volatility forecasting for risk management

Journal of ForecastingJ. Forecast. 22, 1–22 (2003)Published online 9 October 2002 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/for.841

Volatility Forecasting for RiskManagement

CHRIS BROOKS1* AND GITA PERSAND2

1 ISMA Centre, University of Reading, UK2 Department of Economics, University of Bristol, UK

ABSTRACTRecent research has suggested that forecast evaluation on the basis of stan-dard statistical loss functions could prefer models which are sub-optimalwhen used in a practical setting. This paper explores a number of statis-tical models for predicting the daily volatility of several key UK financialtime series. The out-of-sample forecasting performance of various linear andGARCH-type models of volatility are compared with forecasts derived froma multivariate approach. The forecasts are evaluated using traditional met-rics, such as mean squared error, and also by how adequately they performin a modern risk management setting. We find that the relative accuraciesof the various methods are highly sensitive to the measure used to eval-uate them. Such results have implications for any econometric time seriesforecasts which are subsequently employed in financial decision making.Copyright 2003 John Wiley & Sons, Ltd.

KEY WORDS internal risk management models; asset return volatility; Valueat Risk models; forecasting; univariate and multivariateGARCH models

INTRODUCTION

Modelling and forecasting stock market volatility has been the subject of a great deal of debate overthe past fifteen years or so. Volatility, usually measured by the standard deviation of portfolio returns,is uniquely important in financial markets, for it is often taken to represent the portfolio’s risk.Consequently, the literature on forecasting volatility is sizeable and still growing. Akgiray (1989),for example, finds the GARCH model superior to ARCH, exponentially weighted moving average,and historical mean models for forecasting monthly US stock index volatility. A similar resultconcerning the apparent superiority of GARCH is observed by West and Cho (1995) using one-step-ahead forecasts of dollar exchange rate volatility, evaluated using root-mean squared predictionerrors. However, for longer horizons, the model behaves no better than their alternatives.1 Also using

* Correspondence to: Chris Brooks, ISMA Centre, PO Box 242, University of Reading, Whiteknights, Reading RG6 6BA, UK. E-mail: [email protected] The alternative models are the long-term mean, IGARCH, autoregressive models, and a non-parametric model based on the Gaussiankernel.

Copyright 2003 John Wiley & Sons, Ltd.

2 C. Brooks and G. Persand

the same models and data, West et al. (1993) use asymmetric, utility-based criteria for evaluatingthe conditional variance forecasts, finding that GARCH models tend to yield the highest utilities.Pagan and Schwert (1990) compare GARCH, EGARCH, Markov switching regime and three non-parametric models for forecasting monthly US stock return volatilities. The EGARCH followedby the GARCH models perform moderately; the remaining models produce very poor predictions.Franses and van Dijk (1996) compare three members of the GARCH family (standard GARCH,QGARCH and the GJR model) for forecasting the weekly volatility of various European stockmarket indices. They find that the non-linear GARCH models were unable to beat the standardGARCH model. Brailsford and Faff (1996) find GJR and GARCH models slightly superior tovarious simpler models2 for predicting Australian monthly stock index volatility. The conclusionarising from this growing body of research is that forecasting volatility is a ‘notoriously difficulttask’ (Brailsford and Faff, 1996, p. 419), although it appears that conditional heteroscedasticitymodels are among the best that are currently available. In particular, more complex non-linear andnon-parametric models are inferior in prediction to simpler models, a result echoed in an earlierpaper by Dimson and Marsh (1990) in the context of relatively complex versus parsimonious linearmodels. Finally Brooks (1998) uses a measure of market volume in volatility forecasting models,but observes no increase in forecasting power.

Recent papers have also sought to compare the predictive ability of volatility forecasts derivedfrom the market prices of traded options, with those generated using econometric models (see, forexample, Heynen and Kat, 1994 or Day and Lewis, 1992). The general consensus appears to be thatimplied volatility forecasts are more accurate than those derived using pure time series analysis,but also that the latter still contain additional information not embedded in the implied values.

Also over the past decade, there has been rapid development of techniques for measuring andmanaging financial risk, partially motivated by a spate of recent financial disasters involving deriva-tive securities. One of the most popular approaches to risk measurement is by calculating what isknown as an institution’s ‘Value at Risk’ (VaR). Broadly speaking, Value at Risk is an estimationof likely losses which could arise from changes in market prices. More precisely, it is defined asthe money-loss in a portfolio that is expected to occur over a pre-determined horizon and with apre-determined degree of confidence. The roots of VaR’s popularity stem from the simplicity of itscalculation, its ease of interpretation, and from the fact that VaR can be suitably aggregated acrossan entire firm to produce a single number which broadly encompasses the risk of the positions of thefirm as a whole. Jorion (1996) or Dowd (1998) provide thorough introductions to VaR, and Brooksand Persand (2000a,b) present recent discussions of VaR model estimation issues. The value at riskestimate is also often known as the position risk requirement or minimum capital risk requirement(MCRR); we shall use the three terms interchangeably in the exposition below.

Although the academic literature has thus far failed to keep pace with this expansion, evi-denced by the relatively few academic studies that address this topic, one exception is the studyby Jackson et al. (1998), which assesses the empirical performance of various models for VaRusing historical returns from the actual portfolio of a large investment bank. They find that non-parametric, simulation-based techniques yield more accurate measures of the tail probabilities thanparametric models. Alexander and Leigh (1997) offer an analysis of the relative performance ofequally weighted, exponentially weighted moving average (EWMA), and GARCH model forecastsof volatility, evaluated using traditional statistical and operational adequacy criteria. The GARCH

2 The other models employed are the random walk, the historical mean, a short- and a long-term moving average, exponential smoothing,an exponentially weighted moving average model, and a linear regression.

Copyright 2003 John Wiley & Sons, Ltd. J. Forecast. 22, 1–22 (2003)

Volatility Forecasting for Risk Management 3

model is found to be preferable to EWMA in terms of minimizing the number of exceedencesin a backtest, although the simple unweighted average is superior to both. Brooks et al. (2002)investigate the effectiveness of various hedging models when assessed according to their ability tominimize VaR, finding that there is a large role for time-varying volatilities and correlations, but avery minor role for asymmetries.

This paper seeks to combine and advance the two literatures in volatility forecasting and financialrisk management in a number of ways. First, the volatility forecasting debate is re-opened, and theforecasts from the various models evaluated on the basis of how well they perform in a modernrisk management setting, as well as by traditional statistical loss functions. This is important forDacco and Satchell (1999) demonstrate that the evaluation of forecasts from non-linear modelsusing statistical measures can be misleading, and they propose the use of alternative economicloss functions. Here, the relative performances of the forecasting models are evaluated using bothstatistical and economic loss functions, so that a comparison can be drawn between the two. Second,we also directly compare the forecasting performance of univariate and multivariate forecastingmodels for financial asset return volatility. Multivariate GARCH models permit the estimationof the conditional covariances between assets’ returns, and explicit modelling of this interactionmay improve the accuracy of forecasts of volatility for a portfolio comprising these components.Finally, we evaluate forecasts over the 1- 5-, 10- and 20-day horizons. Although many volatilityforecasting papers compare accuracies at daily horizons, it is often the case that financial marketpractitioners require predictions of much lower frequency. For example, the Basle Committee onBanking Supervision rules for the use of VaR models (see, for example, Basle Committee onBanking Supervision, 1998) require the use of a 10-day holding period, which allows reasonabletime for investors to unwind a position, and fund managers typically re-balance their portfolios ona monthly (20 trading days) basis.

The remainder of the paper is organized as follows. The next section presents the data employed inthe study, while the forecasting models are described briefly in the third section. Forecast evaluationmethods are outlined and discussed in the fourth section with results given in the fifth section. Thefinal section summarises the paper, and offers some concluding remarks.

THE DATA

In this study we calculate the VaRs for three different assets—the FTSE All Share Total ReturnIndex, the FTA British Government Bond (over 15 years) Index and the Reuters CommoditiesPrice Index, as well as an equally weighted portfolio containing these three assets.3, 4 The datawere collected from Datastream International, and spans the period 1 January 1980 to 25 March1999. Observations corresponding to UK public holidays were deleted from the data set to avoidthe incorporation of spurious zero returns, leaving 4865 observations, or trading days in the sample.In the empirical work below, we use the daily log return of the original indices. Summary statisticsfor the data are given in Table I. It is evident that the FTSE returns series is the most volatile,

3 Our analysis assumes that we are long all the three assets—both individually and in the portfolio. A similar analysis could be undertakenfor short or netted positions, but we would not expect our conclusions to be markedly altered.4 This portfolio is deliberately highly simplistic relative to a genuine bank’s book, as well as being entirely linear in nature. The use ofa simple portfolio enables us to more easily unravel the various estimation issues and broad aspects of the methodologies. Additionally,the three series that we consider are all fundamental or ‘benchmark’ factor series, to which other series are mapped under the JP Morganapproach.



