1 Introduction On April 4th 2000 around midday (EST), the Nasdaq and Dow Jones stock indexes plunged by 14% and 4.8% respectively from their opening l

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1 CORE DISCUSSION PAPER 2000/45 INTRADAY VALUE-AT-RISK Pierre Giot 1 September 2000 Abstract In this paper, we apply a collection of parametric (Normal, Normal GARCH, Student GARCH, RiskMetrics and high-frequency duration models) and non-parametric (empirical quantile, extreme distributions models) Value-at-Risk (VaR) techniques to intraday data for three stocks traded on the New York Stock Exchange. Because of the small time horizon of the intraday returns (15 and 30 minute returns), intraday VaR can be useful to market participants (traders, market makers) involved in frequent trading. As expected, the volatility features an important intraday seasonality, which must be removed prior to using the VaR models. The estimation and assessment of the VaR techniques indicate that the data displays a high kurtosis (fat tails), and that VaR models should take this important feature into account. More particularly, Student GARCH, empirical quantile and extreme distributions models perform relatively well. Keywords: Intraday volatility, Intraday Value-at-Risk, Duration models, NYSE JEL classification: C22, C41, C53, G10 1 Department of Quantitative Economics, Maastricht University and Center for Operations Research and Econometrics, UCL; giot@core.ucl.ac.be or P.Giot@KE.UNIMAAS.NL While remaining responsible for any errors in this paper, the author would like to thank Luc Bauwens and Franz Palm for useful remarks and suggestions.

2 1 Introduction On April 4th 2000 around midday (EST), the Nasdaq and Dow Jones stock indexes plunged by 14% and 4.8% respectively from their opening levels. At the closing bell (4pm EST), the two indexes had recouped most of their losses, closing down only 1.8% and 0.5%. An outside observer focussing on successive closing prices would conclude that this had been a `normal' trading day, but it certainly was not for most active traders and market makers. Over the last five years, such intraday price variations have occurred more and more frequently, highlighting the importance of intraday price movements in financial assets. 1 Because of the increasing availability of so-called high-frequency databases for prices of stocks and other financial assets 2, there has been a growing number of research papers focussing on intraday volatility: Andersen and Bollerslev (1997, 1998, 1999), Giot (1999), Guillaume (2000), Muller, Dacorogna and Pictet (1996), Guillaume et al. (1995), Guillaume, Dacorogna and Pictet (1997). However, to our knowledge and with the exception of Muller, Dacorogna and Pictet (1996), no study has been made on the extreme intraday price movements and no work has been done on the intraday performance of Value-at-Risk models. In this paper we wish to address this issue by studying large (and extreme) intraday price movements. While this study could be made for a large class of financial assets (stocks traded on the NYSE, Nasdaq, European stocks, bonds, FOREX rates,...), we deal with three actively traded stocks on the NYSE. The main reason is the availability of the appropriate database for intraday prices based on earlier work such as Bauwens and Giot (1997) and Giot (1999). Actually, this work can be viewed as the sequel paper to Giot (1999), where intraday GARCH and duration models such as the Log-ACD were applied to high-frequency data for the IBM stock traded on the NYSE. Intraday volatility was characterized using both class of models and shown to be similar. In this work, we continue this analysis by focussing on the occurrence of large intraday price movements, and we evaluate the performance of several volatility models in `forecasting' one-step ahead bounds for variations in prices. To tackle this problem, we use the VaR framework and study the performance of the one-step ahead VaR predicted by parametric (Normal, Normal GARCH, Student GARCH, RiskMetrics and Log-ACD) and non-parametric (Empirical quantile, extreme returns or Pareto distribution) models. Of couse, our time horizon is extremely short as we deal with 15 and 30 minute returns. This is quite dif- 1 Most of these large intraday price variations can usually be traced back to external events, such as the Asian financial crisis in 1997, the Russian default on some of its bonds in 1998 and the subsequent near-failure of LTCM, a hedge fund. 2 Most exchanges, such asthenew York Stock Exchange (NYSE) or the Paris Bourse, distribute price databases on CD-ROMs which contain most intraday market characteristics, such as the time of the trade, the price, the volume,... Regarding FOREX trading, the Swiss consultancy Olsen & Associates has recorded intraday quoted prices on the Reuters screens for a large time period. 1

