the pre-specified number of tail observations to be included. The selection of m is crucial to obtain an unbiased estimator of the tail index. n is the sample size. [x.sub.i] is the ith increasing order statistic (i=1,2 ... n.). Equation (1) illustrates the Hill estimator measuring the average of the ratios of 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 the magnitude of the fat-tailedness.

However, considerable empirical evidence exists to show that it is biased in relatively small samples and limited to cases in which a larger sample is available. To correct for the bias in small samples, Huisman et al. (2001) modified the Hill estimator by a regression-based approach which is based on an approximation of the asymptotic expected value of m

(2) E([xi](m)) [approximately equal to] 1/[alpha] - cm

where c is a constant depending on the parameters of the distribution and the sample size. If m decreases, then the bias decreases and the expectation approaches to the true value [xi]. The variance of the estimator increases as m decreases.

(3) Var([xi](m)) [approximately equal to] 1/m[[alpha].sup.2]

According to equation (2), Huisman et al. regress [xi](m) on m as follows:

(4) [xi](m) = [[beta].sub.0] + [[beta].sub.1]m + [epsilon](m), m = 1 ... K

The estimated [[beta].sub.0] is an estimator of [xi]. Huisman, et al. propose choosing the threshold value K to equal half the sample (n/2). Although the parameters in equation (4) can be estimated by ordinary least squares (OLS), Equation (3) indicates that the variance of the Hill estimator is not constant for different values of m. The error term [epsilon](m)is heteroscedastic. Accordingly, they proposed a weighted least squares (WLS) approach to correct for the heteroscedasticity and improve the efficiency of the estimator. We apply the modified Hill estimator and obtain the tail index estimates by using both OLS and WLS.

2.3 The Power EWMA Variance Estimator

We then introduce a general Power EWMA estimator proposed by Guermat and Harris (2002), which nests EWMA models that are more robust to the leptokurtosis of the returns, and which is therefore 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

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(6) where [phi] = [{[2.sup.-2/[delta]][GAMMA](1/[delta])/[GAMMA](3/[delta])}.sup.1/2]

and [GAMMA](*) is the gamma function. The power exponential distribution has variance equal to [[sigma].sup.2], zero skewness and a kurtosis coefficient that depends on the value of power parameter [delta]. When [delta] = 2, the power exponential distribution reduces to the normal distribution. When [delta] > 2, the power exponential distribution is thin tailed and platykurtic, and when [delta] < 2, the power exponential distribution is fat-tailed and leptokurtic. When [delta] = 1, the power exponential distribution reduces to the Laplace distribution. Being conditional on the power parameter [delta], the maximum likelihood estimator of the standard deviation of the power exponential distribution is given by

(7) [[sigma].sup.[delta]] = g([delta])1/T[T.summation over.t=1][[absolute value of [r.sub.t]].sup.[delta]]

where

(8) g([delta]) = [delta][[[GAMMA](3/[delta])/[GAMMA](1/[delta])].sup.[delta]/2]

Equation (7) is the unconditional variance estimator that is independent of past information. Guermat and Harris (2002) transformed it into a conditional variance estimator and replaced the unweighted average in (7) with an exponentially weighted average to yield the Power EWMA estimator

(9) [[sigma].sup.k.sub.t+1] = (1 - [lambda])g(k)[infinity.summation over i=0][[lambda].sup.i][[absolute value of r.sub.t-i].sup.k]

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

(10) [[sigma].sup.k.sub.t+1] = [lambda][[sigma].sup.k.sub.t] + (1 - [lambda])g(k)[[absolute value of [r.sub.t]].sup.k]

so that 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 using the fact that [r.sup.2.sub.t+1] = [[sigma].sup.2.sub.t+1] + [[epsilon].sup.2.sub.t+1] where [[epsilon].sup.2.sub.t+1] 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 is given by

(11) [[sigma].sup.2.sub.t+1] = [lambda][[sigma].sup.2.sub.t] + (1 - [lambda]) [r.sup.t.sub.2]

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

(12) [[sigma].sup.2.sub.t+1] = [[alpha].sub.0] + [[alpha].sub.1][[sigma].sup.2.sub.t] + [[beta].sub.1][r.sup.2.sub.t]

where [[alpha].sub.0], [[alpha].sub.-1] and [[beta].sub.-1] are the parameters to be estimated. When [[alpha].sub.0] =.0 and [[beta].sub.1] = 1-[[alpha].sub.1], the GARCH model reduces to the Standard EWMA estimator, alternatively known as Integrated GARCH or IGARCH. When k < 2, the Power EWMA estimator is less sensitive to extreme observations and is thus expected 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