Table I. Summary statistics

Long Govt Bond FTSE All Share Reuters Commodities Portfolio

Mean 0.000233 0.000301 �0.000219 0.000171Variance 6.50 ð 10�6 1.410 ð 10�5 6.210 ð 10�6 3.691 ð 10�6

Skewness 0.0132 �1.063ŁŁ �0.5663ŁŁ �0.291ŁŁKurtosis 3.37ŁŁ 14.654ŁŁ 18.369ŁŁ 4.446ŁŁBera–Jarque Statistic 2300ŁŁ 44400ŁŁ 68700ŁŁ 4080ŁŁ

Notes : The Bera–Jarque statistic is distributed asymptotically as a �2(2) under the null of normality. Ł and ŁŁ indicate significance at the5% and 1% levels respectively.

while the government bond index returns is the least. The benefits from diversification, in terms ofa substantial reduction in variability, are clear, since the variance for the equally weighted portfolioreturns is almost half that of the least volatile component. Also, as one might anticipate, the seriesare all strongly non-normal. All are leptokurtic, while the FTSE All-Share and commodities seriesare also significantly skewed to the left.

FORECASTING VOLATILITY

Construction of forecasts and notationThe total sample of 4865 observations is split into two parts: the first 1250 observations (approx-imately 5 years of daily trading data) are used for estimation of the parameters of the model, andthen one-, two-,. . ., twenty-step-ahead forecasts are calculated. The multi-step-ahead forecasts arethen aggregated to form forecasts of volatility over the next 5, 10, and 20 days. We can thus write5

�2t,N D

N∑nD1

�2t,tCn �1�

where �2t,N denotes the time t aggregated forecast for the next N steps, and �2

t,tCn denotes then-step-ahead forecast made at time t.

In contrast to much previous research in this area, these are not one-, two-, three-. . . twenty-step-ahead forecasts, but rather we aggregate the forecasts for the next 5, 10, and 20 days. Aggregatedforecasts will be the ones of interest to financial market practitioners and risk managers, when theyhave investment horizons longer than one day; they will not be particularly interested in multi-step-ahead one-day volatility forecasts, such as the volatility forecast for day t C 20 made on day t. Thesample is then rolled forward by removing the first observation of the sample and adding one tothe end, and another set of forecasts of the next twenty days’ volatilities is made, and aggregated.This ‘recursive’ modelling and forecasting procedure is repeated until a forecast for observation4865 has been made using data available at time 4845. Computation of forecasts using a rollingwindow of data should ensure that the forecasts are made using models whose parameters have

5 This step is permissible since the variances are additive over time. Another possibility would be to multiply the one-step-ahead forecast bythe desired horizon using an equivalent of the ‘square root of time’ rule, so that, for example, the volatility forecast over the next 20 daysis 20 times the forecast for tomorrow. However, our approach is likely to be superior, since it employs more information while implicitextrapolation of one-step forecasts could be inappropriate for a mean-reverting series.



been estimated using a sufficient span of time, while not incorporating such old vintages that thedata may no longer relevant in the context of an evolving financial market.

Forecasting modelsAlmost all of the forecasting models employed in this study are not new, rather it is the evaluationof the models which is novel. Hence the model descriptions are brief and presented in Table II,with �2

f,tCnj�t denoting the n-step-ahead (n D 1, 2, . . . , 20) forecast for the conditional varianceupon information available at time t, where t runs from observation 1250 to 4845. With onepossible exception, the model equations in Table II are self-explanatory, and readers are referred toBollerslev et al. (1992), Brailsford and Faff (1996), or Brooks (1998), and the references therein,for a more detailed treatment.

The only model which perhaps requires further explanation is the multivariate GARCH model,which has not been employed in previous studies of volatility forecast performance. The particularparameterization used here is of the diagonal VEC form due to Bollerslev, Engle, and Wooldridge(1988), where each element of the conditional variance covariance matrix hjk,t depends on pastvalues of itself and past values of εj,tε0

j,t, which may be written

vec�HtC1� D htC1 D C0 C A1vec�εtε0t� C B1ht �2�

where vec denotes the column stacking operator, A1 and B1 are restricted to be diagonal. Theparameterization for HtC1 conditional upon the information set allows each element of the condi-tional variance–covariance matrix to depend on lags of the squares and of the cross products ofthe elements of εtC1 as well as lags of the elements of HtC1.

EVALUATING VOLATILITY FORECASTS

Standard loss functionsThree criteria are used here to evaluate the accuracy of the forecasts: mean squared error (MSE),mean absolute error (MAE), and proportion of over-predictions. Mean squared error provides aquadratic loss function which disproportionately weights large forecast errors more heavily relativeto mean absolute error, and hence the former may be particularly useful in forecasting situationswhen large forecast errors are disproportionately more serious than small errors. The proportionof over-predictions should give a rough indication of the average direction of the forecast error(compared with the two previous measures which only give some measure of the average size) andwhether the models are persistently over- or under-predicting the ‘true’ value of volatility. Hencethis measure gives an approximate guide as to whether the forecasts are biased.

But what is volatility?Unlike financial asset returns, volatilities are not directly observable from the market. Consequently,when attempting to benchmark the accuracy of volatility forecasting models, researchers are nec-essarily required to make an auxiliary assumption about how the ex post or realized volatilities arecalculated. The vast majority of existing studies, including those listed in the introduction to thispaper, use squared returns of the frequency of the data and analysis, as the measure of realizedvolatility. For example, studies using daily data would assume that the ‘correct’ volatility number



Table II. Description of models used for forecasting

Model Acronym Equations for model Equation

1. Random walk in volatility RW �2f,tCn D �2

t (2)

2. Long-term mean LTM �2f,tCn D 1

1250

t∑jDt�1249

�2t�j (3)

3. Short-term moving average MA5 �2f,tCn D 1

5

4∑jD0

�2t�j (4)

4. Long-term moving average MA100 �2f,tCn D 1

100

99∑jD0

�2t�j (5)

5. Linear regression with one lag AR1 �2f,tCn D ˛0 C ˛1�

2t C εt (6)

6. Linear regression with AIC lags ARAIC �2f,tCn D ˇ0 C

p�1∑jD0

ˇj�2t�j C εt (7)

7. GARCH(1,1) GAR rtC1 D � C εtC1, εtC1 ¾ N�0, �2tC1�, (8)

�2f,tCn D �0 C ϕ1ε

2t C �2�

2t

8. GJR(1,1) GJR �2f,tCn D υ0 C υ1ε

2t C υ2�

2t C υ3S

�t ε2

t (9)S�

t D 1 for εt � 0 and 0 otherwise

9. EGARCH(1,1) EGAR log��2f,tCn� D ω1 C ω2 log��2

t � C ω3εt√�2

t

(10)

Cω4

j εt j√

�2t

�√

2

�

10. Long exponentially weightedmoving average

EMA5 �2f,tCn D �1 � 1�

5∑tD1

t�11 �rt � r� (11)

11. Short exponentially weightedmoving average

EMA100 �2f,tCn D �1 � 1�

100∑tD1

t�11 �rt � r� (12)

12. GARCH with t-distributed errors GART rtC1 D � C εtC1, εtC1 ¾ tk�0, �2tC1�, (13)

�2f,tCn D �0 C ϕ1ε

2t C �2�

2t

13. Multivariate GARCH MGAR See text for model description —

Notes : Forecast equations are given for n D 1 step ahead, and recursions can easily be computed from these for the 2, 3, . . . , 20 step-aheadforecasts. The model order p for ARAIC is determined individually for each forecast iteration by the minimization of Akaike’s informationcriterion, with maximal lag 5. All model parameters are estimated using quasi-maximum likelihood. The exponentially weighted movingaverage coefficients ( i) are chosen to produce the best fit by minimizing the sum of the squared in-sample forecast errors.

on day t is r2t , and it is this value that would be used as an input to the mean squared error calcu-

lation, or as the dependent variable in a Fair–Schiller (1990)-type regression of actual volatilitieson their forecasted values.