3 ferent from the usual VaR models which often work with daily returns and then compute the 10 day VaR (which is the VaR requested for regulatory reasons). Nevertheless, we believe that a VaR study on high-frequency returns can be useful for market participants (such as intraday traders and market makers) involved in frequent intraday trading. It also provides a review of possible parametric and non-parametric models and how to apply them to high-frequency data. Regarding the parametric model, we also (as in Giot, 1999) use a high-frequency duration model on so-called price durations to characterize volatility, which is an alternative toarch type models. All models are estimated on a subsample (estimation sample) of the original datasets and are then evaluated on the remaining portion of the dataset (forecast sample) by comparing the empirical failure rate to the theoretical value. Anticipating on the results, it is shown that `fat tails' models (such as the Student GARCH, empirical quantile and Pareto distribution models) perform much better when extreme returns are to be modelled. Thus, previous results (which recognize the importance of the fat tails of the returns distribution) on daily data seem to extend to intraday data. The rest of the paper is organized in the following way. In Section 2, we briefly review what is VaR. In Section 3, we characterize the intraday VaR and intraday volatility. In Section 4, we present a collection of parametric and non-parametric VaR models that can be applied to our intraday data. These models are then applied to three stocks traded on the NYSE in Section 5. Section 6 concludes. 2 What is Value-at-Risk? Broadly speaking, Value-at-Risk is a quantitative tool whose goal is to assess the possible loss that can be incurred by a trader or bank over a given time period and for a given portfolio of assets. Over the last ten years, this technique has been increasingly used by banks and regulators all over the world as a way to estimate possible losses related to the trading of financial assets. While the VaR can be computed for a large class of assets (stocks, bonds, derivatives such as futures and options,...), we focus on stocks. Further general information about VaR techniques can be found in Jorion (1997), Danielsson and de Vries (1998) or Danielsson, Hartmann and de Vries (1998). From a statistical and quantitative point of view, the VaR can be easily defined if a sample of past returns on the portfolio of assets is available. Indeed, once the time series y i of the returns is known and the VaR level ff is specified, the VaR at level ff for the given sample is simply defined as the empirical quantile at ff%. Hereafter, this empirical quantile is called z ff. Because z ff is such that F (z ff )=ff, where F (x) = R x 1 f(u) du is the cumulative density function of x, itisalsothecase that the probability area right ofz ff is equal to 1 ff. This allowsforanintuitive explanation of the VaR z ff : with probability 1 ff the returns will be larger than the VaR. For example if ff = 5% and if z ff = 1:645 (with daily returns measured in percent), returns larger than will be observed 95% of the times. If a 2

4 sum of W0 is invested by a trader of which returns are given by y i, than the corresponding VaR is equal to z ff W0. For example suppose that W0 =100; 000$. Then 95% of the times the trader will lose less than 1,645$. Contrary to some wide-spread beliefs, the VaR does not specifiy the maximum amount that can be lost. In our example, a loss of 5,000$ cannot be ruled out, albeit it will occur with a very small probability. Because the VaR is tantamount to computing the quantile of the distribution of returns, it is of paramount importance that the returns be identically distributed if the VaR is to be forecasted based on past data. 3 Hereafter we assume that this hypothesis is true. From an empirical point of view, the computation of the VaR of a portfolio of assets thus requires the computation of the empirical quantile at level ff of the distribution of the returns of the portfolio. Because the returns are assumed to be identically distributed, the predicted VaR (i.e. VaR for future returns) can be based on these past returns. Broadly speaking, the VaR can be computed using two kinds of models, parametric and non-parametric models (see van den Goorbergh and Vlaar, 1999, or Jorion, 1997). A parametric model specifies a certain type of distribution for the returns (for example the Normal distribution) and the empirical quantile is directly computed from the theoretical formulae. This case is dealt with in the first part of Section 4. If the non-parametric method is chosen, then the empirical quantile is directly computed from the available data without any model fitted on the returns. This latter case is presented in the second part of Section 4. Prior to detailing these two types of models, we first need to address the issue of intraday VaR (or VaR applied to intraday returns) which is the focus of this paper. 3 Intraday volatility and VaR In this paper we characterize intraday VaR, which is an extension of VaR to intraday returns, i.e. returns defined on intraday prices. As mentionned in the introduction, the main goal of VaR is to assess the possible loss that can be incurred by a trader or financial institution over a given period of time. Because of regulatory reasons, the time horizon is usually a 10 day period and the models are evaluated on daily returns. 4 However, for active market participants such as high frequency traders, floor traders or market makers, the time horizon of their returns is much shorter and the corresponding trading risk must therefore be assessed on such short time intervals. At the New York Stock Exchange for example, all trading in a given stock is monitored by a single market maker (called the specialist). The specialist must maintain an orderly market (which requires buys and sells of the stock by the specialist for his own account) and thus continuously interacts 3 Indeed, if the distribution of returns is different att 1 and t 2,witht 1 <t 2,characteristics of the distribution (such as the quantile) will be different. 4 For an estimation of the 10 day VaR not based on daily returns but on intraday returns, see Beltratti and Morana (1999). 3

