Forecasting One-day-ahead VaR and Intra-Day Realized Volatility in the Athens Stock Exchange Market Timotheos Angelidis∗ Department of Banking and Financial Management, University of Piraeus, 80, Karaoli & Dimitriou street, Piraeus GR- 185 34, Greece Athens Laboratory of Business Administration. Athinas Ave. & 2a Areos street, Vouliagmeni GR-166 71, Greece Stavros Degiannakis∗∗ Department of Statistics, Athens University of Economics and Business, 76, Patision street, Athens GR-104 34, Greece
Managerial Finance, 2005, forthcoming
Abstract Purpose We evaluate the performance of symmetric and asymmetric ARCH models in forecasting both the one-day-ahead Value-at-Risk (VaR) and the realized intra day volatility of two equity indices in the Athens Stock Exchange (ASE). Methodology / Findings Under the VaR framework, we find out that the most appropriate method for the Bank index is the symmetric model with normally distributed innovations, while the asymmetric model with asymmetric conditional distribution applies for the General index. On the other hand, the asymmetric model tracks closer the one-stepahead intra day realized volatility with conditional normally distributed innovations for
∗
Corresponding Author. Tel.: +30-210-8964-736. E-mail address:
[email protected] ∗∗ Tel.: +30-210-8203-120. E-mail address:
[email protected]. The usual disclaimer applies.
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the Bank index but with asymmetric and leptokurtic distributed innovations for the General index. Originality / Value Therefore, as concerns the Greek stock market, there are adequate methods in predicting market risk but it does not seem to be a specific model that is the most accurate for all the forecasting tasks. Keywords: Asymmetric Power ARCH model, Intra Day Realized Volatility, Skewed-t Distribution, Value-at-Risk, Volatility Forecasting. Classification: Research Paper JEL: C32, C52, C53, G15.
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Introduction
Value-at-Risk (VaR) refers to a portfolio's worst outcome that is likely to occur at a given confidence level (95% or 99%) and according to the Basle Committee on Banking Supervision's in January 1996 (Amendment to the Capital Accord to Incorporate Market Risks), the VaR methodology can be used by financial institutions to calculate capital charges in respect of their financial risk. Giot and Laurent (2003a, 2003b) proposed to market practitioners the APARCH skewed Student t model, which accommodates both the skewness and the kurtosis of financial time series. They argued that the overall performance of the model was better than that of the symmetric one, as it described the tails of the empirical distribution more accurately. Huang and Lin (2004) reached to similar conclusions. The issue of the asymmetry in a risk management framework was also considered by Brooks and Persand (2003). They substantiated that model, which does not allow for asymmetries either in the unconditional return distribution or in the volatility specification underestimates the “true” VaR. Andersen and Bollerslev (1998) introduced an alternative volatility measure, the realized volatility, which is based on the idea of using higher frequency data to generate more accurate volatility estimates of lower frequency. As noted by Ebens (1999) and Andersen and Bollerslev (1998) for daily volatility forecasts, the discretely sampled daily returns constitute a noisy estimator, but the accuracy improves as the sampling frequency is increasing. The purpose of the paper is twofold. First, we examine the performance of the most well known parametric volatility models in a risk management framework by estimating the 95% and the 99% VaR numbers for the General and the Bank indices of Athens Stock Exchange. Second, we investigate the ability of ARCH models in
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forecasting the intra day realized variance. Under this framework, we will able to examine whether a model forecasts the observed variance accurately and also generates VaR estimates close to the expected ones. We estimate two volatility specifications, the symmetric GARCH and the asymmetric APARCH processes. Moreover, we extend the processes in order to incorporate the skewness and the excess kurtosis that the asset returns exhibit, by assuming that the conditional innovations are Student t and skewed Student t distributed. The data set is consisted of daily closing prices of the General and the Bank indices from 25th of April 1994 to 19th of December 2003 and their intra day quotation data from May 8th of 2002 to December 19th 2003. The results point out that different models achieve the most accurate VaR and volatility forecasts indicating to portfolio managers the significance of creating a specific model for each case. Specifically, the APARCH - skewed Student t model should be applied in order to estimate the VaR of the General index, while the GARCH–normal is the best performing technique in the case of the Bank index. In contrast, the APARCH models for both indices are preferred over the others in order to estimate the realized volatilities. However, the distributional assumption differs, as in the case of the General index (Bank index) the normal (skewed Student t) distribution generates the best forecasts. Angelidis et al. (2004), who estimated a set of ARCH models in order to predict the one-day-ahead VaR of five international stock indices, have also found that there is not a specific model, which achieves the highest performance in all cases. The volatility forecasting models, the VaR and volatility evaluation methods are presented in the 2nd section. The 3rd section illustrates the results of the study for the Greek stock market and section 4 concludes.
