Fast but not Furious

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-Calculation and Evaluation of an

Intraday Value at Risk

University of Groningen

Master Thesis Econometrics, Operations Research and

Actuarial Studies

author: Florian Walla

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-Calculation and Evaluation of an Intraday Value at

Risk

Florian Walla August 9, 2012

Abstract

Calculating intraday VaR recently came into the focus of research. In this paper different models for forecasting it are taken into account. While an ARMA-GARCH approach is used on the equally spaced grid, log-ACD mod-els are applied on the unequally spaced grid. In addition to the normal innovation distribution which had a bad forecasting performance in the lit-erature, a parametric and a t-distribution are used. The log-ACD models are transformed to an equally spaced grid for testing purposes. Extensive fore-casting performance testing gives evidence that a log-ACD approach with a conditional t-distribution is at least on par with the well-known ARMA-GARCH models as they both pass almost all tests successfully. There is weak evidence the log-ACD approach with the conditional t-distribution outperforms the ARMA-GARCH model.

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List of Figures

1 Raw price durations during the first trading day . . . 8

2 Deseasonalized price durations during the first trading day . . 10

3 30 minutes returns for six trading days . . . 17

4 Absolute values 30 minutes returns for three months . . . 19

5 30 minutes returns for three months . . . 21

6 Seasonality correction factor φ for Mondays . . . 31

7 Kernel density of deseasonalized price durations . . . 34

8 V aR0.01_norm forecast and returns . . . 36

9 V aR0.01_t forecast and returns . . . 37

10 V aR0.01_emp forecast and returns . . . 38

11 V aR0.01_GARCH forecast and returns . . . 40

List of Tables

1 Ljung-Box test for 15 lags . . . 32

2 Summary statistics of deseasonalized data . . . 33

3 Estimation results log-ACD (1,1) . . . 35

4 Estimation results ARMA-ARCH model . . . 39

5 p-values test of unconditional coverage . . . 41

6 Coverage ratio of the models . . . 41

7 p-values test of independent violations . . . 42

8 p-values test of conditional coverage . . . 42

9 p-values Ljung-Box test of 20 lags of the indicator sequence . 43 10 p-values Ljung-Box test of 100 lags of the indicator sequence 43 11 p-Values LR-test of joint significance of all parameters . . . . 44

12 Number of parameters which are statistically significant at the 5% level in the logistic regression . . . 44

A13 AIC values log-ACD model . . . 51

A14 BIC values log-ACD model . . . 51

A15 AIC values for ARMA-model . . . 52

A16 BIC values for ARMA-model . . . 52

A17 AIC values for ARMA(2,2)-GARCH . . . 53

A18 BIC values for ARMA(2,2)-GARCH . . . 53

A19 Regression of price duration on day-of-the-week dummies . . 54

A20 Test results Engle and Manganelli V aR0.05_norm . . . 55

A21 Test results Engle and Manganelli V aR0.025 norm . . . 56

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A23 Test results Engle and Manganelli V aR0.005_norm . . . 58

A24 Test results Engle and Manganelli V aR0.05_t . . . 59

A25 Test results Engle and Manganelli V aR0.025 t . . . 60

A28 Test results Engle and Manganelli V aR0.05_emp . . . 63

A32 Test results Engle and Manganelli V aR0.05_GARCH . . . 67

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1 Introduction

One of the most important tasks of risk managers is to quantify the conse-quences of an adverse market change to certain portfolios. Portfolio man-agers and CEOs need such information as a basis for their investment de-cisions. Equally as important, regulations such as the Basel and Solvency requirements obligate financial institutions to calculate risk measures such as a Value at Risk (VaR) (Basel Committee on Banking Supervision, 1996). It was first suggested by Baumol (1963) as a risk measure in the context of a Markowitz optimization and has gained a huge popularity today both in research and institutional risk management. This may be due to its core function in the Basel regulations and its intuitive and easy interpretation as a quantile of the return distribution.

Albeit some legitimate criticism of its aggregation properties, Artzner et al. (1999) showed that it is not a coherent risk measure since it lacks the property of subadditivity, it is still an important risk measure both for researchers and risk managers. A lot of publications were released in recent decades focusing on the calculation of VaR on a daily basis or longer hori-zons. An overview of these models can be found in Jorion (2000).

One reason for the focus on the daily basis is that the Basel regulations re-quire a bank to estimate a VaR for a 10 day window (Jorion, 2002). However, since market participants such as intraday-traders or market makers have shorter investment periods, risk measures for these horizons are needed. In addition to that, huge intraday movements which happen occasionally are ignored when using aggregated data on a daily basis.1

Intraday VaR models use fine grained data and are able to predict such movements. Therefore not only day traders, but all investors should be interested in risk measures on lower aggregation levels.

Giot (2005) was the first who discussed an intraday VaR. He used a RiskMetrics and ARMA-GARCH approach to estimate a VaR using regu-lar spaced data. In addition, he applied a log-Autoregressive Conditional Duration (ACD) model on the irregular spaced grid to estimate a VaR and transformed the model to the equally spaced grid. The Actual VaR approach of Giot and Grammig (2005) takes the price impact of the liquidation of a portfolio into account. Dionne et al. (2009) use a log-ACD approach to model trade durations and a GARCH approach for the volatility. A Monte-Carlo simulation method is used to estimate an intraday VaR. Our approach

1

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aims at improving the rather disappointing backtesting results of the log-ACD model in Giot (2005): The performance of the model was not sufficient in almost all tested cases. We see possibilities for improvement by allowing for a more general model and using different distributional assumptions. In general we are convinced that due to their high informational content ACD-models are suitable ACD-models for intraday data. Furthermore, our study offers further evidence as to whether ARMA-GARCH models are suitable for the calculation of intraday VaR models. In addition to that, we will apply the models to data from the German stock exchange DAX, which has, to our best knowledge, not been done before.

We are going to estimate intraday VaR models using several methods. First, we will use a log-ACD model which originates from survival analysis for the unequally spaced grid and second we apply well-known models for equidistantly spaced data. To estimate the former class of models, the data has to be aggregated on a certain level. These aggregated data features a lot of the well-known stylized facts of daily data which gives way to the use of standard models. Since the features of unequally spaced data and especially price durations are not so well known, this text will proceed with a description of them in the second section.2 Subsequently econometric models for market risk using unequally respectively equally spaced data will be introduced in section 3. The fourth section describes methods for the evaluation of these VaR forecasts. Section 5 is dedicated to the application of the described methods to the Allianz share. Finally, we conclude.

Since there are a few different mathematical formulations of the VaR concept, we introduce the one used in this paper before closing the intro-duction:

P (rt+h≤ V aRα|Ft) = α. (1)

rt+h denotes the return at time t + h, h the forecasting horizon, and

Ft the information set containing all the information available at time t.

The V aRα is the return which is not exceeded in α percent of the trading periods.