5 with the floor traders and other traders. At thenasdaq (which features a multiple market maker trading mechanism), there is a constant interaction between the market makers and the traders. 5 Furthermore, with the growing use of the Internet as a vehicle of trading, there is an increasing number of individuals engaged in so-called intraday trading. Intraday returns are based on intraday data (prices at which the trades were made or posted quotes by the market makers) of which main feature is that the recorded market characteristics of the trades (prices, traded volume) and quotes (bid and ask prices, depths) are non regularly time-spaced. Indeed, because trades and quotes are recorded continuously, the available observations are no longer equidistantly time-spaced and the times between the trades or quotes may convey important information. As indicated by the recent literature on high-frequency duration models, modelling the times between the market events has become an important topic with applications in econometrics and market microstrucure (Engle and Russell, 1998; Bauwens and Giot, 1999). While the data is presented in Section 5, an important question immediately arises as to the modelling of the returns and their associated volatility. When intraday volatility is to be modelled, two methods can be used (see Giot, 1999). Firstly, the non regularly time-spaced data can be resampled along a prespecified time grid, yielding equidistantly time-spaced observations. Calling P i = (B i + A i )=2 the sampled prices, `raw returns' on the bid-ask quotes can be computed as Y i = ln(p i ) ln(p i 1 ), for i =1:::N,withB i and A i the closest bid and ask prices to time t i. By definition, the prices P i are sampled each s seconds, so that t i t i 1 = s. Then, `standard' volatility models (such as the GARCH model for example) can be used provided that intraday seasonality is taken into account (Andersen and Bollerslev, 1997; Giot, 1999). 6 In this paper, we assume a deterministic seasonality in the intraday volatility and use the method of Giot (1999) to deseasonalize the raw returns Y i. 7 Deseasonalized returns y i are computed as y i = Y i ffi(t i ) where Y i is the raw return and ffi(t i ) is the deterministic intraday seasonal component. The latter is defined as the expected volatility conditioned on time-of-day and on the day of the week (so that, for example, the time-of-day effect of Monday can be different from the time-of-day of Tuesday), where the expectation is com- 5 See Bauwens and Giot (2000) for a review of trading mechanisms in financial markets. 6 One of the main contributions of the papers by Andersen and Bollerslev (1997, 1998, 1999) is the identification of the three sources of intraday volatility: `long term (i.e. daily, but which affect intraday returns because of the aggregation properties of the ARCH model) ARCH effects, intraday seasonality and intraday news announcements. 7 Similar deterministic techniques are also used in Andersen and Bollerslev (1997, 1998, 1999). Beltratti and Morana (1999) define a stochastic seasonality for the intraday volatility (which is much more complicated to estimate) and the results are hardly better than those obtained with the more simple deterministic seasonality. (1) 4

6 puted by averaging the squared raw returns over thirty minutes intervals for each day of the week. Cubic splines are then used on the thirty minutes intervals to smooth the expected volatility. Because intraday seasonality has been taken into account, the volatility and VaR models are then defined on the deseasonalized returns y i. Secondly, intraday volatility canbe directly modelled by using high-frequency duration models applied to the so-called price durations. Price durations X p;i = t 0 i t 0 i 1 are defined as the time needed to witness a given cumulative price change c p in the price of the asset (see Giot, 1999 or Engle and Russell, 1997). 8 Once a high-frequency duration model of the ACD type is fitted on these price durations, it can be shown that intraday volatility isa function of the conditional hazard of the ACD model (see Section 4.1.5). The price durations feature a strong time-of-day effect akin to the intraday seasonnality for the volatility defined on the regularly sampled returns (Engle and Russell, 1998; Giot, 1999). To take into account this deterministic intraday seasonality, we compute time-of-day standardized price durations, which are defined as x p;i = X p;i ffi p (t 0 i) where X p;i is the raw filtered price duration with respect to the minimum price change c p, ffi p (t 0 i) is the time-of-day effect and x p;i denotes the time-of-day standardized price duration. The deterministic time-of-day effect is defined as the expected price duration conditioned on time-of-day and on the day of the week, where the expectation is computed by averaging the durations over thirty minutes intervals for each day of the week. Cubic splines are then used on the thirty minutes intervals to smooth the time-of-day function. The volatility andvar models are then fitted on the newly defined time-of-day standardized price durations. 4 VaR models for intraday data As shown in Section 3, modelling VaR for intraday returns should takeinto account the important intraday seasonality for the volatility and the stochastic behavior of the deseasonalized returns. In this section we present some possible models for computing the VaR of these deseasonalized returns. All but one (the Log-ACD based technique) are used on the resampled data of which intraday seasonality has been removed (i.e. we work with the deseasonalized y i returns) and are direct applications of VaR techniques for daily data. The Log-ACD technique is specifically designed to model irregularly time-spaced data and does not have an equivalent in 8 We use the notation t 0 i to mark the beginning and end of price durations to distinguish them from t i which give the times at which the regularly time-spaced returns are recorded. Thus, the t i are regularly time spaced, while the t 0 i are not. (2) 5