(13) [[sigma].sub.t+1] = (1 - [lambda])[square root of 2][infinity.summation over i=0][[lambda].sup.i]][absolute value of [r.sub.t-i]]

= [lambda][[sigma].sub.t] + (1 - [lambda])[square root of 2][absolute value of [r.sub.t]]

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' 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 in the NGARCH model of Higgins and Bera (1992), given by

(14) [[sigma].sup.k.sub.t+1] = [[alpha].sub.0] + [[alpha].sub.1][[sigma].sup.k.sub.t] + [[beta].sub.1][r.sup.k.sub.t]

When [[alpha].sub.0] = 0 and [[beta].sub.1] = (1 - [[alpha].sub.1])g(k.), 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 method. In contrast, members of the family of Power EWMA estimators have only a single parameter--the decay factor--and consequently their implementation is much more straightforward than that of the more sophisticated NGARCH model. The parameters of the models for the four equity returns are first estimated by maximum likelihood approach, based on power exponential distribution, using the BHHH algorithm (Berndt, et al., 1974) with a convergence criterion of 0.00001 applied to the function value. Next, we apply the likelihood ratio statistics 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 RiskMetrics) as the benchmark to compare the out-of-sample performance of the various Power EWMA estimators.

2.4. Estimating 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 lengths of 250 observations. In addition, VaR is computed for 99% and 95% confidence levels. The VaR of each portfolio in each period t is forecast by the formula

(15) Va[R.sub.t+1] = -[delta]([alpha])[[sigma].sub.t+1]

where [[sigma].sub.t+1] is the standard deviation of the portfolio's return, [r.sub.t+1], that is conditional on the time t information set, and [delta] ([[alpha].sub..])is the [alpha] - quantile of the standardized (i.e. zero mean, unit variance) empirical power exponential distribution and [[alpha].sub..] 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 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 cannot be made, since the latter is unobservable. A variety of evaluation methods have been proposed (see, for instance, Kupiec, 1995; Christofferson, 1998; Lopez; 1998). Up until now, no single definition for VaR model performance has been developed. To evaluate the performance of this family of models we propose a range of statistics that address different aspects of the usefulness of VaR models to risk managers and the supervisory authorities. We focus on three aspects of the models: 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 risk assessment. The larger the VaR value, the more conservative the model. To measure the relative size of VaR among different models, the mean relative bias developed by Hendricks (1996) can be applied. The mean relative bias statistic captures the degree of average bias of the VaR of a specific model from the all-model average. Given T time periods and N VaR models, the MRB of model i can be calculated as:

(16) MR[B.sub.i] = 1/T[T.summation over t=1]Va[R.sub.it]-[bar.Va[R.sub.t]]/[bar.Va[R.sub.t]]

where, [bar.Va[R.sub.t]] = 1/N[N.summation over i=1]Va[R.sub.it]

2.5.2. Accuracy

Different users of the VaR model will focus on different types of inaccuracies. Supervisors may be expected to pay more attention to the underestimation of losses while financial institutions will be more concerned about the over-prediction of losses due to capital adequacy requirements. In this study, we define the accuracy as the rate of failure (or exception) associated with how close each specific model came to the preset level of significance. The three accuracy measures : binary loss function, LR test of unconditional coverage (Kupiec, 1995) and the scaling multiple to obtain coverage are presented below.

a. Binary Loss Function (BLF)

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 failures 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 is equal to 1, with all others being 0. That is

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The aggregate of the number of failures across all dates is divided by the sample size. The BLF obtained is the rate of failure. The closer the BLF value is to the confidence level of the model, the more accurate the model.

b. LR Test of Unconditional Coverage (L[R.sub.uc])

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, is equal to one minus the prescribed confidence level of the VaR model. The model should provide the correct unconditional coverage of loss. Kupiec (1995) proposed a likelihood ratio test based on a binomial process which is 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 [H.sub.0] : p = [alpha] can be stated as