2 Features of price durations

A duration is defined as the time elapsed between two consecutive events. In the context of this text price durations are of particular interest. They

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record the time passing by until a certain cumulative price change cp has

oc-curred. In order to calculate these durations, tick by tick data, also labeled ultra-high frequency data by Engle (2000), is needed. It records information about every buy and sell of a financial instrument. In this sense, the irregu-lar spaced data have a higher informational content then aggregated reguirregu-lar spaced data. Combined with the calculation of price durations, which are closely linked to volatility, models built on this data should be able to out-perform standard models on the equally spaced grid. Figure 1, which plots price durations of the Allianz share during a trading day, gives a first im-pression of the features of intraday data.3

10:00 12:00 14:00 16:00 0 50 100 150 200 250 time−of−the−day seconds

raw price durations

Figure 1: Raw price durations during the first trading day

The remainder of this section gives an overview of the common features of this kind of data.

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2.1 Seasonality of the data

Engle and Russell (1998) note that price durations feature strong intraday seasonalities and give empirical evidence. Giot (2005) backs up this evi-dence. In general, the durations are shorter right after the opening of the market when new information which spread over night is taken into account by investors. Around lunchtime, durations are a bit longer while they tend to decrease before the closing of the market when day traders liquidate open positions in order to avoid huge unexpected overnight changes (Dionne et al., 2009). The described seasonality is clearly depicted in Figure 1.

Regressing the average duration on day-of-the-week dummies Dionne et al. (2009) have to reject the null hypothesis of equality of all coefficients. This is evidence for interday seasonality. When performing the seasonal ad-justment of their raw data, among others Engle and Russell (1998), Giot (2005) and Dionne et al. (2009) take this into account. Moreover, the volatility is high in the period after the opening and before the closing, but plummets in the mid of the trading day (Engle and Russell, 1998). Day-of-the-week effects can be found for the volatility, too. Giot (2005) and Dionne et al. (2009) report this for tick by tick data, whereas Ander-sen and Bollerslev (1997), Bollerslev and Domowitz (1993) and Beltratti and Morana (1999) find the same pattern using regular spaced data from currency trading markets.

2.2 Clustering of durations

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10:00 12:00 14:00 16:00 0 50 100 150 200 250 time−of−the−day deseasonalized price durations

Figure 2: Deseasonalized price durations during the first trading day As can be seen in the plot, long (short) durations still tend to be followed by long (short) durations. Therefore, models used to describe durations should allow for clustering. This has an interesting parallel in the equally spaced grid, where volatility is found to be clustered. As detailed in section 3.2.2 GARCH models account for this stylized fact.

2.3 Overdispersion

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2.4 Zero price durations

Since the smallest time increment used in most transaction datasets is one second, orders fulfilled in the same second have the same time stamp (Pacurar, 2008). Engle and Russell (1998) report that about two thirds of their dataset consists of zero durations. Based on an argument from the market microstructure theory that zero durations correspond to split trans-actions, many authors follow the approach of Engle and Russell (1998) and aggregate them. However, if split transactions do not have the same time stamp, their identification is not straightforward. Grammig and Wellner (2002) consider trades with the same price and a time difference not higher than one second as split transactions.

3 Econometric Modeling

In order to be able to forecast certain characteristics of intraday data, which can be used in the calculation of market risk models, one has to find a suit-able econometric model. Such a model should account for the aforemen-tioned characteristics. In this section, we will first introduce ACD-models for irregular spaced data. Subsequently, we will describe the well-known GARCH-model for regular spaced data.

3.1 Models for irregular spaced data

As defined earlier, price durations measure the time until a certain cumula-tive price change cp has been occurred. They are defined as Xp,i = t

0

i− t

0

i−1,

where t0_i−1 and t0_i denote the beginning and the end of a price duration, re-spectively. Price durations carry important information about little move-ments in the market and they are irregular spaced. Since raw financial data is, in most cases, irregular spaced, they are much closer to the real activi-ties than regular spaced aggregated data. As a good model for forecasting should be as close as possible to reality, the use of such data should give an advantage in forecasting performance.

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3.1.1 Seasonality

The strong intraday and sometimes interday seasonality (see section 2) of financial market data has to be incorporated in a model. The approach of this paper removes seasonality effects first, and then applies a model which accounts for the clustering and overdispersion. Since the seasonality is the main feature of the data, estimating models for volatility directly on the data would bias the estimation results in the models (Giot, 2005).

Following the literature (Engle and Russell, 1998; Giot, 2005; Dionne et al., 2009) a deterministic volatility for the seasonality correction is as-sumed in this paper.4 This implies that the volatility due to seasonal effects is equal for the same time intervals of a weekday in different weeks. There-fore, time-of-the-weekday standardized durations are defined as:

xi= Xi φp(t 0 i) . (2)

Xiis the raw price duration which measures the time elapsed until a price

change of at least cp has occurred. φp(t

0

i) denotes the deterministic

time-of-the-day effect and is calculated as the conditional expected duration. The conditional expectation is computed by taking the mean duration for 30 minutes intervals during the day starting at 9:00 AM and ending at 5:30 PM. These average durations are taken as knots in a cubic spline regression which gives the function for the time-of-the-day effect. The usage of a cubic spline ensures that the volatility does not plummet suddenly at the end of a time interval. To take interday effects into account, this is done separately for each day of the week.

3.1.2 Models for deseasonalized durations

After standardizing the durations the clustering effect might not be fully removed. Often Ljung-Box tests on the first 10 to 15 autocorrelations are much lower but still give evidence for significant autocorrelations (Dionne et al., 2009). The clustering effect can also be seen in Figure 2 on page 10 where short (long) tend to be followed by short (long) durations. There-fore, a model which is able to account for the clustering effect is needed. Furthermore, it should allow for overdispersion. The class of Autoregressive Conditional Duration (ACD) models is suitable for that.

4_{It is also possible to define a stochastic volatility as Beltratti and Morana (1999). Yet}

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There are a lot of different specifications (Pacurar, 2008) and most of them share features of the well-known GARCH models. In their seminal paper Engle and Russell (1998) introduced the ACD model. It assumes that observed durations can be explained by a mixing process:

xi = Φii. (3)

The error term i is identically and independently (i.i.d) Weibull (1, γ)

distributed. Φi is proportional to the conditional expected expectation of

the durations.

Moreover, the model assumes that the conditional expected duration ψi = E(xi|Fi−1), where Fi−1 denotes the information set available at time

ti−1, can be described by:

ψi= ω + n X j=1 λjxi−j + r X j=1 βjψi−j. (4)

ω, λ and β are coefficients while xi denotes the standardized durations

as defined above.

Taking expectations, equations (3) and (4) can be set equal. Recall-ing that the expected value of a Weibull (1, γ) distributed random variable equals Γ(1 +1_γ), this yields:5

ψi= Γ(1 +

1

γ)Φi. (5)

Using again equation (3) to replace Φi and solving for xi, the durations

can be explained by the following relation:

xi=

ψi

Γ(1 + 1/γ)i (6)

As shown in Engle and Russell (1998) GARCH software packages can be used to estimate the original ACD. To ensure the positivity of the con-ditional durations, a natural feature since a duration is always positive, the constraints ω > 0, λj ≥ 0, and βj ≥ 0 should hold. For the existence of

the unconditional mean, the sum over the λj and βj should be smaller than

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one. However, parameter constraints complicate the optimization which is needed to estimate the parameters of the model and make standard inference invalid. Therefore, we will use the log-ACD model introduced by Bauwens and Giot (2000) which does need not any constraints to ensure the positivity of durations. It changes the mixing process defined in equation (3) to:

xi = eΘii. (7)

Θi is proportional to the logarithm of the conditional expectation of

the duration, ψi = ln(E(xi|Fi−1)). Moreover, ψi is again described by the

relation of equation (4). Therefore, equations (7) and (4) can be linked and yield the following relation of the durations:

xi=

eψi

Γ(1 + 1/γ)i (8)

The similarity to equation (6) is striking. The only difference is that eψi _{is used instead of ψ}

i. Therefore, no constraints ensuring the positivity

of the conditional expected durations have to be used in the estimation process. While ACD models are quite similar to GARCH-models, the log-ACD model shares a lot of features with the EGARCH-model introduced by Nelson (1991), especially the nonexistence of parameter constraints.