7 the standard time-series literature. In this case, we use the time-of-day standardized price durations x p;i. For all models except the Log-ACD, we use the following notation. The original sample is S, which is the collection of sampled (and thus equidistantly time-spaced) deseasonalized returns y i (recorded at times t i ) with i = 1 :::N and N is the total number of observed returns. S is split in two sub-samples, an estimation sample S E and a forecast sample S F. The estimation sample contains y i such that i = N0 :::N1. Usually, we have that N0 = 1. Another possibility is to specify N0 = N1 L, in which case a rolling estimation procedure is used. Then L is the length of the window used for estimating the model. Indeed, as N1 is increased, new returns are included in S E but `older' returns are removed. The forecast sample is the set of y i with i = N2 :::N. The parameters of the parametric models (which allow for the computation of the empirical quantile) and the directly computed empirical quantile in the case of the non-parametric models are based on the S E dataset. The performance of the models is tested on the S F dataset (see Section 5). For the Log-ACD model, the original sample is S 0, which is the collection of deseasonalized price durations x p;i (recorded at times t 0 i ) at the prespecified c p threshold, with i = 1 :::N 0 and N 0 is the total number of price durations. S 0 is split in two sub-samples, an estimation sample S 0 E and a forecast sample S 0 F. The estimation sample contains x p;i, i =1:::N1, 0 with N1 0 such that the time index of x 0 p;n coincides with the time index of the last return of S E. The sub-sample S 0 E 1 is then the equivalent of S E, but for price durations; they refer to the same time period. The high-frequency duration model is estimated on the price durations contained in the estimation sample S 0 E and its performance is assessed on the equidistantly time-spaced returns y i in the S F dataset (see Section 5). 4.1 Parametric models Given the returns of S E (or the price durations of S 0 E in the case of the Log-ACD model), a prespecified model is fitted on these returns (or price durations). The empirical quantile is directly given by a deterministic function of the estimated parameters Normal This is the most simple model as it assumes that the returns y i, with i =1:::N1, are Normal: y i ο N(μ; ff 2 ). If z ff is the quantile of the N(0; 1) distribution at the level ff, then the VaR is simply equal to VaR N = bμ + z ff bff (3) where bμ and bff are the empirical counterparts of the theoretical moments. The VaR is thus a function of the unconditional moments of S E. The one-step ahead VaR N (j) forj = N1 + 1 is then equal to VaR N. 6

8 4.1.2 Normal GARCH The ARCH model (Engle, 1982) attempts to model asset returns by allowing temporal dependence between the squares of the returns. The ARCH(1) model introduced by Engle (1982) can be written as: y i = μ + e i (4) e i = ffl i qh i (5) where μ is the expected return, ffl i is drawn from an IID N(0; 1) distribution and h i is defined as: h i =! + ff1e 2 i 1 (6) To guarantee the positivity ofh i,we assume that!>0andff1 > 0. An extension of the ARCH(1) model is the GARCH(1,1) model given by Bollerslev (1986) in which h i =! + ff1e 2 i 1 + fi1h i 1 (7) where!>0, ff1 0andfi1 0. To take into account a possible serial correlation in the returns, one substitutes y i = μ + ffiy i 1 + e i to Equation (4). If the model is estimated on the S E sample, than the one-step ahead VaR for j = N1 + 1 is equal to VaR G (j) =bμ + b ffiy N1 + z ff q bhj : (8) Strictly speaking, VaR G is a conditional VaR as it uses information on the returns up to j 1. 9 Maximum likelihood techniques are usually used to estimate this type of model RiskMetrics The RiskMetrics analysis of VaR was introduced by JP Morgan in 1994 as a simple practical risk model which requires almost no empirical computations. In its most simple form, it can be shown that the basic RiskMetrics model is equivalent to an IGARCH model where the autoregressive parameter fi1 is set at a prespecified value 1 and the coefficient of e 2 i 1 is equal to. Thus it is (4) and (5) with h i = e 2 i 1 +(1 )h i 1 (9) Once bμ and b ffi are known, the one-step ahead VaR for j = N1 +1is equal to 9 This is always true, even if j = N 1 + k, withk 6= 1. For example, if k = 5, the GARCH(1,1) model is estimated on i = 1 :::N 1 but the VaR G (j) uses information on the returns y i with i =1:::(N 1 +4). 7

9 as in the GARCH(1,1) case Student GARCH VaR RM (j) =bμ + b ffiy N1 + z ff q bhj : (10) While the simple GARCH(1,1) model often matches the empirical properties of the data quite well, it usually cannot fully take into account the `fat tails' of the returns distribution. Indeed, it can be shown that ARCH and GARCH models allow for a kurtosis coefficient larger than 3, but empirical daily or intradaily data exhibit a still much larger kurtotis coefficient. To alleviate this problem, the Student GARCH (or t GARCH) is introduced, which specifies the underlying e i as being drawn from a t(0; 1;ν) distribution. If the model is estimated on the S E sample, than the one-step ahead VaR for j = N1 + 1 is equal to VaR t (j) =bμ + b ffiy N1 + t ff;ν q bhj : (11) For similar bμ + ffiy b N1 and h b j, it can immediately be seen that VaR t (j) will be smaller than VaR G (j) ast ff;ν <z ff Log-ACD As indicated above, the Normal, GARCH and RiskMetrics models have been specifically designed for the analysis of time-series which are made up of equidistantly time-spaced returns (for example daily or weekly returns). In this subsection, we highlight an alternative way of modelling intraday volatility which is directly based on the price durations. The Log-ACD model (Bauwens and Giot, 1997) for the standardized price durations (i.e. the price durations of which the intraday seasonality has been removed, see Section 3) is defined by x p;i = e ψ i (1+1=fl) ffl i (12) ψ i =! + ffffl i 1 + fiψ i 1 (13) where the ffl i are IID and follow aweibull(1,fl) distribution and ψ i is the logarithm of the conditional expectation of x i, so that ψ i = ln E(x i ji i 1 ). For covariance stationarity ofψ i, jfij must be smaller than one. As first indicated in Engle and Russell (1998), there is a direct link between the instantaneous intraday volatility and the conditional hazard of the price durations. Giot (1999) provides the link for the Log-ACD models as 10 Of course t ff;ν = z ff when ν = 1. 8