Whilst this method is simple and intuitively plausible, Andersen and Bollerslev (1998, hereafterAB) suggest that ‘same-frequency’ squared returns are an unbiased but extremely noisy measureof the latent volatility factor which underlies financial asset return movements. AB show that a



much better approximation to the latent volatility factor can be obtained by summing the squares ofhigher frequency returns. For example, a superior estimate of volatility on day t to r2

t is given by

r2Łt D

m∑jD1

r2t�1C�j/m� �3�

where m is an intra-day sampling frequency, such as 8 for hourly data.6

Unfortunately, for many applications, the usefulness of this method is limited by the lack ofavailability of a sufficiently long span of higher-frequency returns. In the present paper, however,our analysis focuses upon daily, weekly, bi-weekly, and monthly forecasts. For the latter threehorizons, two methods of calculating ex post volatility are available, both of which are employedin this study. The first of these ex post measures, which may usefully be termed the traditionalmeasure, is to use weekly, bi-weekly or monthly squared returns.7 The second method, would be totake the daily returns, square them, and sum them over the relevant (5-, 10-, or 20-day) horizon.8

As AB note, it is not necessarily the case that the two ex post measures will give the same modelrankings, let alone the same values of the error measures. Thus a comparison of model rankingsunder the two approaches is a relevant question for research, which this paper makes the firstattempt to address.

Value at Risk calculationGiven the voluminous literature which almost unquestioningly evaluates volatility forecasts usingstandard loss functions, three sensible questions to ask are first, what are volatility forecastsuseful for, second, what is an appropriate loss function given this usage, and finally, will alter-native loss functions lead to approval of the same or similar models? Some answers to the firstof these questions are provided in the introduction to this paper. One use of volatility predic-tions, which has grown substantially in importance over recent years, is as an input to financialrisk management. In this paper, we thus employ a relevant ‘risk management’ loss function,which is based upon the calculation of an institution’s value at risk, as defined above in thefirst section. Specifically, we calculate VaR for three individual assets by calculating the follow-ing quantity:

VARit�N, ˛� D [Fi

t,N]�1( ˛

100

)�4�

where VARit is the Value at Risk for a given asset at time t, determined from model i (where

i D 1, 13 are the models as defined above), N is the investment horizon, [Fit,N]�1 is a cumulative

distribution function (cdf) and ˛ is a percentage significance level. The cdf employed in this paperis that of a normal distribution.

A limiting assumption of the analysis in many empirical papers in risk management is the stan-dard assumption of normality, for it is well known that asset returns are not Gaussian. However,the normal approximation is extremely widely used in the risk management field. Fat-tailed returndistributions will lead the delta-normal model to understate the true value at risk (see Jorion, 1996or Huisman et al., 1998). For example, a 5% daily loss is observed to occur approximately once

6 Assuming, of course, that 8 hourly observations are available from the financial market concerned.7 So, for example, the volatility for weekly returns would be given by r2

t D [ln�Pt/Pt�5�]2.8 Obviously for the 1-day horizon, both methods will yield the same ex post measure.



every two years, while if returns were normally distributed, such a change would be expectedonly once every 1000 years (Johansen and Sornette, 1999). A number of methods to incorpo-rate the fat tails have been proposed, most importantly the use of extreme value distributions forreturns (e.g. Embrechts et al., 1999). However, we continue to employ the normality assumptionsince other distributional approaches usually do not directly employ a volatility estimate. There-fore our purpose of comparing between volatility forecasts when used for risk management wouldbe lost.

We employ both the 1% and 5% levels of significance. The former level has been selected bythe Basle Committee (1996) as the focus of attention, although the first percentile of a distributionis more difficult to estimate than the fifth, and thus the latter is the quantity which many securitiesfirms wish to employ (see JP Morgan, 1996). The VaR corresponding to 5% may be defined asthat amount of capital, expressed as a percentage of the initial value of the position, which will berequired to cover 95% of probable losses. In the case of the normal distribution, this quantity maybe calculated as

VARit�N, 5%� D 1.645�i

t,N �5�

where �it,N is the square root of the conditional variance forecast, made at time t for forecast horizon

N (N D 1, 5, 10, 20). We thus forecast volatility for some future period (t, N) and hence we calculatethe amount of capital required to cover expected losses on 95% or 99% of the investment horizons.The 95% confidence level is employed by the popular RiskMetrics risk measurement software,while the regulators require capital to cover 99% of losses.9

The calculation of the value at risk estimates for the individual assets is achieved by followingthe steps outlined above. In the case of the portfolio, however, for all forecasting models except themultivariate GARCH (that is, models 1–12 in Table II), we employ a method known as the ‘fullvaluation approach’. This simply involves the aggregation of the components and the calculationof the portfolio return at each point in time. In this case, the resulting portfolio return series ismodelled in the same way as the individual component assets.

An alternative approach is known as the ‘volatilities and correlations’ method, which has beenpopularized by JP Morgan (1996). Here, the portfolio value at risk is estimated using the volatilitiesof the individual assets which form the MCRR, and the correlations between their returns. Theportfolio value at risk may be written

MCRRP D

√√√√√√a2MCRR2

A C b2MCRR2B C c2MCRR2

CC2ab�ABMCRRAMCRRB

C2ac�ACMCRRAMCRRC

C2bc�BCMCRRBMCRRC

�6�

where A, B, and C denote the bond, stock and commodities series respectively, and a D b D c D1/3. We adopt this approach when using the multivariate GARCH model, but instead of usingthe time-invariant volatility and correlation estimates, we instead use the relevant forecasts of theconditional variances and covariances from the MGARCH model in (6) to derive the VaR.

9 In fact, the 99% VaR is multiplied by a ‘scaling factor’, which is usually 3, so that the actual coverage rate is considerably higher than99%. We do not employ the regulatory scaling factor in our analysis, so as to focus upon forecast adequacy. Multiplying the estimated VaRby 3 has the effect of rendering the forecasted VaRs virtually indistinguishable from one another, since the implied coverage rate is nowmore than 99.99%.



Risk management-based forecast evaluationsIn this paper, we employ three methods for determining the adequacy of the volatility forecaststhat are used as an input to the value at risk calculation. All methods essentially require thecalculation of VaR, and then assuming that the securities firm had employed this much capital, themethods track the actual realized losses during an out-of-sample period. The simplest approach todetermining model adequacy in the risk management framework is to calculate the time until firstfailure (TUFF), defined as the first observation in the hold-out sample where the capital held isinsufficient to absorb that period’s loss, and derived as follows. Following Kupiec (1995), let pdenote the realized probability of observing the first failure of the model in period V, and lettingQR be a random variable that denotes the number of observations until the first failure is recorded,then we may write

Pr� QR D V� D p�1 � p�V�1 �7�

Then QR has a geometric distribution with an expected value of 1/p. This quantity can be interpretedas the expected number of observations until the first failure is observed. In the cases of interestin this paper, if the actual proportion of failures were 5% and 1% respectively, then the time untilfirst failure would be 100 and 20 steps respectively. If we now let p* denote the probability offailure under the null hypothesis, then the following likelihood ratio test can be established:

TUFF �V, pŁ� D �2 log[p Ł �1 � pŁ�V�1] C 2 log

[1

V

(1 � 1

V

)V�1]

�8�

which is �2(1) under the null. Given the appropriate critical value, it is possible to derive a 95%confidence interval for TUFF of (6,439) for the 1% VaR and (�,87) for the 5% VaR.10 Theconfidence intervals can be interpreted as follows. If VaR determined using a 1% significance levelfails before the 6th observation, we can reject at the 5% level the null hypothesis that the model isadequate to cover losses on 99% of occasions. On the other hand, if the actual TUFF is greater than439, then we would conclude that the model was leading to too high a value at risk, and thereforethat the model was not failing as quickly as would be expected given the nominal 1% probabilityof failure.

It is perhaps worth noting that it is desirable from the point of view of the bank or securitiesfirm concerned, for the calculated value at risk to be neither too large nor too small. A value atrisk set too low could imply that the bank does not have sufficient capital to cover future losses,leading at best to regulatory scrutiny, and an increase in the scaling factor (resulting in a substantialincrease in the capital requirement), and at worst to financial distress and possible company failure.Conversely, a VaR set too high, so that it covers more than the nominal percentage of horizons(e.g. an estimated 5% daily VaR which is actually sufficient to cover 99.9% of the out of sampleperiods), probably implies that the firm is tying up too much of its capital unnecessarily in anunprofitable fashion.11

Whilst intuitive and simple to calculate, TUFF has obvious flaws as an evaluation metric. First,it is clearly not using much information from the sample, since all observations after the firstfailure are ignored, resulting in the test being over-sized. Thus, if the start of the out-of-sample

10 It is not possible to establish a lower limit for the 5% VaR interval.11 Particularly in view of the regulatory scaling factor, which multiplies the firm’s own VaR estimate by at least 3.



period occurs at a time of exceptional market turbulence, a model which may have been perfectlyadequate for the rest of the sample and incurring no further failures, would be rejected. Second,the TUFF statistic consequently has low power to reject models which are not adequate—thisis clearly evidenced by the wide confidence intervals for TUFF presented above. For example, a99% nominal coverage rate is expected to result in first failure at observation 100, but even if anexceedence of the VaR is recorded as early as observation 7, we cannot reject the underlying modelat the 1% level; thus TUFF will have low ability to discriminate between volatility forecasts fromdifferent models.