Information criteria like the Akaike information criterion (AIC) or the Schwarz’s Bayesian information criterion (BIC) facilitate the selection of a parsimonious model in equation (4).

So far, we described a model for predicting a conditional price duration. While such an estimation could also be useful itself in risk management, we are more interested in an estimate for the volatility. This estimate could be used in the calculation of a VaR model. Engle and Russell (1998) show that there is a direct link between the instantaneous intraday volatility and the conditional hazard of price durations in the ACD model. The same relationship for the log-ACD model is given in Giot (2000):

ˆ σ2(t|Fi−1) = cp P (t0_i−1) !2 1 eψˆ_φ p(t 0 i−1) . (9) ˆ

σ2(t|Fi−1) is the conditional instantaneous volatility, P (t

0

i−1) is the price

at the begin of the duration, and φp(t

0

i−1) denotes the time-of-the-weekday

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The estimated volatility can be used to define different methods for esti-mating a VaR model. Giot (2005) calculates an intraday VaR by assuming a conditional normal distribution:6

V aRα_norm(t0_i−1) = ˆµ + zα p ˆ σ2_(t|F i−1) (10) ˆ

µ denotes an estimator for the mean, whereas zαdenotes a quantile of the

standard normal distribution. The sample mean will serve as an estimator for the mean in our paper.7 While it is in principle possible to predict for any horizon, we will perform one-step ahead VaR forecasts for every model. Given that it is often found that the normal distribution does not allow to model the fat tails of financial market data adequately, two more approaches using a t(4)- and a non-parametric assumption are taken:

V aRα_t(t0_i−1) = ˆµ + tα(4)

p ˆ σ2_(t|F

i−1) (11)

tα(4) is the α quantile of a t-distribution with 4 degrees of freedom.

Since the parametric distributional assumptions for the innovations made in the calculation of the V aRα_norm and V aRα_t could both be inadequate, we will calculate a VaR using an empirical quantile of the standardized residuals from equation (4). The standardization is done by subtracting the mean and dividing by the standard deviation: ˜i = _sd()i−¯. The VaR is then calculated

as follows: V aRα_emp(t0_i−1) = ˆµ + qα p ˆ σ2_(t|F i−1) (12)

Here qαis the α-quantile of the standardized residuals ˜. The advantage

of the V aRα_emp(t0_i−1) is that its calculation is governed by the data itself and not by a parametric assumption which seems not to be indicated very well by theoretical considerations. However, one assumes, in analogy to the VaR calculation method historical simulation, that past observations are a good description of future values. If a structural break in the marked environment occurred, this assumption may be inadequate.

6_{Giot (2005) is somewhat imprecise concerning how the mean is estimated in his}

method. It is stated that the VaR is estimated for the demeaned series. To ease the evaluation of the VaR we include an unconditional estimate for the mean.

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In order to facilitate a comparison with standard VaR models, the VaR calculated on an unequally spaced grid has to be transformed to an equally spaced grid. Following Giot (2005) we will average the unequally spaced VaR estimates which lie in the time interval of interest to obtain an equally spaced forecast.8 While in principle the aggregation can be done by other methods, we believe that taking an unweighted average is the most suitable. Another possibility would be the use of an weighted average. Guided from the notion that more recent price durations should have a higher im-pact, one could assign them a higher weight. However, we do not believe that finding suitable weights is straightforward in this case. In general, the notion might be correct. Yet to what extent recent durations should get a higher weight may dependent on the volatility of the market environment. Moreover, structural breaks could occur for the weights. In addition to that, it is not clear whether weights should be based on price durations or volatility from the equally spaced grid.

A third way for the transformation would be the use of the median of the unequally spaced predictions instead of the mean. Since the median is more robust against outliers than the mean, this could improve the estimation results. However, we do not believe that the higher outlier-robustness is an advantage in our application. Since financial market data are clustered, the probability that an interval with many outliers is followed by an interval which contains many outliers again, is somewhat higher. The use of the mean takes this into account by being sensitive to outliers, while the median would not account for it.

Therefore, the mean will be used for aggregation in this paper.

While it would be possible to evaluate the performance of the dura-tion based models only, we will calculate well-established models for equally spaced data to facilitate the assessment of the relative performance. The theoretical grounds for the calculation will be described in the next section.

3.2 Models for equally spaced data

To be able to use these models, the original tick data have to be transformed. Raw returns are calculated as continuously compounded returns by ˜Yi =

ln(Pi)−ln(Pi−1), where Pi denotes the transaction price at ti. Subsequently,

the transaction data have to be coerced into an equally spaced grid with intervals of length s. The returns of this grid are calculated as a weighted average using the corresponding transaction volumes as weights. As equally

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spaced data for short aggregation intervals exhibit similar features as their duration counterparts, similar steps are needed to model them. In analogy to the irregular spaced models, seasonal effects will be removed before the stochastic model is fitted.

3.2.1 Seasonality

As can be seen in Figure 3, which plots 30 minutes returns for six adjacent trading days, regular spaced intraday data feature a strong seasonality, too.

Jun 01 09:30 Jun 01 12:00 Jun 01 14:30 Jun 01 17:00

−2e−05

0e+00

Jun 02 09:30 Jun 02 12:00 Jun 02 14:30 Jun 02 17:00

−2e−05

1e−05

Jun 03 09:30 Jun 03 12:00 Jun 03 14:30 Jun 03 17:00

−3e−05

0e+00

Jun 06 09:30 Jun 06 12:00 Jun 06 14:30 Jun 06 17:00

−3e−05

0e+00

Jun 07 09:30 Jun 07 12:00 Jun 07 14:30 Jun 07 17:00

−4e−05

0e+00

4e−05

Jun 08 09:30 Jun 08 12:00 Jun 08 14:30 Jun 08 17:00

−3e−05

0e+00

3e−05

Figure 3: 30 minutes returns for six trading days

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deter-ministic seasonality could be used. Giot (2005) assumes a deterdeter-ministic volatility, too. Therefore, deseasonalized returns are calculated as:

yi =

Yi

pφ_p(ti)

(13) Yi is the return over the interval s, while φp(ti) denotes the

determin-istic intraday seasonal component. It is equal to the conditional expected volatility at the time-of-the-day where the expected volatility is calculated as the weighted average of the squared continuously compounded returns for intervals of length s (Giot, 2005). Yet Figure 3 depicts also that the seasonality seems not be the same for each day of the week. Therefore, as in Giot (2005), the seasonality factor is calculated separately for each day of the week.