10 bff 2 cp 2 1 (tji i 1 )= P (t 0 i 1) eψ b iffip (t 0 i 1) where ff 2 (tji i 1 ) is the conditional instantaneous intraday volatility, P (t 0 i 1) is the bid-ask quote midpoint at time t 0 ψ i 1, e b i is the conditional expectation of price duration x p;i and ffi p (t 0 i 1) is the time-of-day effect. This equation provides a direct estimation of the intraday volatility once the Log-ACD model has been estimated. Strictly speaking, this is the instantaneous volatility as forecasted at time t 0 i 1. In an empirical application, this translates into the discretized volatility forecasted at all times starting and ending a price duration. Thus in our VaR framework, ff 2 (tji i 1 ) is taken to be the forecasted volatility at time t 0 i 1 and valid up to t 0 i, which is the end of price duration x p;i. Using this approximation and the Normal distribution, the VaR forecasted at time t 0 i 1 is (14) VaR Log ACD (t 0 i 1) =z ff q bff 2 (tji i 1 ): (15) As such, it is not possible to compare this VaR to the other measures of VaR given by the models based on regularly time-spaced data. Indeed, the t 0 i are not regularly time-spaced and Equation (15) gives the VaR for the raw returns. To circumvent this difficulties and because we wish to compare this `special' VaR to the VaRs predicted by the other models using regularly time-spaced returns, we resample VaR Log ACD (t 0 i 1) at each times t i defining the regularly time-spaced returns. Then this resampled VaR is divided by q ffi(t i ) to get the VaR for the deseasonalized returns. Thus, if we wish to predict the VaR for the returns on [t i 1 ;t i ] (i.e. y i ), we use VaR Log ACD (t i 1 )=VaR Log ACD (t 0 k)= where t 0 k is the closest (from below) time to t i 1. q ffi(t i 1 ) (16) 4.2 Non-parametric models Historical simulation The historical simulation approach (or empirical quantile model) is the simplest non-parametric model. Given the returns in the S E sample, the empirical quantile Y ff at ff% (which by definition is the VaR at ff%) is directly computed. Because the returns are assumed to be identically distributed, the one-step ahead VaR for j = N1 +1isequal to VaR HS (j) =Y ff : (17) 9

11 4.2.2 Extreme returns (Pareto distribution for the tails) Strictly speaking, this method is semi-parametric as it assumes that the extreme returns follow a Pareto distribution, i.e. that the density distribution of the tails of the returns can be approximated by a Pareto distribution. By definition (see Gouriéroux and Jasiak, 2000), a distribution belongs to the class of Pareto distributions if 1 F (y) ο y fl L(y), where F (y) isthecumulative density distribution at y and L(y) is a slowly varying function at infinity. Because 1 F (y) is the probability area right of y, it implies that the probability mass in the right tail of y decreases as an inverse power of y. When financial returns are to be modelled, it is assumed that they follow a Pareto distribution only for `large' returns, i.e. that 1 F (y) ο y fl L(y) is true when y>c, where c must be properly defined. It can then be shown (Gouriéroux and Jasiak, 2000) that the Hill estimator can be used to estimate bfl with 1=bfl = 1 k 1 X ln(y(j)) ln(c) (18) k j=0 where c is the threshold for the returns above which it is assumed that the y i follow a Pareto distribution, y(j) are the ordered returns and k is the number of returns larger than c. The choice of c is non-trivial as it directly determines bfl. If c is large, few ordered returns y(j) are used in Equation (18) and bfl will have a large variance. If c is too small, too many y(j) are taken into account, many ofwhich do not follow the Pareto distribution. Information on the determination of the threshold level c can be found in van den Goorbergh (1999). As indicated in Gouriéroux and Jasiak (2000), the Pareto distribution is particularly useful for very low quantile VaRs. Indeed, if ff is very small and one wishes to use the historical simulation (or empirical quantile) method, few returns will be available for the computation of Equation (17). However, if one assumes that the Pareto distribution holds for y>c,onehas and for a VaR ff larger than c: 1 F (c) =ff c ο c fl (19) which leads to 1 F (VaR ff )=ff ο VaR fl ff (20) 1=fl VaR ff = c (21) ff Thus, once c and ff c are available (for example by using the historical simulation method for a not too large VaR), VaR ff can be computed for extremely small ff using Equation (21) and with fl estimated by bfl. As such, the Pareto distribution is valid for positive and large returns. In the application to VaR, one must consider negative and large returns. The application of Equation (18) to VaR and to the left tail of the distribution of returns is 10 ffc