Another simple method for determining model adequacy within the risk management frameworkis simply to calculate the percentage of times that the calculated VaR is insufficient to coverthe actual losses, during the rolling out-of-sample period. A good model would be one whoseproportion of out-of-sample exceedences is close to the nominal value of (one minus coverageprobability)% assumed (5% or 1%). We can also formulate a likelihood ratio test for the proportionof failures, in similar vein to (19) above. The probability of observing x failures in an actual sampleof independent observations of size K will be distributed binomially, leading to the following teststatistic distributed �2(1) under the null:

UCF�K, x, pŁ� D �2 log[�1 � pŁ�K�x�pŁ�x] C 2 log[(

1 �( x

K

))K�x ( x

K

)x]

�9�

with notation as above. For ease of interpretation of the results, models are also ranked in thefollowing way. We assume that any model which has a percentage of exceedences in the rolling hold-out sample which is greater than the nominal threshold should be rejected as inadequate. Therefore,the lowest ranking models (classified as worst) are those which have the highest percentage offailures greater than the nominal value.

When these models have been exhausted, we assume further that any model which generates farfewer exceedencess than the expected number is less desirable than a model which generates closerto the nominal number. Thus the best models under this loss function are those which generate lessthan, but closest to, the assumed coverage rate.12

RESULTS

Statistical evaluation criteriaThe results for the volatility forecasts under standard statistical evaluation methods (percentage ofover-predictions, mean squared error, and mean absolute error) are presented in Tables III to VIfor the government bond, FTA All-Share, commodities and portfolio series respectively.

Considering first the one-step-ahead (1-day) forecast horizon, a number of important featuresemerge. As one might anticipate, the random walk in volatility model produces roughly equalnumbers of over-and under-predictions of realized volatility measured by the squared daily returns.On the other hand, all models over-predict volatility on average 70% of the time, except for thetwo EWMAs which over-predict more frequently than they under-predict. In all other respects, therandom walk in volatility produces uniformly poor forecasts.

12 Of course, this could be replaced by a simpler symmetric or any other loss function if the user desired.



Tabl

eII

I.St

atis

tical

loss

func

tions

for

gove

rnm

ent

bond

(sam

efr

eque

ncy

squa

red

retu

rns

asex

post

mea

sure

)

Step

s1

510

20

Mod

els

%O

PR

ank

MSE

Ran

kM

AE

Ran

k%

OP

Ran

kM

SER

ank

MA

ER

ank

%O

PR

ank

MSE

Ran

kM

AE

Ran

k%

OP

Ran

kM

SER

ank

MA

ER

ank

RW

49.8

81

0.03

612

0.08

212

45.3

12

0.92

713

0.42

513

43.0

93

3.14

613

0.85

413

41.5

95

14.2

513

1.90

313

LTM

76.9

713

0.02

01D

0.06

99D

70.8

211

0.48

53D

0.33

710

69.3

511

1.35

73

0.67

36D

67.1

210

6.36

21D

1.49

46

MA

567

.01

30.

021

80.

068

7D59

.36

30.

545

90.

352

1157

.02

41.

693

100.

717

1155

.74

37.

923

101.

617

12M

A10

073

.63

60.

020

1D0.

066

3D67

.23

90.

485

3D0.

329

5D64

.34

91.

406

50.

673

6D62

.42

76.

777

41.

515

9A

R1

76.4

411

D0.

020

1D0.

069

9D70

.90

120.

484

20.

336

969

.37

121.

354

20.

672

567

.15

116.

362

1D1.

493

5A

RA

IC75

.22

100.

020

1D0.

068

7D69

.76

100.

481

10.

334

7D68

.73

101.

349

10.

668

466

.98

96.

393

31.

495

7DG

AR

73.8

57

0.02

01D

0.06

63D

65.9

57

0.48

96

0.32

74

60.8

95

1.44

04

0.66

13

57.3

34

7.07

16

1.48

24

GJR

76.4

411

D0.

024

10D

0.07

611

62.7

85

0.54

710

0.34

32

54.3

51

1.56

58

0.67

48D

46.5

72

7.61

88

1.48

02

EG

AR

69.9

65

0.17

013

0.13

513

48.0

11

0.79

612

0.41

212

37.8

36

2.00

012

0.77

212

28.7

612

8.32

411

1.64

010

EM

A5

43.4

82

0.02

29

0.05

61

37.8

94

0.54

28

0.30

21

35.8

68

1.62

89

0.62

71

34.3

88

7.87

09

1.44

01

EM

A10

031

.77

40.

024

10D

0.06

12

26.8

713

0.59

711

0.33

47D

25.7

613

1.86

011

0.70

210

24.5

313

8.76

512

1.57

611

GA

RT

74.1

68

0.02

01D

0.06

63D

67.2

08

0.48

53D

0.32

95D

63.4

27

1.43

06

0.67

48D

60.3

36

6.92

05

1.49

57D

MG

AR

74.8

39

0.02

01D

0.06

76

63.6

76

0.49

67

0.32

13

56.5

82

1.48

67

0.65

42

51.0

21

7.41

87

1.48

13

Tabl

eIV

.St

atis

tical

loss

func

tions

for

equi

ties

(sam

efr

eque

ncy

squa

red

retu

rns

asex

post

mea

sure

)

Step

s1

510

20

Mod

els

%O

PR

ank

MSE

Ran

kM

AE

Ran

k%

OP

Ran

kM

SER

ank

MA

ER

ank

%O

PR

ank

MSE

Ran

kM

AE

Ran

k%

OP

Ran

kM

SER

ank

MA

ER

ank

RW

50.2

11

0.42

09D

0.17

511

45.5

41

18.2

613

1.03

212

43.1

46

103.

213

2.30

813

41.0

36

440.

213

5.34

413

LTM

75.1

913

0.41

57D

0.15

69D

69.0

112

15.0

26

0.90

68D

65.9

012

78.8

73

1.92

59

62.3

99

313.

34

4.44

29

MA

563

.95

30.

389

50.

148

357

.89

415

.26

90.

904

754

.88

479

.81

81.

913

850

.38

131

5.9

54.

333

4M

A10

069

.37

60.

412

60.

156

9D63

.03

915

.07

70.

906

8D60

.25

779

.38

41.

977

1156

.44

531

3.0

2D4.

352

5A

R1

71.8

511

0.38

74

0.15

15

67.8

711

14.5

71

0.88

65

65.8

111

78.2

11

1.90

87

62.2

08

313.

02D

4.43

28

AR

AIC

70.3

28

0.41

57D

0.15

37

67.0

110

14.9

74

0.89

06

65.4

810

78.7

72

1.90

76

62.1

17

312.

71

4.42

77

GA

R70

.68

90.

342

10.

142

261

.25

814

.93

30.

842

355

.36

579

.47

61.

808

344

.70

431

8.9

74.

275

1G

JR72

.05

120.

420

9D0.

149

457

.52

315

.13

80.

830

146

.98

180

.05

91.

796

234

.94

1032

3.8

104.

363

6E

GA

R66

.15

40.

796

120.

274

1245

.40

215

.54

100.

929

1136

.97

880

.81

101.

984

1230

.63

1232

3.5

94.

610

12E

MA

543

.98

22.

859

130.

443

1339

.50

617

.38

121.

147

1335

.10

980

.89

111.

897

532

.63

1132

4.1

114.

517

10E

MA

100

32.1

85

0.47

511

0.15

58

29.3

513

15.9

111

0.92

210

26.9

813

81.3

712

1.94

310

24.5

613

325.

712

4.53

311

GA

RT

70.1

87

0.34

32

0.13

91

60.8

37

14.9

02

0.83

42

54.7

43

79.5

27

1.79

31

44.9

83

319.

88

4.28

62

MG

AR

71.7

410

0.37

03

0.15

26

60.2

05

15.0

15

0.85

74

54.2

42

79.4

25

1.85

04

45.7

02

317.

26

4.32

13



Tabl

eV

.St

atis

tical

loss

func

tions

for

com

mod

itie

s(s

ame

freq

uenc

ysq

uare

dre

turn

sas

expo

stm

easu

re)

Step

s1

510

20

Mod

els

%O

PR

ank

MSE

Ran

kM

AE

Ran

k%

OP

Ran

kM

SER

ank

MA

ER

ank

%O

PR

ank

MSE

Ran

kM

AE

Ran

k%

OP

Ran

kM

SER

ank

MA

ER

ank

RW

50.0

71

0.14

611

0.09

110

45.4

05

2.68

313

0.44

612

43.5

92

10.7

813

0.95

612

40.9

54

45.6

313

2.12

413

LTM

78.2

513

0.08

21D

0.07

62D

73.6

313

0.83

13

0.33

47

69.8

510

3.08

64

0.70

56D

66.3

18

16.1

74

1.57

55D

MA

567

.23

40.