3.2.2 Model for deseasonalized returns

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0.0 0.2 0.4 0.6 0.8 1.0 1.2

Absolute values of 30 minutes returns of deseasonalized data

retur

n

01−06 13−06 23−06 05−07 15−07 27−07 08−08 18−08 30−08

Figure 4: Absolute values 30 minutes returns for three months One can notice volatile periods (e.g. in the right end of the plot as of mid-August) and rather involatile periods (e.g. between mid and end of July). Therefore, the volatility of the periods before the upcoming period carries valuable information about the volatility to be expected. A suitable model should account for this. The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models which were introduced by Bollerslev (1986) and are a generalization of seminal ARCH models (Engle, 1982) can be applied in this context. GARCH models describe the conditional volatil-ity hi by lags of squared residuals of the conditional expectation and lags

of the conditional volatility itself. Therefore, they take the aforementioned structure of the data into account:

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τ , η and ν denote coefficients in the GARCH model, while ei is the

residual of the following relation:

yi= Ω + ei. (15)

Ω is an intercept. In addition, it is assumed that the ei can be described

by ei = di

√

hi, where the di are i.i.d and are drawn from some

paramet-ric distribution. Taking into account the trade-off between specifying the conditional distribution of these innovations as correct as possible and still retaining a parsimonious model, a t-distribution is selected. Its degrees of freedom are estimated using a Maximum Likelihood approach. One par-ticular reason for this decision is the fact that the t-distribution is able to account for the fat tails of the conditional distribution. The order of p and q in equation (14) can be determined using information criteria like the AIC or BIC.

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−1.0

−0.5

0.0

0.5

1.0

30 minutes returns of deseasonalized data

retur

n

01−06 13−06 23−06 05−07 15−07 27−07 08−08 18−08 30−08 returns sample mean

Figure 5: 30 minutes returns for three months

The plot shows the 30 minutes returns as dashed line, while the flat line is the sample mean which could be used as an estimator for the conditional mean. The plot makes the bad performance of this estimator obvious. Yet one can again see a tendency that high (low) returns are followed by high (low) returns. In analogy to the volatility, the return of the periods be-fore the current period contains important information about return to be expected.

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˜

yi denotes a trend stationary representation of the original time series

yi. Given the original time series is not trend stationary, an I-times

differ-entiation has to be applied to obtain a trend stationary series. It is often found that returns of financial markets are stationary and no differentiation is needed. ξ, δ and θ are coefficients in the ARMA model.

Following a modified version of the Box-Jenkins approach (Box and Jenk-ins, 1970), the order of m and l in equation (16) has to be determined. As before, this can be done by means of information criteria.

Subsequently to the estimation of its components, the intraday VaR can be calculated as follows: V aRα_GARCH(j) = ˆξ q φ(tj) + m X k=1 ˆ δkY˜i−k q φ(tj) + l X j=1 ˆ θjei−j q φ(tj) + tα q ˆ hφ(tj). (17) tα is the α quantile of a t-distribution which degrees of freedom are

estimated within the Maximum Likelihood approach.

4 Evaluation of the VaR forecasts

Having forecasted the VaR, it is natural to evaluate and validate the model. While a qualitative analysis of the models and their assumptions should not be neglected, this paper focuses on the not less important quantitative anal-ysis. In this context, several approaches were made in the recent literature. The unconditional coverage and the independence of violations of a VaR are the main features of interest (Campbell, 2005; Christoffersen, 1998).

The former refers to the question whether the percentage of coverage which is implied by the model (i.e. the α-level of the VaR) is met in the empirical application. To exemplify that, suppose a V aR0.01. The model implies that in 1% of the trading periods the realized returns will be lower than the VaR and a violation occurs. If the empirical coverage rate is lower, the model underestimates the risk.9 _{In contrast, if it is higher, the model}

is too conservative. Both situations are undesirable. The VaR often de-termines the buffer capital to be held available by a financial institution. If the forecasted absolute value is too low, the institution runs the risk of bankruptcy. In contrast, if it is too high, too much buffer capital is provided which increases the capital costs.

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4.1 Evaluation of the unconditional coverage

To assess the unconditional coverage, it can be tested whether the empirical and theoretical coverage rates are statistically distinguishable from each other. The classical statistical test in this context has been suggested by Kupiec (1995). However, it has two major drawbacks: (1) It is not defined if no violations occur and (2) it cannot detect patterns in the violations and therefore the independence of violations cannot be tested. The first disadvantage can be solved by using a test for unconditional coverage as suggested by Christoffersen (1998). The solution for the second problem is described in the following sections.

Taking the VaR forecasts as one-sided interval forecasts, an indicator sequence which takes the value one in case of a correct VaR forecast and zero in case of a violation can be defined:

It=

(

1, if yt≥ V aR

0, otherwise . (18)

As explained above, the empirical coverage ratio should be as close as possible to the implied coverage ratio of the model. Therefore, a VaR model is efficient with respect to the information set Ft−1 if

E[It|Ft−1] = 1 − α = p ∀t (19)

for the sequence of forecasts {V aRt}Tt=1 (Christoffersen, 1998). In other

words, a good VaR model should produce a ratio of ones in the indicator sequence which is equals as exact as possible its ex ante implied value of 1 − α. To ease notation, p refers to 1 − α. This should hold for all periods t. The information set Ft−1 which will be used in this test consists of the

past realizations of the indicator sequence: Ft−1= {It−1, It−2, ..., I1}.

Therefore, we can write:

E[It|Ft−1] = E[It|It−1, It−2, ..., I1] = p. (20)

Recalling that the expected value of a Bernoulli variable, which takes value one with probability p and zero with probability 1 − p, equals p, equa-tion (20) can be written as:

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Applying the definition of the Bernoulli distribution on equation (21), one can notice that testing if E[It|Ft−1] = p is equivalent to testing whether

the sequence {It}Tt=1 is i.i.d. Bernoulli with parameter p (Christoffersen,

1998).

As stated in Christoffersen (1998), a likelihood ratio test can be used to test the null hypothesis of E[It] = 1 − α = p against its alternative E[It] 6= p

assuming independence. The likelihood of the null hypothesis writes:

L(p; I1, I2, ..., IT) = (1 − p)n0pn1. (22)

n0and n1 denote the number of zeros and ones contained in the indicator

sequence, respectively. The first factor of the product in equation (22) is the likelihood contribution of the violations, i.e. the observations which indica-tor has value zero. The second facindica-tor comes from the correctly forecasted observations.

The likelihood of the alternative hypothesis is calculated as follows:

L(π; I1, I2, ..., IT) = (1 − π)n0πn1. (23)

π is the empirical coverage ratio and needs to be estimated. Its Maximum Likelihood estimate equals ˆπ = n1/(n1+ n0) and equals the percentage of

correct forecasts.

Using the likelihoods defined in equations (22) and (23) the test can be constructed as (Christoffersen, 1998): LRuc= −2 log L(p; I1, I2, ..., IT) L(ˆπ; I1, I2, ..., IT) a ∼ χ2(k − 1) = χ2(1). (24)

k denotes the number of the different possible outcomes of the sequence. While this test is a powerful tool for the detection violations of the correct coverage ratio, it has no power against clustered violations of the VaR. To test the independence of the violations, different tests have to be used. The construction of these tests will be described in the sections 4.2, 4.4 and 4.5.