12 immediate if the y denote the opposite of the observed returns. See Section 5 for the practical implementation of this method on the high-frequency returns. 5 Empirical application 5.1 Data and intraday seasonality We use the dataset of Giot (1999) extended to the first five months of This dataset is based on the TAQ CD-ROMs of the NYSE and contains intraday trade and quote data for some selected stocks. In addition to the original irregularly time-spaced trades and quotes, the dataset also contains regularly sampled quotes and related trade information (volume traded, number of trades,...). In this paper we focus on three actively traded stocks on the NYSE, BOEING, EXXON and IBM, for the January-May 1997 period. We make use of both types of datasets as most VaR models presented in Section 4 need the regularly time-spaced data, while the Log-ACD model is estimated on the irregularly spaced data. As detailed in Section 4, the original sample is split into an estimation sample S E (and S 0 E for the price durations) and a forecast sample S F (and S 0 F for the price durations). In our case, this means that the five months dataset for the three stocks is split in a three months long estimation sample and a two months long forecast sample. We deal with two sampling frequencies which yield 15 and 30 minute returns respectively and use the deseasonalization procedures given in Section 3 to transform the raw returns Y i into intraday seasonally adjusted (isa) returns y i. Hereafter, the terms `data' and `returns' refer to the deseasonalized returns y i. Information on the data is given in Table 1. As indicated in Table 1, the mean intraday return is extremely small for all stocks, and much lower than its standard deviation. 11 Moreover, intraday returns exhibit fat tails as their kurtosis is higher than 3. In Figure 1 we plot the density and cumulative density functions for the 15 minute returns of the IBM stock. To allow an easy comparison with hypothetical returns drawn from the Normal distribution, we also plot the density and cumulative density functions for the simple Normal distribution fitted on these returns (i.e. the Normal distribution of which μ and ff 2 are given by the empirical unconditional mean and variance of the 15 minute returns). While the empirical density and cumulative density functions closely track the Normal ones, fat tails are immediately apparent as indicated by the larger probability massin the tails of the density distribution (see the bottom left and right figures which present a zoom of the density and cumulative density functions) Because the mean intraday return is so small, all reference to the mean of the returns (μ) could be removed in the preceding VaR formulae and the results would be almost identical. 12 We only report results for the left tail of the density function as VaR usually deals with extreme negative events. Similar figures are available for the right tail of the density distribution and for the other two stocks. 11

13 Focussing again on the 15 minute returns for the IBM stock, we plot in Figure 2 the autocorrelation functions (ACF) of the raw returns (Y i ), squared raw returns (Y 2 i ), isa returns (y i ) and squared isa returns (y 2 i ). These figures are similar to those given in Giot (1999) and Andersen and Bollerslev (1997, 1998, 1999) and clearly indicate the need to deseasonalize the data prior to estimating any stochastic model. As expected, the squared raw returns exhibit cycles in their ACF due the strong seasonality in the volatility (because of the recurrent increasing volatility at the open and close of trading). The deseasonalized returns y i do not feature such cycles and exhibit a slowly decreasing ACF. Table 1: Information on the intraday returns BOEING EXXON IBM 15 minute returns N 2,703 2,703 2,703 N1 1,586 1,586 1,586 Mean S.E Kurtosis minute returns N 1,351 1,351 1,351 N Mean S.E Kurtosis Information on the intraday15minute and 30 minute returns. The returns are defined on equidistantly sampled intraday data extracted from the January- May 1997 TAQ CD-ROMs. 5.2 Estimated intraday VaR In this section, we compute different measures of the intraday VaR for the deseasonalized returns sampled at several frequencies. For each model presented in Section 4 and for two sampling frequencies (15 and 30 minute returns), we compute the one-step ahead VaR on the isa returns y i in S F. All models (except the 12