095

7D0.

084

8D61

.53

71.

049

90.

369

959

.11

53.

736

90.

781

1155

.58

217

.51

71.

712

10M

A10

076

.97

80.

083

50.

079

5D72

.13

100.

835

40.

345

868

.71

93.

076

10.

721

864

.23

515

.89

11.

616

8A

R1

77.4

111

0.08

21D

0.07

51

73.3

511

0.83

02

0.33

35D

69.9

613

3.08

53

0.70

56D

66.2

06

16.1

63

1.57

67

AR

AIC

77.2

59

0.08

21D

0.07

62D

73.4

612

0.82

91

0.33

35D

69.9

011

D3.

084

20.

704

566

.23

716

.14

21.

575

5DG

AR

76.5

56

0.09

57D

0.08

17

54.9

46

0.89

48

0.29

63

41.9

83

3.34

97

0.64

12

29.0

710

17.8

58

1.53

41

GJR

77.7

512

0.10

89

0.08

48D

46.2

04

0.88

77

0.29

02

34.4

48

3.34

16

0.64

33

23.8

413

17.8

69

1.55

33

EG

AR

72.6

35

0.34

412

0.15

613

53.1

03

1.47

210

0.43

010

41.6

44

3.78

310

0.75

19

32.8

59

18.2

811

1.63

59

EM

A5

55.9

72

0.12

410

0.11

511

50.5

41

1.79

611

0.53

013

47.8

41

5.45

512

0.99

313

44.4

03

16.6

85

1.78

512

EM

A10

037

.36

30.

353

130.

125

1231

.15

91.

838

120.

434

1130

.10

11D

3.78

611

0.75

810

27.6

511

18.8

712

1.71

311

GA

RT

77.2

710

0.08

21D

0.07

74

51.4

12

0.86

55

0.28

41

39.0

37

3.37

48

0.63

91

26.2

912

18.0

010

1.54

62

MG

AR

76.8

07

0.09

36

0.07

95D

66.6

88

0.88

16

0.33

04

60.4

26

3.28

95

0.69

74

51.4

31

17.0

76

1.56

54

Tabl

eV

I.St

atis

tical

loss

func

tions

for

port

foli

o(s

ame

freq

uenc

ysq

uare

dre

turn

sas

expo

stm

easu

re)

Step

s1

510

20

Mod

els

%O

PR

ank

MSE

Ran

kM

AE

Ran

k%

OP

Ran

kM

SER

ank

MA

ER

ank

%O

PR

ank

MSE

Ran

kM

AE

Ran

k%

OP

Ran

kM

SER

ank

MA

ER

ank

RW

49.8

81

0.03

612

0.08

212

47.5

71

0.39

913

0.24

512

44.8

43

1.63

713

0.52

012

42.1

73

6.63

713

1.14

412

LTM

76.9

713

0.02

01D

0.06

99D

70.4

010

0.20

01

0.19

67

68.4

610

0.75

61

0.40

86

62.8

77

2.92

71

0.90

04D

MA

567

.01

30.

021

80.

068

7D60

.81

40.

259

100.

209

1158

.03

40.

955

100.

441

1053

.21

13.

420

80.

969

9M

A10

073

.63

60.

020

1D0.

066

3D66

.98

70.

201

20.

193

663

.89

70.

764

20.

405

558

.72

43.

035

40.

914

8A

R1

76.4

411

D0.

020

1D0.

069

9D69

.65

90.

212

50.

197

8D68

.26

90.

772

30.

409

762

.92

82.

950

30.

902

6A

RA

IC75

.22

100.

020

1D0.

068

7D69

.40

80.

218

6D0.

197

8D68

.01

80.

777

40.

410

862

.70

62.

930

20.

900

4DG

AR

73.8

57

0.02

01D

0.06

63D

60.8

95

0.21

04

0.18

23D

51.6

31

0.82

37

0.38

74

40.3

95

3.39

97

0.89

43

GJR

76.4

411

D0.

024

10D

0.07

611

54.0

82

0.22

48

0.16

11

39.9

45

0.78

45

0.32

71

28.1

010

3.06

45

0.79

51

EG

AR

69.9

65

0.17

013

0.13

513

44.7

03

0.21

86D

0.17

12

36.1

66

0.86

48

0.38

02

29.7

99

3.58

110

0.90

77

EM

A5

43.4

82

0.02

29

0.05

61

28.4

311

0.23

19

0.19

15

25.4

011

0.90

09

0.41

59

22.0

312

3.70

011

0.97

110

EM

A10

031

.77

40.

024

10D

0.06

12

20.2

212

0.28

912

0.20

310

18.4

712

1.16

412

0.44

511

15.9

413

3.88

812

0.99

411

GA

RT

74.1

68

0.02

01D

0.06

63D

62.8

76

0.20

53

0.18

23D

54.2

72

0.80

26

0.38

33

43.3

92

3.33

46

0.88

52

MG

AR

74.8

39

0.02

01D

0.06

76

87.8

213

0.28

211

0.34

613

83.2

013

0.99

711

0.65

913

75.7

411

3.48

99

1.26

113



No clear ‘winners’ emerge at the 1-day horizon, with different models being preferred for eachseries. MSE is clearly not a good discriminator at the top end, with many models ranked equallyas the best. MAE, on the other hand, selects EWMA models for the bond and portfolio series,while for the share and commodity series, the GJR and autoregressive volatility models are pre-ferred. In terms of the least accurate next-day forecasting models, the random walk in volatilityand EGARCH models emerge as the worst performers, followed by the EWMAs for commoditiesand shares, although the latter proved the most accurate for the other two series.

An extension of the forecast and investment horizon to the one (trading) week, two-week, and one-month range does not markedly alter the relative model rankings, although the broad disagreementbetween criteria for a given series and model is still apparent. For example, the autoregressivemodel, which ranks only seventh by MSE for the equities series at the one-day horizon, ranks firstwhen the investment horizon is extended to one month.

However, as Andersen and Bollerslev (1998) have shown, the use of low-frequency squaredreturns is often not a useful way to evaluate volatility forecasts, and it is quite possible that whensums of higher-frequency squared returns are used instead as the ex post volatility measure, notonly the values of the error measures but also the model rankings could change substantially.Thus for the 5-, 10-, and 20-day periods we also evaluate the forecast accuracies using the sum ofsquared daily returns. Results are presented for the bond, share, commodities, and portfolio series inTables VII to X respectively.13 Comparing the results for the low-frequency squared returns versusthe high-frequency sums of squared returns, we note first that the values of the error measures areas expected reduced considerably.14

The GARCH model with t-distributed errors now emerges as the clear winner, producing themost accurate forecasts according to MAE, for three of the four series (bonds, stocks, and theportfolio). Only for the commodities return series does GARCH-t perform poorly. For the latterseries, the long-term mean and autoregressive volatility models prove to be the best under bothsquared and absolute error measures. Interestingly, the worst models seem invariant to both the useof a same-frequency or higher-frequency ex post measure, and to whether the errors are squaredor the absolute values taken; a bad model appears to be a bad model whatever. Models which fitinto this category are the random walk in volatility, the exponential GARCH, and the exponentiallyweighted moving average model.

Risk management evaluation criteriaThe corresponding evaluations for the forecasts when used in a risk management context are givenin Tables XI to XVIII. Volatility forecasts can be employed for the production of 99% and 95%nominal coverage rates for the value at risk estimates. In other words, forecasts are generatedin respect of the amount of capital required to cover expected losses on 99% and 95% of daysrespectively. The results for these two sets of nominal coverage rates are provided in Tables XI toXIV and XV to XVIII respectively for the 1-day, 1-week, 2-week and 1-month horizons. Threestatistics are presented in each table—the time until first failure (TUFF), the proportion of failures(FT), and the test statistic associated with whether this proportion of failures is significantly higherthan the nominal rate (UCF). Also given are the model rankings according to FT and UCF15 asdescribed above.

13 Of course, the results for the one-step-ahead evaluations will be identical to those of Tables III to VI.14 Mean squared errors are reduced by roughly an order equivalent to the forecasting horizon, while absolute errors are reduced by a factorof around two for all horizons.15 The rankings according to FT and UCF will, of course, by definition be identical.



Tabl

eV

II.