4.2 Evaluation of the independence of violations

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clusters, the model might not be able to adjust fast enough to a changed market environment. A general problem for all kinds of independence tests is to formulate an explicit hypothesis of dependence. Therefore, several sug-gestions for independence tests can be find in the literature. A Markov-chain based approach suggested by Christoffersen (1998) is able to detect adjacent violations. Moreover, Portmanteau tests of the indicator sequence as sug-gested in Berkowitz et al. (2011) and a logistic regression on the indicator sequence are powerful alternatives. All three tests will be used in this paper and their construction is described below.

The first alternative models a potential dependence in the indicator se-quence by means of a first-order Markov-chain. One underlying assumption of a c-order Markov chain is that the temporal dependence is of order c. To exemplify this, suppose that a weather forecast will be constructed us-ing a Markov-chain. Assumus-ing the order one the modeler implies that the weather of today is only influenced by the weather of yesterday and not the day before yesterday. Transferred to the VaR evaluation framework this means that a potential dependence of the violations is assumed to be present in adjacent periods.

Since the indicator sequence features two possible outcomes, either a violation or correct forecast, and a first-order Markov-chain is used, the matrix of the transition probabilities can be written as:

Π1=1 − π01 π01

1 − π11 π11

. (25)

πij = P (It = j|It−1 = i) is the probability that in the current period

the indicator sequence takes value j, given that it took value i in the pe-riod before. By construction, this matrix allows for temporal dependence. The probability of taking value one given that a violation as occurred (i.e. the indicator sequence takes value zero), which can be found in the second column of the first row, needs not to be equal to the probability of taking value one given value zero before, which can be found in the second column of the second row. Since the indicator sequence can only take two possible outcomes, the first column of the transition matrix can be constructed by subtracting the respective element of the row in the second column from one.

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L(Π1: I1, I2, ..., IT) = (1 − π01)n00πn0101(1 − π11)n10πn1111 (26)

where nij denotes the number of observations i followed by j.

Maximizing the log-likelihood and solving for the parameter yields the following estimator of the transition matrix:

b Π1 =   n00 n00+n01 n01 n00+n01 n10 n10+n11 n11 n00+n01  . (27)

Using this result, a first-order Markov-chain can be estimated on the indicator sequence {It}.

However, the hypothesis of independence still needs to be operational-ized. Given independence, the instant in time of a violation should not give rise to any information about later or former violations. Transferred to the first order Markov-chain framework, this means that the transition matrix is written as: Π2 = 1 − π2 π2 1 − π2 π2 . (28)

Unlike the matrix given in equation (25) this matrix does not allow for temporal dependence. Since both rows of the transition matrix are equal, the probability of the occurrence of a violation does by construction not depend on former values of the indicator function. Therefore, the transition probability π2 does not use any temporal index. The likelihood under the

null hypothesis is

L(Π2; I1, I2, ..., IT) = (1 − π2)(n00+n10)π2(n01+n11). (29)

The Maximum Likelihood estimator for _cπ2 equals (n01 + n11)/(n00+

n10+ n01+ n11). Again, the first factor contains the likelihood contribution

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is the number of correct forecasts.10 As before, an asymptotically χ2 (1)-distributed likelihood ratio can be constructed:

LRind= −2 log

L( bΠ2; I1, I2, ..., IT)

L( bΠ1; I1, I2, ..., IT) a

∼ χ2_{(k − 1) = χ}2_(1). ₍₃₀₎

As can be seen easily in equation (30) the test does not depend on the true coverage p, and therefore has no power to test violations of the model‘s implied coverage ratio. A test which heals this disadvantage will be described in the next section. In some situations however, it may be useful to test only the temporal independence and not the coverage ratio.

Moreover, it can be criticized that using a Markov-chain of order one does not detect temporal dependence of higher orders. If the order would be known, a Markov-chain of higher order could be used. However, the number of transition probabilities to be estimated increases substantially with the order of the chain. Alternative independence tests which heal this disadvantage are described in section 4.4 and 4.5. First, Christoffersen´s test of conditional coverage will be described in the next section to complete his testing framework.

4.3 Evaluation of the conditional coverage

Since both unconditional coverage and independence are important features of a VaR model, yet none of the described test statistics has power to test both attributes, a joint hypothesis of conditional coverage should be tested. The approach of Christoffersen (1998) allows testing such a hypothesis.

Combining both above described tests, the null hypothesis of uncondi-tional coverage will be tested against the alternative of the independence test. Therefore, the likelihood ratio writes:

LRcc= −2 log

L(p; I1, I2, ..., IT)

L( bΠ1; I1, I2, ..., IT)

. (31)

As proved in Christoffersen (1998), this likelihood is asymptotically χ2(1) distributed. The astute reader will notice that we used both likelihoods

be-10

Instead of n00+n10(n01+n11) one could also write n0(n1). Yet to keep the consistence

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fore. If the first observation is ignored in the test of unconditional coverage, the conditional likelihood can be written easily as:11

LRcc = LRuc+ LRind. (32)

Knowing the test for conditional coverage, one could legitimately won-der whether the separate tests of unconditional coverage and independence are needed. However, the conditional coverage test has a decreased abil-ity to detect inadequate VaR models which perform well in one of the two features Campbell (2005). This becomes intuitive if one takes again a look at equation (32). If a model performs well in the unconditional test, the corresponding likelihood ratio LRuc is relatively low. If it performs not as

good in the independence test, LRind will be relatively high. Since booth

likelihood ratios are summed in the conditional coverage test, there is the possibility of a compensation. Therefore the conditional test has a decreased ability to detect models which perform well in only one of the two basis tests (Campbell, 2005). To avoid this disadvantage, all three tests will be used in the empirical application in the next section.

Having completed the testing framework suggested by Christoffersen (1998), we will describe two alternative test for independence in the next sections.

4.4 Ljung-Box Test

If the violations occur independently, it is a necessary condition that the demeaned indicator sequence {It− p}Tt=1 does not feature any

autocorrela-tion. The Portmanteau test, also known as Ljung-Box test, can be applied (Berkowitz et al., 2011) in this context. It allows to test whether the first autocorrelations are significantly different from zero (Ljung and Box, 1977). The number of autocorrelations in the test should be high enough to cap-ture possible dependent lags. Transferred to the framework of this paper, that means that at least as many lags as daily seasonal correction intervals should be used.

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4.5 Logistic regression

The aforementioned feature of Christoffersen’s test of accounting only for temporal dependence of adjacent observations, leads to a crucial disadvan-tage in the intraday VaR framework. If violations would frequently occur at the same time of the day, the independence test would not reject the model. Due to the fact that the VaR is estimated a few times a day, these violations do not appear adjacently but in a different regular pattern depending on the forecast horizon of the VaR. Yet the first-order Markov-chain is not able to grasp such dependencies. Therefore, a different test should be used which allows the inspection of this feature.

A regression can be run on the indicator sequence of lags of the indicator sequence and former values of the VaR :

It= κ + g X i=1 ρiIt−i+ z X j=1 ζjV aRt−j (33)

κ, ρ and ζ are regression coefficients, whereas V aRt−j denotes the VaR

forecast which was made j periods before t.