(18) L[R.sub.uc] = -2ln[[(1 - p).sup.T-N] [p.sup.N]] + 2ln [[(1 - N/T).sup.T-N][(N/T).sup.N]] ~[[chi square].sub.1, [alpha]]

Under the null hypothesis of correct unconditional coverage, the L[R.sub.uc] 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 the VaR estimate, we compare the multiple to obtain coverage as proposed by Engel and Gizycki (1999). The multiple equivalent, [X.sub.i], of risk measure for model is calculated so that

(19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where the [F.sub.i] is equivalent to the total number of failures. [T.sub.i] is the sample size and u is the significance level of the model. [DELTA][P.sub.i,t+1] denotes the realized profit or loss on t+1 day. The closer the MOC value is to the value of one, the more accurate the model.

2.5.3 Efficiency

Mean Relative Scaled Bias (MRSB): Efficiency is important, since VaR measures are used by both the supervisory authorities and internal management of financial institutions to influence investors' incentives. A more efficient VaR model can provide more precise resource allocation signals to the financial institutions. Hence we address the efficiency aspect related to the ability of a model to provide adequate risk coverage with minimum capital. The MRSB (Engel and Gizycki ,1999) aimed at evaluating a model which, once it obtains the desired risk coverage level, would produce the smallest VaR measurement. Calculating the MRSB measure involves two steps. First, the scaling was calculated by multiplying the VaR for each model by the multiple needed to obtain the 95% or 99% coverage as described in the MOC measure. Following this, we compare the scaled VaR measurements with the average relative size to the all-model average. The MRSB measure is given as follows:

(20) MRS[B.sub.i] = 1/T [T.summation over t=1] [X.sub.i] x Va[R.sub.i,t] - [bar.X x Va[R.sub.t].bar]/[X x Va[R.sub.t].bar]

where, [bar.X x Va[R.sub.t].bar] = 1/N [N.summation over i=1] [X.sub.i] x Va[R.sub.i,t]

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 kurtosis 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.

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 indices of the right-tail, left-tail and both tails, 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 see 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 less than 8.5, indicating that all the equity return distributions exhibit fatter tails than the normal distribution, as is commonly found in the literature. Further, the DJIA has the highest degree of fat-tailedness, which is also consistent with the findings of the preliminary statistics. If these estimates differ significantly over both tails, it is inappropriate to use the estimate obtained from their combined information. However, our results show that the left-tail estimates are a little fatter than those of the right-tail, for the majority of the equity returns.

3.3. Estimation of Power EWMA

The NGARCH model, and the Power EWMA estimator that it nests, are based on 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 [[alpha].sub.1] and [[alpha].sub.2] is 0.99168 for the TAIEX, 0.97781 for the FTSE 100, 0.99096 for the DJIA and 0.98757 for the portfolio, which are all very close, but not equal to 1.The second column reports the results for the Power EWMA estimator which imposes restrictions of [[alpha].sub.0] = 0 and [[alpha].sub.2] = 1 - [[alpha].sub.1]. The estimated power parameters of the conditional distribution, [delta], in the Power EWMA model is 1.52346 for the TAIEX ,1.57267 for the FTSE 100, 1.24451 for the DJIA and 1.50543 for the portfolio. The estimated power parameters 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 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 the FTSE 100, for all four series is closer to 1 than 2, suggesting that the Robust EWMA estimator may be expected to perform better than the Standard EWMA estimator. The estimated decay factors of the Standard 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 was suggested by JP Morgan. The estimated power parameters of the conditional distribution, [delta], varying between 1.22393 and 1.60166, are all smaller than 2 which again confirms our previous findings that all the equity returns exhibit a fat tailed and leptokurtic distribution.

3.4. Testing Restrictions on the Nested Models

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 significance 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 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 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 the portfolio at a significance level of 1%; with the exception of the DJIA, the Robust EWMA estimator can be rejected only at a significance level of 5%. A closer inspection of the results show that, due to the different degrees of fat-tailedness, the extent of rejection of the 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. On the whole, all of the restrictions imposed on the EWMA models can not be supported by the sample asset returns. The unconstrained model, the NGARCH, is the most suitable one for our data.

3.5 Model Evaluation

For each model, the average value of the performance criteria across sample assets for the 95th and 99th percentiles, and the average of the two, are summarized in Table 5. The mean relative bias tends to fall between 7.5% and -7% for the average, indicating that there is little difference in the magnitude of risk measurement across the models. From the viewpoints of the supervisory authorities, the most conservative model is the Robust EWMA, which produces the largest average VaR estimate, while the RiskMetrics[R] is the least conservative model which produces the lowest average VaR estimate.