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The ARCH Volatility Forecasting Techniques
Let yt = ln (Pt Pt −1 ) denotes the return series, where Pt is the price of an equity index at day t . The asymmetric power ARCH, or APARCH, model, which was introduced by Ding et al. (1993), is presented in the following framework:
yt = c0 + ε t
ε t = ztσ t (1)
zt ~ f (0,1) i .i .d .
σ tδ = a0 + a1 ( ε t −1 − γε t −1 ) + b1σ tδ−1 , δ
where c0 is a constant parameter, ε t is the innovation process, σ t is the conditional standard deviation, zt is an i.i.d. process, f (.) is the probability density function and
F (.) is the cumulative density function with a 0 > 0 , a1 ≥ 0 , b1 ≥ 0 , δ ≥ 0 and
γ < 1 . The model imposes a Box and Cox (1964) power transformation in the conditional standard deviation process and the asymmetric absolute innovations. In the APARCH model, good news, (ε t −i > 0 ) , and bad news, (ε t −i < 0 ) , have different predictability for future volatility, because the conditional variance depends not only on the magnitude but also on the sign of ε t . For δ = 2 and γ = 0 the APARCH model reduces to Bollerslev’s (1986) representation of the generalized ARCH, or GARCH, model. Surveys of Bera and Higgins (1993), Bollerslev et al. (1992), Bollerslev et al. (1994), Degiannakis and Xekalaki (2004), Gourieroux (1997) and Poon and Granger (2003) cover a wide range of ARCH presentations. In the influential paper of Engle (1982), the density function of z t , f (.) , was the standard normal distribution. Bollerslev (1987) tried to capture the high degree of leptokurtosis that is presented in high frequency data and proposed the Student t distribution in order to produce an unconditional distribution with thicker tails. Lambert and Laurent
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(2000) suggested that not only the conditional distribution of innovations may be leptokurtic, but also asymmetric and proposed the skewed Student t density function:
Γ((v + 1) 2)
⎛ 2s ⎞⎛ sz t + m − dt ⎞ ⎜ ⎟⎜1 + f ( z t ; v, g ) = g ⎟ −1 v−2 Γ(v 2) π (v − 2) ⎜⎝ g + g ⎟⎠⎝ ⎠
−
v +1 2
, v > 2,
(2)
where g is the asymmetry parameter, Γ(.) is the gamma function, d t = 1 if
(
z t ≥ −m / s and d t = −1 otherwise, m = Γ((v − 1) 2 ) (v − 2 ) Γ(v 2 ) π and s =
) (g − g ) −1
−1
g 2 + g −2 − m 2 − 1 are the mean and the standard deviation of the non-
standardized skewed Student t distribution, respectively. Note that for g = 1 , the skewed Student t distribution reduces to Bollerslev’s (1987) specification. The one-day-ahead VaR is computed as
(
)
VaRt +1|t = F {εˆt +1-τ |t σˆ t +1-τ |t }τm=1 ; a σˆ t +1|t ,
(3)
where F (.; a ) is the corresponding a th quantile (5% or 1%) of the assumed distribution and σˆ t +1|t is the one-day-ahead conditional standard deviation forecast given the information that is available at time t . In order to evaluate the adequacy of the realized VaR forecasts in a risk management environment, the VaR violations, that occur if the predicted VaR is not able to cover the realized loss, must be statistically equal to the expected one and independently distributed. In the former case, the financial institution does not use its capital efficiently, while in the latter there are indications that the risk model is misspecified. Christoffersen (1998) proposed a conditional coverage test, which jointly examines the two hypothesises. His statistic is computed as:
LRcc = −2 ln[(1 − p)T − N p N ] + 2 ln[(1 − π 01 ) n00 π 01n01 (1 − π 11 ) n10 π 11n11 ] ∼ χ 2 (2),
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(4)
where nij is the number of observations with value i followed by j , for i, j = 0,1 while π ij = nij
∑n j
ij
are the corresponding probabilities, while p = N / T denotes
the observed exception frequency. Finally, N =
T
∑I t =1
t
are the observed number of