Using a sufficient order of g an z heals the above described disadvantage. If the VaR model uses all available information efficiently, the regression should not yield any significant coefficients.

In this test, a linear regression is used in a rather unusual context. Often researchers use it to get empirical evidence for their postulated model by obtaining significant coefficients. Significant coefficients are, at least in most cases, are positive evidence for the model. However, in our framework a good model does produce an independent indicator sequence. By definition, an independent sequence cannot be explained by any covariates. Therefore, significant parameters in the regression equation 33 is evidence against the model which produced the sequence. The test was suggested by Clements and Taylor (2003) and Engle and Manganelli (2004).

Since the dependent variable in this regression is a binary variable, esti-mating this regression with ordinary least squares would cause heteroskedas-ticity. To facilitate valid inference, one would have to use robust standard errors. Using a logistic regression however is more suitable in this context. Then standard inference can be used to test the single and joint significance of parameters.

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5 Empirical Application

This section describes the application of the described models to data from the German Stock exchange DAX in 2011. The dataset contains the order book of the online trading platform XETRA and has been obtained from the KKMDB (capital markets database Karlsruhe.12) The DAX consists of the 30 biggest German companies. Trading is only possible via the online platform.

While the models described in the first part of this paper can be applied to all stocks, the Allianz stock was selected in this illustration. Allianz is a German insurance company and one of the most heavily traded stocks in the index. The estimation period used in this paper spans over the months June, July and August, while September and October are used as evaluation period. If this periods are changed, the estimation results change a bit but all general conclusion remain the same.

5.1 Data preparation

5.1.1 Filtering and calculation of price durations

Before the described models can be applied, the dataset needs to be pre-pared. After selecting trades which are affiliated with the Allianz share, all but continuous trading observations were deleted. As described in Engle and Russell (1998), the use of non-continuous trades which are observed e.g. in the opening or closing auction, distorts the normal price process.

In addition to that, a few more idiosyncrasies of the dataset had to be taken into account. In contrast to most datasets used in the literature (Pacu-rar, 2008), this dataset did not record the time of a trade in seconds, but in hundredths of seconds. Moreover, the regulations of the XETRA platform offer an incentive to traders to split transactions in order to facilitate their execution. Therefore, similar to Grammig and Wellner (2002), an algorithm was used to identify split transactions. If two or more consecutive trades within one second are executed at the same price, the trades are regarded as one split trade. Trading volumes were summed up and the timestamp of the first trade was used. Subsequently, price durations were calculated. Given that the trading time was recorded in a hundredth of a second, zero trade durations played a minor role and were removed.

12

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5.1.2 Seasonal adjustment

As described in the theoretical part of the paper, intraday data feature a strong seasonality. The plot of the price durations of the first trading day in Figure 1 which can be found in section 2 on page 7 gives a nice graphical representation of this stylized fact.

It is striking to see that price durations are rather low in the morning and before the end of the trading day, but tend to be longer around lunchtime.

Since a regression of the price duration on day-of-the-week dummies yielded individual and joint significance of the parameters, a seasonal cor-rection factor was estimated for each day of the week.13 Figure 6 depicts this factor for Mondays which was calculated as described in section 3.1.1.

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 5000 10000 15000 20000 25000 20 30 40 50

time in seconds from market opening

φ

(

t

)

Figure 6: Seasonality correction factor φ for Mondays

The shape in general is similar to corresponding plots in the literature

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which can be found in Engle and Russell (1998) and Giot (2000). However, the second hump after the maximum of the day around lunchtime seems to be typical for the DAX data. It can be found for every day of the week and also for different stocks and different periods.

These factors were used to obtain diurnally adjusted price durations as described in equation (2).

It is interesting to see that the seasonal adjustment of the data consid-erably lowered Ljung-Box χ2 test statistics for the first 15 autocorrelations as depicted in the first and third column of Table 1.

trading day

price du-rations

p-value des. du-rations p-value residuals log-ACD model p-value 1 144.97 0.00 61.19 0.00 8.64 0.80 2 473.94 0.00 113.33 0.00 21.57 0.06 3 604.64 0.00 387.34 0.00 20.33 0.09 4 109.09 0.00 42.42 0.00 13.40 0.42 5 262.11 0.00 23.12 0.08 15.57 0.27 6 335.58 0.00 45.62 0.00 17.45 0.18 7 360.12 0.00 223.99 0.00 9.15 0.76 8 183.38 0.00 109.54 0.00 9.16 0.76 9 117.17 0.00 37.68 0.00 14.98 0.31 10 172.60 0.00 57.97 0.00 11.85 0.54 Table 1: Ljung-Box test for 15 lags

As can be seen in Figure 3 on page 17 which shows 30 minutes returns for 6 adjacent trading days, intraday seasonality is not restricted to unequally spaced data. A daily pattern, which seems to differ for different days of the week, is visible. Therefore, a seasonal correction factor as described in section 3.2.1 is estimated.

5.2 Description of the data

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returns 30 min price durations c=0.02 N 1853 249371 mean -0.02 0.64 sd 0.18 0.88 max 1.63 19.74 min -1.09 0.0003 1. Quartile -0.11 0.14 median -0.01 0.34 3. Quartile 0.08 0.79

Table 2: Summary statistics of deseasonalized data

the number of 30 minutes intervals. This means that a VaR based on price durations is updated much more often than its equally spaced counterpart. This is backed up by the fact that the average non-deseasonalized price duration is around 13 seconds. The evaluation part will reveal whether this leads to an advantage of the unequally spaced models.

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0 1 2 3 4 5 6

0.0

0.5

1.0

1.5

Kernel density of deseasonalized price durations

N = 249371 Bandwidth = 0.03578

Density

Figure 7: Kernel density of deseasonalized price durations

As can be seen, almost all probability mass lies in the interval between zero and one.14 Having an approximate idea about the data we can continue with the estimation of the models.

5.3 Estimation of the VaR for unequally spaced data

Table 1 shows that the diurnally adjustment considerably lowered the Ljung-Box test statistics, but they are high enough to provide evidence for sig-nificant autocorrelations. Therefore, the model for the price durations is estimated. The information criteria AIC and BIC are used to determine the order of n and r. As can be seen in Tables A13 and A14, both informa-tion criteria suggest a log-ACD(2,2) model. Since the performance of this

14

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model was not so good in the latter described evaluation phase, we esti-mated the more parsimonious log-ACD(1,1) model. The use of the more parsimonious model is in line with the results of Dionne et al. (2009). A Maximum Likelihood approach was used to estimate the parameters of this model. The log-likelihood function used in this approach writes (Bauwens and Giot, 2000): N X i=1 ln(γ) − ln(xi) + γ ln[xiΓ(1 + 1 γ)] − γψi− xiΓ(1 + 1_γ) eψi . (34)

An exponential transformation of γ was used in the estimation to ensure its positivity and facilitate the correctness of standard inference. Different starting values were used in order to get evidence for a global optimum. The estimation results can be found in Table 3. They are similar for different estimation periods and stocks. Furthermore, they are similar to findings from the literature (Giot, 2000).