Stat

istic

allo

ssfu

nctio

nsfo

rgo

vern

men

tbo

nd(s

umof

daily

squa

red

retu

rns

asex

post

mea

sure

)

Step

s1

510

20

Mod

els

%O

PR

ank

MSE

Ran

kM

AE

Ran

k%

OP

Ran

kM

SER

ank

MA

ER

ank

%O

PR

ank

MSE

Ran

kM

AE

Ran

k%

OP

Ran

kM

SER

ank

MA

ER

ank

RW

49.8

81

0.03

612

0.08

212

35.2

76

0.55

313

0.34

413

32.1

87

2.05

213

0.66

013

30.4

09

8.04

213

1.30

813

LTM

76.9

713

0.02

01D

0.06

99D

71.0

710

0.14

37

0.23

28

68.6

59

0.37

66

0.40

68

68.2

96

1.04

25

0.72

57

MA

567

.01

30.

021

80.

068

7D51

.18

10.

181

80.

231

746

.62

20.

561

100.

407

943

.26

31.

716

90.

788

9M

A10

073

.63

60.

020

1D0.

066

3D64

.17

50.

137

50.

214

461

.14

50.

359

40.

369

460

.67

41.

016

30.

667

3A

R1

76.4

411

D0.

020

1D0.

069

9D71

.10

110.

141

60.

230

668

.51

80.

372

50.

403

768

.32

71.

036

40.

722

6A

RA

IC75

.22

100.

020

1D0.

068

7D70

.38

90.

136

40.

222

568

.82

100.

354

30.

387

568

.35

81.

001

10.

705

5G

AR

73.8

57

0.02

01D

0.06

63D

62.4

54

0.13

02

0.20

12

56.0

53

0.34

72

0.34

11

47.4

31

1.07

76

0.63

32

GJR

76.4

411

D0.

024

10D

0.07

611

57.4

12

0.18

610

0.23

39

43.8

74

0.47

28

0.40

16

29.7

410

1.49

58

0.76

18

EG

AR

69.9

65

0.17

013

0.13

513

32.1

08

0.42

112

0.33

312

20.4

711

0.90

812

0.60

312

15.0

811

2.69

012

1.18

012

EM

A5

43.4

82

0.02

29

0.05

61

18.8

912

0.18

49

0.23

710

14.6

312

0.54

49

0.45

810

12.7

412

1.73

710

0.89

610

EM

A10

031

.77

40.

024

10D

0.06

12

9.04

013

0.22

811

0.27

711

6.89

813

0.73

111

0.54

911

6.37

013

2.53

511

1.09

311

GA

RT

74.1

68

0.02

01D

0.06

63D

65.3

17

0.12

81

0.20

43

61.2

56

0.33

61

0.34

52

55.1

62

1.00

82

0.62

81

MG

AR

74.8

39

0.02

01D

0.06

76

58.8

03

0.13

53

0.20

01

47.9

01

0.37

77

0.34

73

35.7

45

1.20

47

0.67

14

Tabl

eV

III.

Stat

istic

allo

ssfu

nctio

nsfo

req

uiti

es(s

umof

daily

squa

red

retu

rns

asex

post

mea

sure

)

Step

s1

510

20

Mod

els

%O

PR

ank

MSE

Ran

kM

AE

Ran

k%

OP

Ran

kM

SER

ank

MA

ER

ank

%O

PR

ank

MSE

Ran

kM

AE

Ran

k%

OP

Ran

kM

SER

ank

MA

ER

ank

RW

50.2

11

0.42

09D

0.17

511

36.4

76

9.77

513

0.77

412

34.7

26

40.4

013

1.55

013

33.0

24

161.

313

3.14

713

LTM

75.1

913

0.41

57D

0.15

69D

72.1

611

4.83

37

0.58

29

70.7

77

13.0

06

1.06

59

71.9

98

34.2

47

2.00

18

MA

563

.95

30.

389

50.

148

351

.18

14.

898

90.

522

546

.43

413

.17

80.

937

541

.47

133

.72

31.

734

4M

A10

069

.37

60.

412

60.

156

9D64

.03

74.

891

80.

541

762

.75

513

.28

90.

991

758

.67

232

.30

11.

657

3A

R1

71.8

511

0.38

74

0.15

15

71.1

08

4.61

41

0.55

28

70.6

38

12.6

21

1.02

98

71.5

76

33.8

44

1.96

17

AR

AIC

70.3

28

0.41

57D

0.15

37

71.2

99

4.72

94

0.53

16

71.8

210

12.6

93

0.98

56

71.7

97

33.6

72

1.89

96

GA

R70

.68

90.

342

10.

142

259

.39

54.

677

30.

450

247

.96

112

.67

20.

789

126

.90

934

.12

51.

587

1G

JR72

.05

120.

420

9D0.

149

451

.41

24.

773

60.

464

329

.93

913

.10

70.

888

46.

6512

36.7

99

2.03

19

EG

AR

66.1

54

0.79

612

0.27

412

27.9

610

5.25

710

0.67

811

20.7

511

14.0

211

1.27

512

17.5

810

37.9

311

2.49

512

EM

A5

43.9

82

2.85

913

0.44

313

20.8

312

7.32

812

0.86

713

14.0

812

13.8

810

1.14

510

12.0

211

37.8

210

2.27

010

EM

A10

032

.18

50.

475

110.

155

810

.35

135.

540

110.

663

107.

455

1314

.06

121.

204

116.

175

1338

.01

122.

369

11G

AR

T70

.18

70.

343

20.

139

159

.05

44.

674

20.

443

147

.34

312

.79

50.

792

228

.93

534

.35

81.

603

2M

GA

R71

.74

100.

370

30.

152

658

.47

34.

743

50.

475

447

.84

212

.74

40.

867

336

.89

334

.16

61.

741

5



Tabl

eIX

.St

atis

tical

loss

func

tions

for

itco

mm

oditi

es(s

umof

daily

squa

red

retu

rns

asex

post

mea

sure

)

Step

s1

510

20

Mod

els

%O

PR

ank

MSE

Ran

kM

AE

Ran

k%

OP

Ran

kM

SER

ank

MA

ER

ank

%O

PR

ank

MSE

Ran

kM

AE

Ran

k%

OP

Ran

kM

SER

ank

MA

ER

ank

RW

50.0

71

0.14

611

0.09

110

34.5

57

2.42

513

0.42

312

30.5

24

8.99

913

0.82

812

27.4

07

34.2

113

1.63

013

LTM

78.2

513

0.08

21D

0.07

62D

73.8

312

0.50

81D

0.26

85D

71.1

37

1.05

91

0.46

63

68.5

74

2.29

33

0.80

23

MA

567

.23

40.

095

7D0.

084

8D51

.13

10.

735

90.

320

945

.31

21.

739

100.

587

940

.72

13.

746

91.

044

6M

A10

076

.97

80.

083

50.

079

5D68

.23

80.

523

40.

289

864

.70

31.

120

40.

508

660

.36

22.

557

40.

890

5A

R1

77.4

111

0.08

21D

0.07

51

73.2

79D

0.50

81D

0.26

74

71.2

48

1.06

02D

0.46

51D

68.6

55

2.29

22

0.80

12

AR

AIC

77.2

59

0.08

21D

0.07

62D

73.2

79D

0.50

81D

0.26

85D

71.1

06

1.06

02D

0.46

51D

68.7

66

2.29

11

0.80

01

GA

R76

.55

60.

095

7D0.

081

741

.25

20.

586

80.

255

216

.63

101.

306

60.

501

53.

143

123.

378

61.

086

7G

JR77

.75

120.

108

90.

084

8D26

.70

110.

575

6D0.

260

37.

677

131.

312

70.

542

82.

337

133.

472

81.

152

9E

GA

R72

.63

50.

344

120.

156

1336

.33

41.

156

100.

396

1019

.94

91.

758

110.

634

1011

.60

93.

902

101.

195

10E

MA

555

.97

20.

124

100.

115

1135

.13

51.

556

120.

519

1329

.46

53.

817

120.

891

1324

.42

84.

358

121.

329

12E

MA

100

37.3

63

0.35

313

0.12

512

14.4

713

1.51

011

0.42

111

11.2

712

1.68

89

0.65

511

10.1

310

4.20

811

1.25

011

GA

RT

77.2

710

0.08

21D

0.07

74

34.9

96

0.55

05

0.24

91

13.0

711

1.31

38

0.52

07

4.53

411

3.45

67

1.11

38

MG

AR

76.8

07

0.09

36

0.07

95D

61.3

13

0.57

56D

0.27

17

50.0

41

1.24

05

0.47

74

38.6

13

2.92

75

0.88

14

Tabl

eX

.St

atis

tical

loss

func

tions

for

port

foli

o(s

umof

daily

squa

red

retu

rns

asex

post

mea

sure

)

Step

s1

510

20

Mod

els

%O

PR

ank

MSE

Ran

kM

AE

Ran

k%

OP

Ran

kM

SER

ank

MA

ER

ank

%O

PR

ank

MSE

Ran

kM

AE

Ran

k%

OP

Ran

kM

SER

ank

MA

ER

ank

RW

49.8

81

0.01

29D

0.04

511

35.8

65

0.25

713

0.19

912

33.1

04

0.99

713

0.38

812

31.4

36

3.85

013

0.76

512

LTM

74.9

110

0.00

91D

0.03

87D

71.1

09

0.08

82

0.12

36

69.0

47D

0.20

81

0.21

26

68.1

05

0.49

42

0.36

83

MA

565

.09

30.