14 Log-ACD model) are estimated using the isa returns y i in S E while the Log-ACD model makes use of the price durations given in S 0 E. More precisely: (1) The GARCH (Normal and Student) models are estimated on the returns y i in S E with N0 = 1 and N1 fixed at 1,586 (15 minute returns) and 793 (30 minute returns), yielding the estimated parameters bμ, b ffi, b!, bff1, b fi1 and bν. Equation 8 (for the Normal GARCH) and equation 11 (for the Student GARCH) are then used on the returns y i in S F to compute the one-step ahead VaR. (2) The Normal model is estimated on the returns y i in S E with N0 = 1 and N1 increasing from 1,586 to 2,702, so that the `new' information is included in the estimation sample once it is available. This corresponds to the real world situation where the researcher makes use of all information available at time t to forecast the next one-step ahead VaR. Equation 3 is then used to compute the one-step ahead VaR. The preceding discussion implies that the estimation sample is fixed for the GARCH models (N1 is fixed) while it is increasing for the other methods (N1 increases up to N 1). An alternative method would be to estimate all models with N1 increasing, but this requires the estimation of the GARCH models at each step (which is computationally intensive). As the estimated coefficients do not change much if this latter method is used, we do not report the results here. Moreover, the GARCH models do make use of the `new' information as equations 8 and 11 use the most recent returns to forecast the VaR (hence the name of conditional VaR, even if the parameters are estimated on a fixed length sample). (3) For the RiskMetrics method, μ and ffi are set at their estimated values from the Normal GARCH model and =0:94. Equation 10 is then used on the returns y i in S F to compute the one-step ahead VaR. The starting value for the iterative computation of h i is set at the unconditional variance of the returns in S E. (4) To computethevar of the Log-ACD model, we use the methodology given in Section First, we estimate the Log-ACD model on the price durations in SE. 0 Then we use the estimated parameters to determine VaR Log ACD (t 0 i 1) at all price durations in S 0 F. In a third step, we compute VaR Log ACD (t i 1 ) for all times t i 1 in S F using Equation (16). This gives us the regularly time-spaced VaR. (5) The historical simulation method is very similar to the Normal model. The empirical quantile is directly computed on the returns y i in S E with N0 =1 and N1 increasing from 1,586 to 2,702. Equation 17 is then used to compute the one-step ahead VaR. (6) The method based on the Pareto distribution for the tails uses Equation (21) for small ff, with ff c =2:5% and c computed using the historical simulation 13

15 method as detailed in [5]. This VaR is thus computed only for ff smaller than 2.5%. All parametric models are estimated using maximum likelihood (CML estimation in GAUSS) and diagnostic tests are (successfully) performed on the residuals of the Normal GARCH, Student GARCH and Log-ACD models. The performance of each model is then assessed by computing the failure rate 13 for the returns y i in S F. It should be stressed that all models (including the Log-ACD model) are evaluated with this criteria. Thus, the Log-ACD based method (which deals directly with the irregularly spaced data) is assessed using regularly spaced returns. An alternative would be to define and assess a VaR on irregularly spaced prices. The failure rates are given in Table 2 (BOEING), Table3(EXXON) and Table 4 (IBM). Generally speaking, the results are quite good for all three stocks and forbothtypes of returns as the empirical failure rates are close to their theoretical counterparts. Nevertheless, a statistical test is needed in order to ascertain the quality of the estimation methods. Because the computation of the empirical failure rate defines a sequence of yes/no observations, it is possible to test H0 :f = ff against H0 : f 6= ff, where f is the failure rate (estimated by f, b the empirical failure rate). 14 At the 5% level and if T yes/no observations are available, a confidence interval for b f is given by h b f 1:96 q bf(1 b f)=t ; b f +1:96 q bf(1 b f)=t i. In Tables 2, 3 and 4, empirical failure rates are set in bold when the theoretical quantile does not belong to the corresponding confidence interval: this denotes a failure of the VaR model. 15 A summary of the number of failures for each model as a function of the level of ff is given in Table 5. These results indicate that: (1) When ff is large, there are no key differences between the models. Most models fail only once; the Normal GARCH is rejected twice because it is too conservative and the Log-ACD is rejected in most cases (see below for comments regarding this model). (2) When ff» 1%, the Normal, Normal GARCH and RiskMetrics models clearly underperform. When ff = 0:5% for example, these models are rejected 5, 4 and 4 times out of 6 respectively. In contrast, the Student GARCH and empirical quantile models never fail and the Pareto model fails once. When ff = 0:25%, the last three models never fail while the first three fail most of the time. Of course this was to be expected as the Student GARCH, empirical quantile and Pareto models all take into account the `fat tails' feature of the distribution of returns. The first one uses explicitely the Student 13 By definition, the failure rate is the number of times returns exceed the forecasted VaR. If the VaR model is correctly specified, the failure rate should be equal to the prespecified VaR level. 14 In the literature on VaR models, this test is also called the Kupiec LR test, if the hypothesis is tested using a likelihood ratio test. See Kupiec (1995). 15 Because this is a bilateral test, a model can fail if it under or overestimates the true VaR. Thus, too conservative models are also rejected. 14