Estimate Std. error t value Pr(> t) ˆ ω -0.0638 0.0020 -32.6565 0.0000 ˆ λ 0.0623 0.0019 32.3707 0.0000 ˆ β 0.9877 0.0009 1056.8377 0.0000 d ln(γ) -0.0421 0.0031 -13.4554 0.0000 Table 3: Estimation results log-ACD (1,1)

First of all, it is striking that all parameters are significantly different from zero. For covariance stationarity, |β| has to be smaller than one. A t-test gives evidence that this is the case. The relatively high value of β underlines the economic significance of the lagged price duration. While the innovation parameter λ is statistically distinguishable from zero, its economic significance is rather low. In general, a ceteris paribus higher λ increases the overdispersion of the fitted durations (Bauwens and Giot, 2000). As can be seen in Table 2, the deseasonalized data feature a moderate overdispersion. In this sense, the estimation result lies within the expected range.

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simplified to the exponential distribution. The Ljung-Box statistic is sub-stantially reduced as can be seen in the two last columns of Table 1. Pacurar (2008) notes in her overview article that even after fitting a model of the ACD-class many authors report autocorrelations. Since the parsimonious model used above does not face this problem, it is again underlined that the lower order of the lags should be sufficient.

These point estimates can be used for the calculation of the VaR as described in equations (10) to (12). A regular spaced interval of s 30 minutes is used to calculate the final VaR estimates which are depicted in Figure 8 to 10. −2e−04 0e+00 2e−04 4e−04 01−09 08−09 15−09 22−09 29−09 06−10 13−10 20−10 27−10 return VaR

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−4e−04 −2e−04 0e+00 2e−04 4e−04 01−09 08−09 15−09 22−09 29−09 06−10 13−10 20−10 27−10 return VaR

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−1e−04 0e+00 1e−04 2e−04 3e−04 4e−04 5e−04 01−09 08−09 15−09 22−09 29−09 06−10 13−10 20−10 27−10 return VaR

Figure 10: V aR0.01_emp forecast and returns

The first graphical impression is that the e − V aRα_log−ACD is violated too often while the norm − V aRα_log−ACD seems to perform quite well but may be violated a bit too often. The plot of the t − V aRα_log−ACD depicts that it is not violated very often and therefore the model may perform well in the subsequent statistical tests. In order to get evidence for this impression, the above described tests of Christoffersen will be applied in section 5.5.

5.4 Estimation of the VaR for equally spaced data

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used to estimate the mean process. In order to determine the order of the ARMA part, the information criteria AIC and BIC are used. As can be seen in Tables A15 and A16 in the appendix, the AIC suggests an ARMA(5,4)-model whereas the BIC suggests an ARMA(2,2) ARMA(5,4)-model. Since a Ljung-Box Test on the residuals of an ARMA(2,2) model reveals no evidence for signif-icant autocorrelations of the first 15 lags (p-value 0.19), we opt for the more parsimonious model.

Subsequently the GARCH order has to be determined. Using the ARMA(2,2) for the mean, the same information criteria are used to determine the order. As can be seen in Tables A17 and A18, the AIC suggests an GARCH(1,6) model whereas the BIC suggests an ARCH(1) model without a GARCH part. Again a Ljung-Box test on the squared residuals gives evidence (p-value 0.36) that the more parsimonious model removes the autocorrelation successfully and therefore we opt for it. The estimation results, which were obtained in a Maximum-Likelihood approach, can be found in Table 4.

Estimate Std. Error t value Pr(>|t|) ˆ Ω -0.01 0.00 -1158.06 0.00 ˆ δ1 -0.25 0.00 -9820.81 0.00 ˆ δ2 0.75 0.00 30013.52 0.00 ˆ θ1 0.32 0.0001 3242.64 0.00 ˆ θ2 -0.71 0.0001 -7315.44 0.00 ˆ τ 0.02 0.003 9.99 0.00 ˆ η 0.12 0.06 2.07 0.04 b df t-distribution 4.10 0.49 8.41 0.00 Table 4: Estimation results ARMA-ARCH model

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−1e−03 −5e−04 0e+00 5e−04 01−09 08−09 15−09 22−09 29−09 06−10 13−10 20−10 27−10 return VaR

Figure 11: V aR0.01_GARCH forecast and returns

The plot shows that the VaR forecasts follow the changing volatility of the returns quite well and from the plot it is expected that the V aRα_GARCH will perform well in the statistical tests in the next section.

5.5 Evaluation of the Forecasts

In addition to the graphical impression, it is important to run statistical tests in order to assess the performance of the model. As described in the theoretical part of the paper, test statistics suggested by Christoffersen (1998), a Portmanteau test and a logistic regression are useful in this context.

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α 0.05 0.025 0.01 0.005 V aRα_norm 0.0134 0.0000 0.0000 0.0000 V aRα t 0.7911 0.8645 0.8005 0.7229 V aRα_emp 0.0000 0.0000 0.0000 0.0000 V aRα_GARCH 0.6616 0.9479 0.9076 0.3423

Table 5: p-values test of unconditional coverage

Table 5 corresponds to the test of unconditional coverage. The null hypothesis states a correct unconditional coverage. As can be seen easily, the null hypothesis has to be rejected for the V aRα_norm and the V aRα_empfor almost all α levels. Only for α = 5%, it cannot be rejected for the V aRα

norm

on a 0.01 level. The V aRα_GARCH and the V aRα_t however perform very well in this test. The null hypothesis cannot be rejected on every tested level of α.

While the test results for the V aRα_emp, the V aRα_t and V aRα_GARCH could be expected from the plots, the results for the V aRα_norm are a bit surprising. The coverage ratios of the models tabled in Table 6 give a useful insight into the reasons for this test result.

α 0.05 0.025 0.01 0.005 V aRα_norm 0.9289 0.9453 0.9562 0.9631 V aRα_t 0.9521 0.9740 0.9891 0.9959 V aRα_emp 0.8126 0.8222 0.8222 0.8249

V aRα_GARCH 0.9535 0.9754 0.9904 0.9973

Table 6: Coverage ratio of the models

Ideally the coverage ratio would equal 1 − α. As can be seen in the table, the V aRα_norm always has a lower coverage ratio than necessary. A financial institution which uses such a VaR as a part of the buffer capital calculation has a higher risk of bankruptcy than implied by the α-level.

The table clearly reveals that the V aRα_empforecasts are way out of line. This is due to the fact that qα is relatively small compared to the quantiles used

in the other models.

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Having tested the unconditional coverage, it is interesting to know whether the violations come up independently. The p-values of this test can be found in Table 7. α 0.05 0.025 0.01 0.005 V aRα_norm 0.4880 0.3453 0.7089 0.9989 V aRα_t 0.5547 0.3136 0.6737 0.8750 V aRα_emp 0.0760 0.0484 0.0484 0.0303 V aRα GARCH 0.6039 0.3401 0.7127 0.9165

Table 7: p-values test of independent violations

They offer a more positive evidence for the use of the models. The null hypothesis of the independence of violations is rejected on the 0.05 level only for three of the four V aRα_emp models. All other models perform very well and cannot be rejected.