With the exceptions of the RiskMetrics[R] in the 99th VaR estimate and the Robust EWMA in the 95th VaR estimate, the VaR measures produce rates of failure (i.e. BLF) close to the benchmarks of 0.01 and 0.05, respectively. The RiskMetrics[R], based on normal distribution, appears to be much more accurate when forecasting the 95th VaR than the 99th VaR, which suggests that it may be less sensitive to the outlying observations than other models based on exponential power distribution and then lead to underestimating the extreme risk.. The most conservative model, the Robust EWMA, thereby produces the lowest BLF in both the 95th VaR estimate and the 99th 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 benchmark of 3%, indicating that these models understate risk; the others are 3.20% for the Standard EWMA and 3.46% for the RiskMetrics[R], which are higher than the benchmark of 3%, indicating that these models overstate risk. The Power EWMA provides the BLF which is closest to the benchmark, suggesting that the Power EWMA is the most accurate model.

The results of the likelihood ratio test show the numbers of the NGARCH, Power EWMA, Standard EWMA, Robust EWMA and RiskMetrics models, which rejecte the null hypothesis, for the 99th VaR estimates to be 2, 0, 1, 2 and 4, and for the 95th VaR, estimates to be 2, 1, 0 and 4, respectively. The Power EWMA, which rejects the null hypothesis only once and has the lowest average LR statistics, achieve the most accurate results. Next to the Power EWMA, the diminishing accuracy sequence is Standard EWMA, NGARCH, Robust EWMA and RiskMetrics[R].

For the 99th VaR estimate, the multiples needed to obtain coverage are all larger than one, except for the Robust EWMA model. Stated another way, the Robust EWMA model overestimates the risk while the other models under-estimates the risk. A closer inspection of the results reveals that for the different percentile VaR estimates, the Standard EWMA and the RiskMetrics[R], with the power parameter of 2, are more accurate at the 95% confidence level (less extreme risk) than at the 99% confidence level (more extreme risk). For the average, the Power EWMA requires the multiple closest to one, consistent with the results of previous accuracy measures, to achieve the highest accuracy. Following, in diminishing accuracy sequence are the Standard EWMA, NGARCH, Robust EWMA and RiskMetrics[R]. Worth noting, is that the powder parameter in the EWMA model is the important factor in relation to the degree of the fat-tailedness of asset return. The higher the degree of the fat-tailedness of the asset return, the larger powder parameter the EWMA model should be with. The results of evaluation of model accuracy highlight that due to the flexibility of the power parameters of the models, the Power EWMA model achieves the highest accurate results in terms of different accuracy measures and the majority of our models are accurate.

Comparing all models, we find that the NGARCH exhibit a much different efficiency performance. It provides the highest efficiency performance at 99th measure whereas provides the lowest efficiency performance at 95th measure among these models. The RiskMetrics[R] model dominates the other models for the 95th VaR estimate. On the average, the NGARCH is still the most efficient model, providing adequate risk coverage with minimum capital for the financial institutions.

4. CONCLUSION

The RiskMetrics[R], Standard EWMA, based on normal distribution, is widely used to forecast the variance for conditional distribution of asset returns. This is appropriate, when the asset returns are drawn from a normal distribution; however, there is considerable evidence suggesting that the distribution of most financial returns is not well approximated by normal distribution, even conditionally. The conditional distribution of asset returns is typically found to be leptokurtic, and to have fatter tail than that of normal distribution. To improve the efficiency of EWMA estimators, we introduce the power exponential distribution to construct a series of EWMA family estimators to forecast the VaR. Considering the different aspects of the usefulness of the VaR model to both risk managers and supervisory authorities, we focus on three aspects--the conservativeness, accuracy and efficiency of the models--and propose a range of statistics based on these criteria in order to evaluate the performance of this family of models.

From the results of descriptive statistics, estimated tail-index and the estimated power parameter of power exponential distribution, we obtain the consistent findings showing that all equity returns have a significantly fat-tailed and leptokurtic distribution. The estimated decay factors of the family of EWMA models are all very close to the value of 0.94 suggested by JP Morgan.