010

4D0.

039

951

.07

20.

136

100.

140

946

.62

10.

379

100.

250

743

.53

10.

837

90.

462

7M

A10

071

.54

50.

009

1D0.

037

4D64

.62

60.

090

30.

118

363

.17

30.

217

40.

206

360

.06

20.

543

40.

371

4A

R1

72.3

59

0.01

04D

0.03

87D

69.6

88

0.09

34

0.12

25

68.9

96

0.21

42

0.21

05

67.7

34

0.49

53

0.36

42

AR

AIC

71.9

36D

0.01

811

0.04

010

69.2

17

0.09

76

0.12

14

69.0

47D

0.21

63

0.20

94

67.3

73

0.49

01

0.36

21

GA

R72

.21

80.

010

4D0.

037

4D50

.77

10.

095

50.

109

232

.99

50.

242

60.

203

214

.35

90.

669

60.

439

6G

JR75

.05

110.

044

130.

046

1236

.41

40.

111

80.

137

822

.31

90.

275

80.

269

916

.36

80.

770

70.

538

8E

GA

R60

.33

20.

010

4D0.

033

117

.86

100.

100

70.

125

74.

089

120.

273

70.

261

80.

362

130.

788

80.

548

9E

MA

530

.71

40.

010

4D0.

035

28.

567

110.

112

90.

159

105.

953

110.

309

90.

317

105.

007

110.

917

100.

636

10E

MA

100

22.0

912

0.01

29D

0.03

63

4.36

713

0.16

311

0.17

511

3.28

213

0.53

112

0.35

211

2.97

612

0.96

411

0.65

511

GA

RT

71.9

36D

0.00

91D

0.03

74D

54.1

33

0.08

71

0.10

71

37.5

82

0.21

95

0.19

11

16.9

47

0.61

25

0.40

75

MG

AR

91.3

213

0.03

212

0.08

113

94.5

512

0.17

312

0.29

913

91.2

910

0.48

411

0.52

513

85.7

910

1.27

612

0.91

013



Tabl

eX

I.G

over

nmen

tbo

nd(r

isk

man

agem

ent

eval

uatio

n—

1%V

aR)

Step

s1

510

20

Mod

els

TU

FFFT

UC

FR

ank

TU

FFFT

UC

FR

ank

TU

FFFT

UC

FR

ank

TU

FFFT

UC

FR

ank

RW

312

.41

640.

712

111

.77

588.

612

110

.96

524.

610

19.

990

450.

111

LTM

331.

613

5.00

47D

396

0.96

70.

011

143

70.

974

0.01

11D

429

1.16

80.

424

1DM

A5

233.

255

50.3

39

983.

227

49.2

99

942.

726

31.9

48

33.

004

41.2

57

MA

100

231.

530

3.81

14D

221.

307

1.35

94

221.

335

1.60

35

429

1.47

43.

095

4A

R1

331.

613

5.00

47D

397

0.94

60.

047

243

70.

974

0.01

11D

429

1.16

80.

424

1DA

RA

IC33

1.53

03.

811

4D39

71.

196

0.57

13

437

0.94

60.

047

342

91.

196

0.57

13

GA

R23

1.50

23.

445

2D22

1.58

64.

590

626

71.

808

8.30

96

428

2.28

118

.99

6G

JR23

1.44

62.

762

122

2.78

233

.73

839

73.

922

77.4

99

145.

814

172.

99

EG

AR

121

3.39

455

.65

1020

8.73

435

9.3

106

12.1

361

7.9

123

14.5

282

1.2

12E

MA

515

11.2

154

4.2

1120

11.6

657

9.6

1121

11.4

956

6.3

114

9.84

743

9.7

10E

MA

100

318

.55

1195

131

18.0

511

4713

118

.11

1152

131

16.1

696

9.0

13G

AR

T23

1.50

23.

445

2D22

1.33

51.

603

543

41.

224

0.73

84

141.

558

4.19

35

MG

AR

231.

558

4.19

36

212.

031

12.9

07

212.

420

22.7

67

903.

727

69.1

68

Tabl

eX

II.

Equ

ities

(ris

km

anag

emen

tev

alua

tion

—1%

VaR

)

Step

s1

510

20

Mod

els

TU

FFFT

UC

FR

ank

TU

FFFT

UC

FR

ank

TU

FFFT

UC

FR

ank

TU

FFFT

UC

FR

ank

RW

911

.79

590.

812

3010

.99

526.

712

4410

.24

468.

910

649.

708

429.

410

LTM

444

1.69

76.

334

744

01.

752

7.29

32

713

1.50

23.

445

370

31.

975

11.6

83

MA

544

2.42

022

.76

912

23.

700

67.9

99

397

3.47

758

.93

711

73.

978

79.9

35

MA

100

231.

641

5.43

25D

302.

197

16.8

64

713

1.72

56.

806

470

32.

197

16.8

64

AR

123

1.64

15.

432

5D44

01.

780

7.79

43

713

1.44

62.

762

270

31.

947

11.0

82

AR

AIC

397

1.78

07.

794

844

01.

613

5.00

41

713

1.25

20.

925

170

31.

780

7.79

41

GA

R23

1.39

12.

148

213

22.

420

22.7

65

397

2.86

536

.48

611

84.

840

120.

86

GJR

231.

419

2.44

63

303.

366

54.5

78

126

4.59

010

8.4

910

97.

594

281.

89

EG

AR

163.

115

45.2

010

308.

651

353.

410

4413

.05

694.

012

6515

.83

938.

512

EM

A5

2310

.79

511.

611

3011

.13

537.

611

119

12.1

061

5.6

1111

410

.85

515.

911

EM

A10

016

17.1

610

6213

3017

.25

1070

1368

17.8

311

2513

108

16.3

398

4.5

13G

AR

T23

1.47

43.

095

413

22.

531

25.9

66

397

2.69

831

.06

511

85.

007

129.

37

MG

AR

231.

307

1.35

91

302.

587

27.6

27

121

3.53

361

.15

810

96.

203

195.

48



Tabl

eX

III.

Com

mod

ities

(ris

km

anag

emen

tev

alua

tion

—1%

VaR

)

Step

s1

510

20

Mod

els

TU

FFFT

UC

FR

ank

TU

FFFT

UC

FR

ank

TU

FFFT

UC

FR

ank

TU

FFFT

UC

FR

ank

RW

115

.02

865.

612

116

.69

1018

1255

17.3

910

8312

4520

.17

1355

9LT

M1

1.69

76.

334

136

1.28

01.

132

1D67

1.72

56.

806

159

2.39

221

.99

1M

A5

364.

312

95.0

810

364.

256

92.4

96

615.

035

130.

75

537.

510

276.

45

MA

100

12.

114

14.8

38

361.

864

9.37

94

692.

142

15.4

94

355

2.61

528

.47

4A

R1

11.

864

9.37

92D

361.

280

1.13

21D

671.

752

7.29

32D

592.

420

22.7

62D

AR

AIC

11.

864

9.37

92D

361.

280

1.13

21D

671.

752

7.29

32D

592.

420

22.7

62D

GA

R1

2.00

312

.28

636

6.23

119

7.0

760

13.4

172

4.7

850

22.8

416

3010

GJR

361.

947

11.0

85

368.

679

355.

49

3616

.86

1033

1136

26.1

519

8812

EG

AR

363.

282

51.3

89

368.

790

363.

210

3614

.38

808.

99

3622

.06

1548

8E

MA

51

9.51

341

5.1

1136

10.8

551

5.9

1136

12.7

166

6.1

736

15.2

788

8.0

7E

MA

100

119

.69

1307

1336

23.3

916

8913

3624

.45

1802

1336

26.2

620

0113

GA

RT

11.

919

10.5

04

366.

843

233.

98

3614

.99

863.

110

3624

.42

1799

11M

GA

R36

2.03

112

.90

736

3.25

550

.33

536

5.48

015

4.4

636

8.73

435

9.3

6

Tabl

eX

IV.

Port

folio

(ris

km

anag

emen

tev

alua

tion

—1%

VaR

)

Step

s1

510

20

Mod

els

TU

FFFT

UC

FR

ank

TU

FFFT

UC

FR

ank

TU

FFFT

UC

FR

ank

TU

FFFT

UC

FR

ank

RW

1212

.24

626.

911

1211

.63

577.

411

3610

.74

507.

310

3610

.24

468.