exceptions over a T period, where I t = 1 if a violation occurs and I t = 0 otherwise. The values i, j = 1 denote that an exception has been made, while i, j = 0 indicates the opposite. The main advantage of this test is that it can reject a VaR model that generates either too many or too few clustered violations. In the case of evaluating the predictive performance of a conditional volatility model, we should use a measure of the distance between estimations and realizations that takes into account the non-linearity of the data. Symmetric loss functions, such as the mean squared error or the mean absolute error, are not appropriate in evaluating volatility models. The Logarithmic Error (LE) function, which was introduced by Pagan and Schwert (1990), is robust to non-linearity and heteroskedasticity. Denoting the one-day-ahead forecasting variance by σ t2+1t| , and the realized intra-day variance by ht2+1 , the LE loss function was considered as: T
(
)
LE = T −1 ∑ ln ht2+1 σ t2+1|t ,
(5)
t =1
where T is the number of the one-day-ahead volatility forecasts. In order to evaluate the one-day-ahead volatility forecasts accuracy, we have to compare them with the realized daily volatility, which is unobserved. The most popular measure of daily volatility, among practitioners in financial markets, is the average of squared daily returns. However, as noted by Ebens (1999), the squared daily return is an unbiased but a noisy estimator. Andersen and Bollerslev (1998) introduced the realized intra-day volatility, which is computed as the summation of
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the squared finely sampled high frequency data. The realized intra-day volatility of day t is computed as: 2
⎛ ⎛ P ( i +1 j ) ⎞⎞ , h = ∑ ⎜⎜ ln⎜ t (i j ) ⎟ ⎟ Pt ⎠ ⎟⎠ ⎝ i =1 ⎝ j −1
2 t
(6)
where Pt ( j ) is the discretely observed series of prices of an asset at day t with j observations per day.
Empirical Results: The case of the ASE market
The data set consists of daily closing prices for the Bank index and the General index of ASE from 25th of April 1994 to 19th of December 2003. Figure 1 plots the daily log-returns and reveals the volatility clustering. According to Table 1, which provides the summary statistics and the Jarque-Bera statistic, the normality is rejected at any level of significance, as there is evidence of significant excess kurtosis and non-zero skewness relative to that of the standard normal distribution. From the histograms of Figure 2 is apparent that the fat tails are non-symmetric and therefore the non-symmetric skewness must be parameterized. The APARCH and the GARCH models are estimated under the normal, the Student t and the skewed Student t conditional distributions. We use a rolling sample of 2000 observations, leaving 405 trading days in order to evaluate the one-dayahead volatility forecasting accuracy. The parameters of the models are re-estimated every trading day. Maximum likelihood estimates of the parameters are obtained by numerical maximization of the log-likelihood function using the Marquardt (1963) algorithm, which is an updated modification of the well-known BHHH method of Berndt et al. (1974).
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Forecasting One-day-ahead VaR
In the sequel, the LRcc statistic is computed to evaluate the adequacy of the models in predicting the next day’s VaR forecast. Based on Table 2, we infer that the APARCH model with conditionally skewed Student t distributed innovations is the most adequate in predicting the one-day-ahead VaR number of the General index. As concerns the Bank index, we find out that the GARCH model with conditionally normally distributed innovations has the best performance in forecasting the one-dayahead VaR number. To sum up, as Giot and Laurent (2003a, 2003b) noted the APARCH-skewed Student t model should be applied by the risk managers in order to calculate the VaR number for both confidence levels, since it captures the asymmetry and the “fat tails” of the empirical distribution. However, this is not the case for the Bank index as the “simple” GARCH–normal is characterized as the most adequate risk model and therefore there is no a unique risk model that can be employed for all equity indices.