16 distribution, and hence allows for a large kurtosis, while the latter two are based directly on the empirical properties of the data. (3) The Log-ACD model performs rather poorly when the one-step ahead VaR is to be forecasted. As indicated in Table 5, the price durations based model fails most of the time and for all stocks. Indeed, the empirical results given in Tables 2, 3 and 4 and the plot 16 of the Log-ACD VaR vs GARCH VaR in Figure 3 show that the Log-ACD VaR tracks quite well the GARCH VaR while being always somewhat lower, hence the larger failure rate. Thus, while Equation 14 allows the computation of the instantaneous volatility based on the price durations, there does seem to be a slight discrepancy with the volatility computed from discrete returns. However, one should notice that (a) the instantaneous volatility has to be discretized to be compared with the observed returns and can only be computed when a price duration ends, (b) the instantaneous volatility is based on the Normal approximation and hence uses the critical values of this distribution. (4) Quite surprisingly, the most simple model (empirical quantile) performs remarkably well. Thus, in our framework, the more complicated parametric models do not bring about significant improvements compared to a simple (empirical) quantile estimation. 6 Conclusion In this paper, we apply VaR techniques to intraday returns for three stocks actively traded on the NYSE. While VaR models are usually applied to daily data to help manage the financial risks of banks and financial institutions, we use seven VaR models to assess the trading losses on 15 and 30 minute intraday returns. Our time horizon is thus much shorter than what is usually considered in the VaR literature but our framework can be helpful to market participants engaged in frequent trading (such asmarket makers or day traders). More precisely, we study the performance of five parametrics models (Normal, Normal GARCH, Student GARCH, RiskMetrics and Log-ACD) and two non parametric ones (empirical quantile and extreme returns distribution). Regarding the parametric models, we thus also provide an application of high-frequency duration models (applied to irregularly spaced data) to forecasting intradayvolatility, which is an alternativeto high-frequency GARCH models (applied to regularly spaced data). As in previous papers by Giot (1999) or Andersen and Bollerslev (1997, 1998, 1999), the empirical returns feature a strong intraday seasonality in the volatility, whichmust be taken into account prior to using the VaR models. 16 We show the plot for the EXXON stock and for the first 200 observations. Similar plots are available for the other stocks and for other samples. 15

17 Table 2: VaR results for BOEING 15 minute returns 5% 2.5% 1% 0.5% 0.25% Normal N GARCH t GARCH RiskMetrics Log-ACD Quantile Pareto minute returns 5% 2.5% 1% 0.5% 0.25% Normal N GARCH t GARCH RiskMetrics Log-ACD Quantile Pareto Failure rates for the parametric (Normal, Normal GARCH, t GARCH, RiskMetrics and Log-ACD) and non-parametric (Empirical quantile, Pareto distribution) measures of intraday VaR. The estimation sample is the January-March 1997 period, and the forecast sample is the April-May 1997 period (BOEING stock). A bold figure indicates that the corresponding VaR is significantly different (LR test) from the theoretical value. 16

18 Table 3: VaR results for EXXON 15 minute returns 5% 2.5% 1% 0.5% 0.25% Normal N GARCH t GARCH RiskMetrics Log-ACD Quantile Pareto minute returns 5% 2.5% 1% 0.5% 0.25% Normal N GARCH t GARCH RiskMetrics Log-ACD Quantile Pareto Failure rates for the parametric (Normal, Normal GARCH, t GARCH, RiskMetrics and Log-ACD) and non-parametric (Empirical quantile, Pareto distribution) measures of intraday VaR. The estimation sample is the January-March 1997 period, and the forecast sample is the April-May 1997 period (EXXON stock). A bold figure indicates that the corresponding VaR is significantly different (LR test) from the theoretical value. 17

19 Table 4: VaR results for IBM 15 minute returns 5% 2.5% 1% 0.5% 0.25% Normal N GARCH t GARCH RiskMetrics Log-ACD Quantile Pareto minute returns 5% 2.5% 1% 0.5% 0.25% Normal N GARCH t GARCH RiskMetrics Log-ACD Quantile Pareto Failure rates for the parametric (Normal, Normal GARCH, t GARCH, RiskMetrics and Log-ACD) and non-parametric (Empirical quantile, Pareto distribution) measures of intraday VaR. The estimation sample is the January-March 1997 period, and the forecast sample is the April-May 1997 period (IBM stock). A bold figure indicates that the corresponding VaR is significantly different (LR test) from the theoretical value. 18

20 Table 5: Number of failures at given percentage level 5% 2.5% 1% 0.5% 0.25% Normal NGARCH t GARCH RiskMetrics Log-ACD Quantile Pareto Number of failures for each VaR model, for all three stocks and for 15 and 30 minute returns combined, i.e. number of times the empirical failure rate is significantely different from the corresponding percentage level. The Normal, Normal GARCH, Student GARCH, RiskMetrics, empirical quantile and extreme returns distribution models are applied to the deseasonalized 15 and 30 minute returns and their performance is assessed by computing their empirical failure rates. The Log-ACD model is applied to the price durations and its performance is assessed in a regularly time-spaced framework, i.e. its empirical failure rate is computed on the same 15 and 30 minute returns. Among the seven models, two stand out as `winners' as far as the performance of their empirical failure rates is concerned: the Student GARCH and the empirical quantile models. The extreme returns distribution model is a second best choice. Not surprisingly, these three models explicitely take into accout the `fat tails' feature of the data, which is an important characteristic of daily and intradaily returns. Broadly speaking, once intraday seasonality is taken into account, there do not seem to be any key differences between daily and intradaily VaR models as the results given in this paper show that usual VaR results on daily data extend to intradaily returns. When assessed in a regularly time-spaced framework, the high-frequency duration model (Log-ACD model) performs rather poorly. Although the forecasted intraday volatility and VaR are quite close to their regularly time-spaced counterparts 17, the empirical failure rate of the model is too high and the model is often rejected. An alternative would be to define an irregularly time-spaced VaR, 17 For example, Prigent, Renault and Scaillet (1999) successfully price stock options using this method. 19

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