From the aspect of concern to the supervisory authorities, we find that the most conservative model is the Robust EWMA, while the RiskMetrics[R] is the least conservative model. As expected, the RiskMetrics[R] model based on the normal distribution appears to be less sensitive to the outlying observations than other models based on the exponential power distribution and then lead to underestimating the extreme risk. Worth noting, is that the powder parameter in the EWMA model is a critical factor in relation to the degree of the fat-tailedness of asset return. The higher the degree of the fat-tailedness of the asset return, the smaller power parameter the EWMA model should be with. The results of evaluation of model accuracy highlight that the Power EWMA model achieves the highest accurate results in terms of different accuracy measures and the majority of our models are accurate. Another finding is that the NGARCH is the most efficient model, providing adequate risk coverage with minimum capital for the financial institutions.

Overall, the back-testing results demonstrate that the power exponential distribution can properly capture the fat-tailedness characteristic of the asset return distributions, so that the family of EWMA estimators, based on a power exponential distribution, rather than normal distribution, offer superior coverage for extreme risk over the RiskMetrics[R] estimator. Due to the flexibility of the power parameters of the models, the Power EWMA performs with greater accuracy in VaR estimation than the other EWMA estimators.

TABLE 1 DESCRIPTIVE STATISTICS

                        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     -0.813053    -2.75528     -0.737642
                      (0.00000)    (0.00000)    (0.00000)    (0.00000)
Kurtosis               4.87460     13.52859     66.52633     11.43150
                      (0.00000)    (0.00000)    (0.00000)    (0.00000)
Jarque-Bera           815.3398     22547.77     805907.4     17156.62
                      (0.00000)    (0.00000)    (0.00000)    (0.00000)
Sample size             5289         4768         4757         5620

Note: P-values are reported in parentheses for the skewness, kurtosis
and the Jarque-Bera statistics.

TABLE 2 TAIL--INDEX ESTIMATES

                          TAIEX      FTSE 100       DJIA

Both [tail.sub.OLS]      5.56616      3.59839      3.59066
                        (0.00000)    (0.00000)    (0.00000)
Both [tail.sub.WLS]      4.93622      3.8069       3.87973
                        (0.00000)    (0.00000)    (0.00000)
Sample size               2,644        2,383        2,378
Left [tail.sub.OLS]      5.46890      3.38424      3.24098
                        (0.00000)    (0.00000)    (0.00000)
Left [tail.sub.WLS]      4.80790      3.63574      3.52230
                        (0.00000)    (0.00000)    (0.00000)
Sample size               1,310        1,155        1,171
Right [tail.sub.OLS]     5.87510      3.87090      4.00489
                        (0.00000)    (0.00000)    (0.00000)
Right [tail.sub.WLS]     5.19921      3.98977      4.22635
                        (0.00000)    (0.00000)    (0.00000)
Sample size               1,333        1,228        1,206

                        Portfolio     Normal

Both [tail.sub.OLS]      4.50852      8.41893
                        (0.00000)    (0.00000)
Both [tail.sub.WLS]      4.68356        --
                        (0.00000)
Sample size               2,808       24,999
Left [tail.sub.OLS]      4.33279      8.49762
                        (0.00000)    (0.00000)
Left [tail.sub.WLS]      4.62216        --
                        (0.00000)
Sample size               1,380       12,527
Right [tail.sub.OLS]     4.93776      8.74279
                        (0.00000)    (0.00000)
Right [tail.sub.WLS]     4.95584        --
                        (0.00000)
Sample size               1,428       12,472

Note: P-values are reported in parentheses for the estimate of the tail
index [alpha] = 1/[etha]

TABLE 3: ESTIMATES OF EWMA MODELS

                      NGARCH          Power EWMA

                                 TAIEX

[[alpha].sub.0]      0.00001             [0]
                    (0.23825)
[[alpha].sub.1]      0.90227           0.92967
                    (0.00000)         (0.00000)
[[alpha].sub.2]      0.08941      [1-[[alpha].sub.1]
                    (0.00000)
       k             1.64705           1.34356
                    (0.00000)         (0.00000)
    [delta]          1.56534           1.52346
                    (0.00000)         (0.00000)
     LOGL          18216.77374       18194.93186