99

LTM

374

1.14

00.

298

2D37

01.

113

0.19

32

367

0.94

60.

047

1D35

91.

140

0.29

81

MA

516

3.03

242

.22

975

3.17

147

.23

813

22.

893

37.4

26

117

3.22

749

.29

6M

A10

023

1.50

23.

445

837

01.

168

0.42

45

368

1.25

20.

925

535

91.

391

2.14

85

AR

123

1.33

51.

603

637

01.

140

0.29

83

367

0.94

60.

047

1D35

91.

168

0.42

42D

AR

AIC

361.

474

3.09

57

370

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738

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918

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93

359

1.16

80.

424

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231.

140

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370

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53.

588

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122

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100

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140

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1.89

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5.53

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40.

139

18.4

31

370

0.41

76.

862

136

70.

556

3.69

84

355

1.16

80.

424

2D



Tabl

eX

V.

Gov

ernm

ent

bond

(ris

km

anag

emen

tev

alua

tion

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VaR

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Step

s1

510

20

Mod

els

TU

FFFT

UC

FR

ank

TU

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315

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943.

611

115

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114

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801.

610

112

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689.

410

LTM

233.

922

77.4

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223.

477

58.9

36

396

3.56

162

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442

83.

255

50.3

34

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53

6.62

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0.3

91

6.98

224

2.6

91

6.06

418

7.3

73

6.00

818

4.1

7M

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022

4.72

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5.2

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4.50

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4.3

221

4.00

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214

3.95

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57.8

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4.42

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Tabl

eX

VI.

Equ

ities

(ris

km

anag

emen

tev

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tion

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VaR

)

Step

s1

510

20

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els

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UC

FR

ank

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915

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211

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81

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3.64

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23.

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Tabl

eX

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isk

man

agem

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eval

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n—

5%V

aR)

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119

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366.

314

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2.3

550

13.5

773

8.9

5M

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01

4.70

111

3.8

336

4.89

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3.6

136

5.64

716

3.6

436

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327

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75

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6.78

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9E

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51

14.7

283

8.4

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310

7811

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736

22.0

315

467

EM

A10

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24.8

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4513

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31.1

325

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T1

4.67

311

2.4

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13.5

773

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22.5

616

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3631

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4.78

411

7.9

236

7.17

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4.9

536

10.6

550

0.8

636

15.5

291

0.7

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Tabl

eX

VII

I.Po

rtfo

lio(r

isk

man

agem

ent

eval

uatio

n—

5%V

aR)

Step

s1

510

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els

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UC

FR

ank

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1215

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311

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011

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610

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233.

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77.4

96

342

3.50

560

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235

23.

060

43.2

13

342

2.86

536

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4M

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166.

871

235.

79

746.

843

233.

98

685.

925

179.

36

116

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125

1.4

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4.53

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5.7

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4.45

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1.7

135

23.

644

65.6

91

342

4.11

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1A

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234.

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42

342

3.39

455

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115

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02

342

3.00

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352

3.00

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4.28

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100

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234.

395

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563

158.

96

696.

759

228.

87

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346

837

00.

695

1.63

85

366

1.02

90.

013

511

92.

754

32.8

35



The first point to note is that if the objective is to cover 99% of future losses, then almost none ofthe models are adequate. The proportion of exceedences for the bond, share and commodity assetsis always considerably in excess of 1%—typically 1.4–2%. Thus for example, even the best modelat the 1-day horizon for the commodities data, which is the long term mean, has nearly 70% moreviolations of value at risk in the hold-out sample than would be expected under the null. Also forthis series, the majority of models have a TUFF statistic that takes on a value of one—that is, theyfail at the first observation! Almost none of the models for any of the four asset classes makes itto the hundredth observation, the expected time until first failure. Consequently, the UCF statisticrejects all models for all individual asset series at all horizons.

Matters are improved somewhat for the portfolio of assets, presumably as a result of the benefitsof diversification in reducing the number of extreme observations that lead to an exceedence ofthe VaR. The typical proportion of exceedences is reduced to around 1.2%, and although onlythe multivariate GARCH model has fewer than 1% exceedences, several models are acceptableaccording to the UCF test statistic. Similar patterns are revealed at the 1-day and longer horizons.The models fare much better when only 95% coverage is desired; more than half of the modelsachieve their nominal rate. In terms of model rankings, the long-term mean and the linear regressionin volatility models seem preferable, although again, there is no uniformly most accurate model.The GARCH model seems to provide reasonably accurate VaR estimates, evidenced by its actualcoverage rate being close to the nominal rate, although there is a tendency to over-estimate theVaR, a result also observed by Brooks, Clare and Persand (2000).

CONCLUSIONS

This paper has sought to re-examine the volatility forecasting literature in the context of a relativelynew use of volatility forecasts—for financial (market) risk assessment. A number of our resultsare worthy of further note. First, the gain from using a multivariate GARCH model for forecastingvolatility, which has not been previously investigated, is minimal. This result is true both understandard statistical and risk management evaluation measures. Given the complexity, estimationdifficulties, and computer-intensive nature of MGARCH modelling, we conjecture that unless theconditional covariances are required, the estimation of multivariate GARCH models is not worthwhile. In the context of portfolio volatility, more accurate results can be obtained by aggregating theportfolio constituents into a single series, and forecasting that, rather than modelling the individualcomponent volatilities and the correlations between the returns.

Second, it appears that some models are poor performers irrespective of both the series on whichthey are estimated and the loss function used to evaluate their forecasts. The random walk involatility, the EGARCH and to a lesser extent the EWMA models, fall into this category.

When it comes to selecting the ‘best’ model for forecasting, however, the particular evaluationmeasure employed plays a predominant role. Whilst there seems to be little difference in the modelrankings when the ex post measure is changed from low-frequency to high-frequency squaredreturns, the differences between rankings under statistical and risk management procedures aresubstantial. Although generalizing across data series (asset classes) and investment horizons isdifficult, overall the statistical measures preferred the GARCH(1,1) model over simpler techniquesand over its extensions and variants. On the other hand, when evaluated in the context of VaRestimates which achieve an appropriate out-of-sample coverage rate, the simplest models, such asthe long-term mean (historical average) or the autoregressive volatility model, are preferred. We



thus concur with Dacco and Satchell (1999) in arguing that the choice of loss function can havean over-riding effect upon volatility forecasting accuracies; thus the debate on superior volatilityforecasting models should be considered far from resolved.

ACKNOWLEDGEMENTS

The authors would like to thank an anonymous referee for useful comments on a previous versionof this paper. The usual disclaimer applies.

REFERENCES

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Alexander CO, Leigh CT. 1997. On the covariance models used in Value at Risk models. Journal ofDerivatives 4: 50–62.

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18: 1–16.Day TE, Lewis CM. 1992. Stock market volatility and the information content of stock index options. Journal

of Econometrics 52: 267–287.Dimson E, Marsh P. 1990. Volatility forecasting without data snooping. Journal of Banking and Finace 14:

399–421.Dowd K. 1998. Beyond Value at Risk: The New Science of Risk Management. Wiley: Chichester.Embrechts P, Resnick SI, Samorodnitsky G. 1999. Extreme value theory as a risk management tool. North

American Actuarial Journal 3: 30–41.Fair RC, Shiller RJ. 1990. Comparing information in forecasts from econometric models. American Economic

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Jackson P, Maude DJ, Perraudin W. 1998. Testing Value at Risk approaches to capital adequacy. Bank ofEngland Quarterly Bulletin 38: 256–266.

Johansen A, Sornette D. 1999. Critical crashes. Risk 12: 91–95.Jorion P. 1996. Value at Risk: The New Benchmark for Controlling Market Risk. Irwin: Chicago, IL.Morgan JP. 1996. Riskmetrics Technical Document 4th edn.Kupiec P. 1995. Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives

2: 73–84.Pagan AR, Schwert GW. 1990. Alternative models for conditional stock volatilities. Journal of Econometrics

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Authors’ biographies:Chris Brooks is Professor of Finance at the ISMA Centre, University of Reading, where he also obtained hisPhD. He has published over forty papers in leading academic and practitioner journals in finance, econometricsand economics, and is also author of the textbook Introductory Econometrics for Finance, published byCambridge University Press.

Gita Persand is a Lecturer in Finance at the Department of Economics, University of Bristol. She obtainedher PhD from the ISMA Centre, University of Reading. Her research interests are in the field of financial riskmanagement, and she has published in outlets such as Journal of Business, Journal of Banking and Finance,and Journal of Risk.

Authors’ addresses:Chris Brooks, ISMA Centre, University of Reading, PO Box 242, Whiteknights, Reading RG6 6BA, UK.

Gita Persand, Department of Economics, University of Bristol, Bristol B58, 1TH, UK.


volatility forecasting for risk management

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