α 0.05 0.025 0.01 0.005 V aRα_norm 0.0370 0.0000 0.0000 0.0000 V aRα t 0.8109 0.5931 0.8864 0.9275 V aRα_emp 0.0000 0.0000 0.0000 0.0000 V aRα GARCH 0.7942 0.6330 0.9282 0.6335

Table 8: p-values test of conditional coverage

Before a final conclusion is drawn within the framework of Christoffersen, the joint hypothesis of a correct conditional coverage should also be taken into account. As can be seen in Table 8, the V aRα_GARCH and the V aRα_t perform again well in this category. The correct coverage hypothesis of the V aRα

norm is only not rejected for an α of 0.05 on a 1% level. As expected

from the results obtained so far, the V aRα_emp has the worst performance of all models.

Before an overall final conclusion is drawn, the test results of the Ljung-Box and the logistic regression should be taken into account. The test results of the former can be find in Table 9

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Table 9: p-values Ljung-Box test of 20 lags of the indicator sequence

models has significant autocorrelations at the 5% level. According to this result, none of the models has to be rejected.

Due to the seasonality of the data it would also be possible that time-of-the-weekday dependencies are present. To test this, 100 lags were used and the results are tabled in Table 10.

α 0.05 0.025 0.01 0.005 V aRα_norm 0.2259 0.1502 0.7963 0.8645 V aRα_t 0.4444 0.1123 0.9940 0.7745 V aRα_emp 0.0622 0.0498 0.0631 0.0754

V aRα_GARCH 0.8206 0.7981 0.4175 1.0000

Table 10: p-values Ljung-Box test of 100 lags of the indicator sequence

Apart from the V aRα_emp, no significant autocorrelations can be found. Moreover it is striking to see that the p-values of this model are considerably lower compared to Table 9. This could be caused by time-of-the-weekday dependencies.

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Table 11: p-Values LR-test of joint significance of all parameters

Since significant parameters are evidence against the independence of violations, the parameters would ideally be jointly insignificant.

As can be seen from the table, the independence hypothesis has to be rejected for one of the V aRα

norm models and for all V aRαempon a 5%

signif-icance level. However, the V aRα_t and V aRα_GARCH models perform well for all levels of α.

In addition to the joint significance, the single significance of the pa-rameters can be inspected. To save space here, the results can be found in Tables A20 to A35 in the appendix. Coefficients significant at the 5% level are printed in bold face.

Table 12 summarizes these inference results. It tables the number of parameters which are significant at the 5% level in the logistic regression.

α 0.05 0.025 0.01 0.005 V aRα_norm 1 0 0 0

V aRα_t 1 1 2 0

V aRα_emp 0 0 1 1

V aRα_GARCH 0 2 4 0

Table 12: Number of parameters which are statistically significant at the 5% level in the logistic regression

Ideally, no parameter would be significant. However, as can be seen from the table no model reaches this on all α-levels. While the two models which performance was rather unsatisfactory in the test results obtained so far, the V aRα_norm and the V aRα_emp, do not feature many significant param-eters, the V aRα

t and the V aRαGARCH do not perform so well. Especially the

V aR0.01_GARCH yields 4 significant parameters. If one inspects Table A34 on

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are significant. Since 17 estimation intervals are used during one day, this is evidence that the model has especially problems at certain time of the days. However, one should be careful to draw conclusions from the number of significant parameters. First, this number depends on the selected signifi-cance level. Second, there is no objective rule which can decide whether a model with many significant parameters at a relatively high significance level (e.g. 0.1) or a model with less significant parameters at a lower level (e.g. 0.01) should be preferred. All in all one can conclude that there is evidence against the independence of violations for all models. However, this does not mean that the models in this application perform not as good as models in comparable applications in the literature. Since the usage of this test is barely found, it would be interesting to apply these test in applications where the models passed all tests used by their authors.

In this sense, our test results are a more promising for the use of log-ACD models for intraday VaR calculation than the ones obtained by Giot (2005). While he does not test the independence of the violations, his test of correct unconditional coverage rejects his model (a V aRα_norm model) in almost all cases.15 However, our test results of the V aR_tα model are very promising. It perform as least as well as the ARMA-GARCH models in all tests. And according to the logistic regression the V aRα_t has fewer problems with the independence of violations.

The test results underline the importance of testing different hypotheses. Since not all the models perform well in each of the tests, it depends on the requirements of the investor if a model can be used. The V aRα_empperformed bad in all test and should not be used at all. While in most applications a correct conditional coverage might be the most important feature, in others the independence of the violations might be particularly interesting. A very risk averse investor whose utility function is very sensitive to adjacent vio-lations of the VaR might prefer this, especially if the model with the correct conditional coverage did not pass the unconditional coverage. In this case, the V aRα

norm might serve as a useful alternative. Yet one has to be aware

that the coverage ratio was too low. Therefore the ARMA-GARCH and the V aRα_t model, which passed all but one tests successfully, are probably the models preferred by all investors. In the last test, the V aRα

t was even

one step ahead of the V aRα_GARCH. Therefore, we recommend the use of the V aRα_t for the calculation of an intraday VaR. More applications in differ-ent datasets should be done to see whether our results are restricted to our

15

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dataset. Due to the high informational content we believe that the V aRα_t will also perform well in other datasets.

6 Conclusion

We inspected different methods for calculating an intraday VaR in this pa-per. The log-ACD approach using a conditional normal distribution per-forms quite well in terms of independence of violations. Its performance is not as good as in the unconditional coverage but there seems to be room for improvement in future research. One problem might be the transformation from the unequally to the equally spaced grid. In addition to that, other choices for the conditional distribution might address this disadvantage. A nonparametric approach used in this paper did not perform very well. The coverage ratio suggested that the problem of the V aRα_norm is rather speci-fying the conditional mean correctly than predicting the correct conditional volatility. A hybrid approach, which used an ARMA model to forecast the mean and a V aRα_norm for the volatility, could not improve the performance. Specifying the conditional mean correctly in the unequally spaced grid could be a path for future research. Moreover, another parametric distribution for the residuals of the model could improve the performance.

The ARMA-GARCH and V aRα_t approach passed almost all tests suc-cessfully and are the landmark to beat for all upcoming models. A logistic regression on the correct forecast indicator sequence gave weak evidence that the V aRα_t performs better. Due to the higher informational content used in this model compared to the V aRα_GARCH we are optimistic that this results holds in different datasets, too. Future research should find ways to estimate the degrees of freedom of the t-distribution on grounds of the dataset.

In this sense, both the V aRα_t and the V aRα_GARCH are fast since they are able to adopt quickly to a changing market environment, but not furious since they forecast correctly in an involatile market environment, too.

Acknowledgements

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Furthermore I would like to thank my mentor Roland Ismer for encour-aging me to study a full program abroad. Regina Riphahn and Ingo Klein made me familiar with the great science of econometrics.

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Fast but not Furious

-Calculation and Evaluation of an

Intraday Value at Risk

University of Groningen

Master Thesis Econometrics, Operations Research and

Actuarial Studies

-Calculation and Evaluation of an Intraday Value at

Risk

Abstract

Contents

List of Figures

List of Tables

1

Introduction

2

Features of price durations

3

Econometric Modeling

4

Evaluation of the VaR forecasts

5

Empirical Application

6

Conclusion

Acknowledgements

References