                               FTSE 100

[[alpha].sub.0]      0.00001             [0]
                    (0.00000)
[[alpha].sub.1]      0.89057           0.93125
                    (0.00000)         (0.00000)
[[alpha].sub.2]      0.08724      [1-[[alpha].sub.1]
                    (0.00000)
       k             1.79268           1.32207
                    (0.00000)         (0.00000)
    [delta]          1.60166           1.57267
                    (0.00000)         (0.00000)
     LOGL          18859.79677       18834.25421

                                  DJIA

[[alpha].sub.0]      0.00002             [0]
                    (0.31736)
[[alpha].sub.1]      0.93652           0.95328
                    (0.00000)         (0.00000)
[[alpha].sub.2]      0.05444      [1-[[alpha].sub.1]
                    (0.00000)
       k             1.33747           1.10735
                    (0.00000)         (0.00000)
    [delta]          1.25417           1.24451
                    (0.00000)         (0.00000)
     LOGL          18856.36568       18846.36007

                               Portfolio

[[alpha].sub.0]      0.00003             [0]
                    (0.00000)
[[alpha].sub.1]      0.92574           0.94236
                    (0.00000)         (0.00000)
[[alpha].sub.2]      0.06183      [1-[[alpha].sub.1]
                    (0.00000)
       k             1.24067           1.24380
                    (0.00000)         (0.00000)
    [delta]          1.52604           1.50543
                    (0.00000)         (0.00000)
     LOGL          23680.06706       23665.59646

                     Standard EWMA          Robust EWMA

                                     TAIEX

[[alpha].sub.0]           [0]                   [0]

[[alpha].sub.1]         0.92986               0.92670
                       (0.00000)             (0.00000)
[[alpha].sub.2]    [1-[[alpha].sub.1]    [1-[[alpha].sub.1]

       k                  [2]                   [1]

    [delta]             1.55402               1.43966
                       (0.00000)             (0.00000)
     LOGL             18190.44856           18174.73734

                                     FTSE 100

[[alpha].sub.0]           [0]                   [0]

[[alpha].sub.1]         0.93792               0.92505
                       (0.00000)             (0.00000)
[[alpha].sub.2]    [1-[[alpha].sub.1]    [1-[[alpha].sub.1]

       k                  [2]                   [1]

    [delta]             1.57728               1.51066
                       (0.00000)             (0.00000)
     LOGL             18824.40090           18811.12709

                                     DJIA

[[alpha].sub.0]           [0]                   [0]

[[alpha].sub.1]         0.95535               0.95236
                       (0.00000)             (0.00000)
[[alpha].sub.2]    [1-[[alpha].sub.1]    [1-[[alpha].sub.1]

       k                  [2]                   [1]

    [delta]             1.25896               1.22393
                       (0.00000)             (0.00000)
     LOGL             18828.91324           18843.71799

                                     Portfolio

[[alpha].sub.0]           [0]                   [0]

[[alpha].sub.1]         0.94212               0.94124
                       (0.00000)             (0.00000)
[[alpha].sub.2]    [1-[[alpha].sub.1]    [1-[[alpha].sub.1]

       k                  [2]                   [1]

    [delta]             1.51714               1.44407
                       (0.00000)             (0.00000)
     LOGL             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.

TABLE 4 LR TEST OF RESTRICITIONS ON THE NESTED MODELS

                      [H.sub.0]           TAIEX      FTSE 100

Power EWMA       [[alpha].sub.0] = 0    43.68376     51.08512
     vs.         [[alpha].sub.1] +
NGARCH           [[alpha].sub.2] = 1    (0.00000)    (0.00000)
Standard EWMA                            8.96660     19.70662
     vs.         k = 2                  (0.00275)    (0.00001)
Power EWMA
Robust EWMA                             40.38904     46.25424
     vs.         k = 1                  (0.00000)    (0.00000)
Power EWMA

                   DJIA       Portfolio

Power EWMA       20.01122     28.94120
     vs.         (0.00005)    (0.00000)
NGARCH
Standard EWMA    34.89366     24.40276
     vs.         (0.00000)    (0.00000)
Power EWMA
Robust EWMA      5.28416      30.66794
     vs.         (0.02152)    (0.00000)
Power EWMA

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.) P-values are reported in parentheses.