Forecasting One-day-ahead intra-day Volatility
In order to evaluate the ability of the ARCH processes in forecasting the oneday-ahead realized volatility, we compute the intra-day variance as the summation of the squared 5-minutes sampled log-returns. The 5-minutes sampling frequency were also used by Andersen and Bollerslev (1998), Andersen et al. (2000), Andersen et al. (2001), Degiannakis (2004) and Kayahan et al. (2002) among others. The transaction data used in this study are drawn from the intra day files of the Athens Stock Exchange. This dataset contains for all trades, the time-stamped prices to the nearest second and the relative volumes from May 8th of 2002 to December 19th 2003. According to Table 3, that presents the values of the LE loss function for the
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six estimated processes and for both under study indices, the APARCH model provides the most accurate estimate of the 5-minute realized intra-day variance. However, the normal and the skewed Student t are considered as the most appropriate conditional distributions for the General and the Bank index, respectively. For robustness purposes, the study was repeated by computing the intra-day variance as the summation of the squared 15-minutes log-returns. We reach to similar findings, which demonstrate that the results of our study are not affected by the measure of realized volatility.
Conclusion
In this paper we examined the performance of various volatility models in forecasting the realized volatility based on intra day data and in calculating the 95% and 99% VaR numbers based on daily returns. The evaluation of the risk management techniques was made on two grounds. First, the volatility forecasts were compared with the intra-day realized variance based on the 5-minute intra day returns. Second, under the VaR framework, we simultaneously examine whether the exception rate is statistically equal to the expected one and the independence of failures hypothesis is valid. For both equity indices, no model can forecast both the VaR number and the realized volatility. For the ASE General index the exception rate of the GARCH– normal model is statistically equal to the expected one, while for the Bank index this is not the case, since the APARCH-skewed Student t model estimates the VaR number for both confidence levels more accurately. On the other hand, the APARCH models estimate the realized intra-day volatility better than the GARCH framework. However, as in the case of the VaR predictions, there is not a particular model that can be applied for both indices. Thus, as concerns the volatility of the Greek stock
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market, although it is predictable, there is not an explicit model, which is the most accurate for all the forecasting tasks.
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Table 1. Descriptive statistics of the Bank index and the General index of ASE from 25th of April 1994 to 19th of December 2003. Statistics
General index
Bank index
Mean
0.04%
0.06%
Maximum
7.66%
10.69%
Minimum
-9.69%
-9.53%
Std. Dev.
1.68%
1.92%
Skewness
-0.04
0.22
Kurtosis
6.80
6.57
1453.72
1298.50
Probability
0.00
0.00
Observations
2412
2412
Jarque-Bera
14
Table 2. Probability values of the conditional coverage VaR test for the estimated ARCH processes of the Bank and the General indices. The sample period runs from 25th of April 1994 to 19th of December 2003. The best performing models are boldfaced. 95% VaR Confidence
99% VaR Confidence
Level
Level
General
Bank
General
Bank
index
index
index
index
GARCH – Normal
2.26%
77.60%
81.94%
81.94%
GARCH–Student t
0.45%
0.01%
1.59%
1.59%
GARCH - Skewed Student t
0.45%
0.01%
1.59%
1.59%
APARCH – Normal
2.26%
16.92%
81.94%
49.92%
APARCH – Student t
0.00%
0.00%
1.59%
1.59%
APARCH - Skewed Student t
19.23%
0.00%
84.99%
1.34%
Process
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Table 3. The LE loss function for the estimated ARCH processes, for the daily log-returns of the Bank and the General indices. The sample period runs from May 8th of 2002 to December 19th 2003. The best performing models are boldfaced. Process
General index
Bank index
GARCH – Normal
1.9332
1.049
GARCH – Student t
1.9701
1.096
GARCH - Skewed Student t
1.9720
1.092
APARCH – Normal
1.9240
1.032
APARCH – Student t
2.0999
1.141
APARCH - Skewed Student t
4.4522
0.899
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Figure 1. Daily log-returns of the General index of ASE market from 25th of April 1994 to 19th of December 2003. 0.10
0.05
0.00
-0.05
-0.10 500
1000
1500
2000
Daily log-returns of the Bank index of ASE market from 25th of April 1994 to 19th of December 2003. 0.15
0.10
0.05
0.00
-0.05
-0.10 500
1000
17
1500
2000
Figure 2. Histogram of daily log-returns of the General index of ASE market from 25th of April 1994 to 19th of December 2003. 600 500 400 300 200 100 0 -0.100 -0.075 -0.050 -0.025 0.000 0.025 0.050 0.075
Histogram of daily log-returns of the Bank index of ASE market from 25th of April 1994 to 19th of December 2003. 800
600
400
200
0 -0.10
-0.05
0.00
18
0.05
0.10