TABLE 5 THE PERFORMANCE MEASURES OF MODELS

                      NGARCH        Power EWMA      Standard EWMA

                                      99 VaR

Conservativeness
MRB                  0.02537          0.01204          -0.02654
Accuracy
BLF                   0.98%            1.04%            1.25%
MOC                  0.98286          1.02196          1.06662
[LR.sub.uc]         2.26346(2)      0.75996(0)        2.74555(1)
Efficiency
MRSB                 -0.00650        -0.00005          0.00448

95 VaR

Conservativeness
MRB                  0.01729         -0.00128          -0.03687
Accuracy
BLF                   4.46%            4.66%            5.15%
MOC                  0.96373          0.97798          1.01458
[LR.sub.uc]        6.40036(2)       1.64764(1)         1.01450(0)
Efficiency
MRSB                 0.00214         -0.00065          -0.00021

Average

Conservativeness
MRB                  0.02133          0.00538          -0.03170
Accuracy
BLF                   2.72%            2.85%            3.20%
MOC                  0.98330          0.99997          1.04060
[LR.sub.uc]         4.33191(4)       1.20380(1)        1.88002(1)
Efficiency
MRSB                 -0.00218        -0.00035          0.00213

                   Robust EWMA    RiskMetrics[R]

Conservativeness
MRB                  0.08799         -0.09886
Accuracy
BLF                   0.80%            1.72%
MOC                  0.95036          1.14936
[LR.sub.uc]         3.16126(2)      17.16379(4)
Efficiency
MRSB                 0.00008          0.00199

95 VaR

Conservativeness
MRB                  0.06299         -0.04213
Accuracy
BLF                   3.92%            5.19%
MOC                  0.91985          1.01823
[LR.sub.uc]        11.83509(4)       0.89814(0)
Efficiency
MRSB                 0.00072         -0.00201

Average

Conservativeness
MRB                  0.07549         -0.07049
Accuracy
BLF                   2.36%            3.46%
MOC                  0.93511          1.08380
[LR.sub.uc]        7.49817(6)        9.03097(5)
Efficiency
MRSB                 0.00040         -0.00001

Note: The numbers of the models which reject the null hypothesis
are reported in parentheses.

REFERENCE

Alexander, C., and Leigh, C. T., "On the Covariance Matrices Used in Value at Risk Models", Journal of Derivatives, Vol. IV (3), 1997, 50-62.

Baillie, R., and Bollerslev, T., "The Message in Daily Exchange Rates: A Conditional-Variance Tale", Journal of Business and Economic Statistics, Vol. VII, 1989, 297-305.

Baillie, R., and de Gennaro, R., "Stock Returns and Volatility", Journal of Financial and Quantitative Analysis, Vol. XXV, 1990, 203-14.

Beine, M., Laurent, S. and Lecourt, C., "Accounting for Conditional Leptokurtosis and Closing Days Effects in FIGARCH Models of Daily Exchange Rates", Applied Financial Economics, Vol. XII, 2002, 589-600.

Berndt, E. K., Hall, B. H., Hall, R. E. and Hausman, J. A., "Estimation and Inference in Nonlinear Structural Models", Annals of Economic and Social Measurement, Vol. IV, 1974, 653-65.

Bollerslev, T., "Generalized Autoregressive Conditional Heteroskedasticity", Journal of Econometrics, Vol. XXXI, 1986, 307-27.

Bollerslev, T., "A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return", Review of Economics and Statistics, Vol. LXIX, 1987, 542-47.

Bollerslev, T., Chou, R. and Kroner, K., "ARCH Modeling in Finance", Journal of Econometrics, Vol. LII, 1992, 5-59.

Boudoukh, J., Richardson, M. and Whitelaw, Robert. F., "Investigation of a Class of Volatility Estimators", Journal of Derivatives, Vol. IV (3), 1997, 63-72.

Christoffersen, P., "Evaluating Interval Forecasts", International Economic Review, Vol. XXXIX, 1998, 841-62.

Dowd, K., Beyond Value-at-Risk: The New Science of Risk Management, New York: Wiley, 1998.

Engel, J. and Gizycki, M., "Conservatism, Accuracy and Efficiency: Comparing Value-at-Risk Models", Workinq Paper 2, 1999, Australian Prudential Regulation Authority.

Engle, R., "Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation", Econometrica, Vol. L, 1982, 987-1007.

Fama, E., "The Behaviour of Stock Market Prices", Journal of Business, Vol. XXXVIII, 1965, 34-105.

Greene, William. H., Econometric Analysis, Prentice Hall, 2000.

Guermat, C., and Harris, Richard D. F., "Robust Conditional Variance Estimation and Value at Risk", Journal of Risk, Vol. IV, 2002, 25-41.

Hendricks, D., "Evaluation of Value at Risk Models Using Historical Data", Federal Reserve Bank of New York Economic Policy April 1996, 39-69.

Higgins, M., and Bera, A., "A Class of Nonlinear ARCH Models", International Economic Review, Vol. XXXIII, 1992,137-158.

Hill, Bruce M., "A Simple General Approach to Inference about the Tail of a Distribution", Annals of Statistics Vol. III, 1975, 1163-74.

Hsieh, D., "Modeling Heteroskedasticity in Daily Foreign Exchange Rates", Journal of Business and Economic Statistics, Vol. VII, 1989, 307-17.

Huisman, R., Koedijk, Kees G., Kool, Clemens J. M. and Palm, F., "Tail-Index Estimates in Small Samples", Journal of Business and Economic Statistics, Vol. XIX, 2001, 208-16.

Hull, J., and White, A., "Value at Risk When Daily Changes in Market Variables Are Not Normally Distributed", Journal of Derivatives, Vol. V, 1998, 9-19.

Jansen, D., and de Vries, C., "On the Frequency of Large Stock Returns: Putting Booms and Busts into Perspective", The Review of Economics and Statistics, Vol. LXXIII, 1991, 18-24.

Jorion, P., "On Jump Processes in the Foreign Exchange and Stock Markets", The Review of Financial Studies, Vol. LXVIII, 1988, 165-176.

Jorion, P., Value at Risk: The New Benchmark for Managing Financial Risk, McGraw-Hill, 2000.

J.P. Morgan, RiskMetics[R] Technical Document, 4th ed., Morgan Guaranty Trust Company, 1996.

Kaiser, T., "One-Factor-GARCH Models for German Stocks--Estimation and Forecasting", Working Universiteit Tubingen, 1996.

Kearns, P., and Pagan, A., "Estimating the Density Tail Index for Financial Times Series", The Review of Economics and Statistics, Vol. LXXIX, 1997, 171-75.

Koedij, Kees. G., and Kool, C., "Tail Estimates and the EMS Target Zone", Review of International Economics, Vol. II, 1994,153-65.

Kupiec, Paul. H., "Techniques for Verifying the Accuracy of Risk Measurement Models", Journal of Derivatives, Vol. III, 1995, 73-84.

Linden, M., "A Model for Stock Return Distribution", International Journal of Finance & Economics, Vol. VI, 2001, 159-69.

Lopez, J., "Methods for Evaluating Value-at-Risk Estimates", Federal Reserve Bank of New York Research Paper, No. 9802, 1998.

Loretan, M., and Phillips, P., "Testing the Covariance Structure of Heavy-Tailed Time Series", Journal of Empiric, a/Finance, Vol. I, 1994, 211-48.

Mandelbrot, B., "The Variation of Certain Speculative Prices", Journal of Business, Vol. XXXVI, 1963, 394-419.

Nelson, D., and Foster, D., "Asymptotic Filtering Theory for Univariate ARCH Models", Econometrica, Vol. LXII, 1994, 1-41.

Varma, Jayanth R., "Rupee-Dollar Option Pricing and Risk Measurement: Jump Processes, Changing Volatility and Kurtosis Shifts", Journal of Foreign Exchange and International Finance, Vol. XIII (1), 1999, 11-33.

Vlaar, P., and Palm, F., "The Message in Weekly Exchange Rates in the European Monetary System: Mean Reversion, Conditional Heteroskedasticity and Jumps", Journal of Business and Economic Statistics, Vol. XI, 1993, 351-60.

Author Profile

Dr. Mei-Ying Liu earned her Ph.D. at the National Chiao-Tung University, Taiwan in 1994. Currently she is an associate professor of department of business administration at Soochow University, Taiwan.

Chi-Yeh Wu earned his MBA degree at Soochow University, Taiwan in 2003. Currently he is a manager of Information Department at WK Technology Fund, Taiwan.

Dr. Hsien-Feng Lee earned his Ph. D. at the Bielefield, Germany in 1993. Currently he is an associate professor of department of economics at National Taiwan University, Taiwan.