Quantile forecasts using the Realized GARCH-EVT approach

Samit Paul (Department of Finance and Control, Indian Institute of Management Calcutta, Calcutta, India)
Prateek Sharma (Department of Finance and Accounting, Indian Institute of Management Udaipur, Udaipur, India)

Studies in Economics and Finance

ISSN: 1086-7376

Publication date: 1 October 2018

Abstract

Purpose

This study aims to implement a novel approach of using the Realized generalized autoregressive conditional heteroskedasticity (GARCH) model within the conditional extreme value theory (EVT) framework to generate quantile forecasts. The Realized GARCH-EVT models are estimated with different realized volatility measures. The forecasting ability of the Realized GARCH-EVT models is compared with that of the standard GARCH-EVT models.

Design/methodology/approach

One-step-ahead forecasts of Value-at-Risk (VaR) and expected shortfall (ES) for five European stock indices, using different two-stage GARCH-EVT models, are generated. The forecasting ability of the standard GARCH-EVT model and the asymmetric exponential GARCH (EGARCH)-EVT model is compared with that of the Realized GARCH-EVT model. Additionally, five realized volatility measures are used to test whether the choice of realized volatility measure affects the forecasting performance of the Realized GARCH-EVT model.

Findings

In terms of the out-of-sample comparisons, the Realized GARCH-EVT models generally outperform the standard GARCH-EVT and EGARCH-EVT models. However, the choice of the realized estimator does not affect the forecasting ability of the Realized GARCH-EVT model.

Originality/value

It is one of the earliest implementations of the two-stage Realized GARCH-EVT model for generating quantile forecasts. To the best of the authors’ knowledge, this is the first study that compares the performance of different realized estimators within Realized GARCH-EVT framework. In the context of high-frequency data-based forecasting studies, a sample period of around 11 years is reasonably large. More importantly, the data set has a cross-sectional dimension with multiple European stock indices, whereas most of the earlier studies are based on the US market.

Keywords

Citation

Paul, S. and Sharma, P. (2018), "Quantile forecasts using the Realized GARCH-EVT approach", Studies in Economics and Finance, Vol. 35 No. 4, pp. 481-504. https://doi.org/10.1108/SEF-09-2016-0236

Download as .RIS

Publisher

:

Emerald Publishing Limited

Copyright © 2018, Emerald Publishing Limited


1. Introduction

Quantile measures of financial returns distributions, such as Value-at-Risk (VaR) and expected shortfall (ES), are useful for the computation of regulatory capital requirements for the financial institutions and for risk management applications. The generalized autoregressive conditional heteroskedasticity (GARCH) models are frequently used for estimating conditional moments of the returns distribution, which are subsequently used to forecast conditional quantiles. The standard GARCH model uses squared daily returns as the estimate of the daily conditional variance. But a single daily return contains little information about the current level of volatility, and hence it provides a very noisy estimate of the daily conditional variance (Andersen and Bollerslev, 1998). A better approach is to use the realized variance (RV) measures that estimate the latent conditional variance more accurately by exploiting high-frequency intraday returns’ data. Several studies document that incorporating realized measures in volatility forecasting models leads to substantial statistical and economic gains (Andersen et al., 2001a, 2001b; Andersen and Bollerslev, 1998; Barndorff-Nielsen, 2002; Barndorff-Nielsen and Shephard, 2002; Christoffersen et al., 2014; Fleming et al., 2003; Koopman et al., 2005; Pong et al., 2004; Vortelinos, 2013; Vortelinos and Thomakos, 2012; Sharma and Vipul, 2016b). Recently, Hansen et al. (2012) introduced the Realized GARCH model that jointly models the conditional moments and the RV process. Their empirical results suggest that the Realized GARCH model significantly improves the out-of-sample empirical fit as compared to the daily returns-based standard GARCH models. Using the S&P500 index data, Watanabe (2012) showed that the Realized GARCH model provides superior quantile forecasts than the exponential GARCH (EGARCH) model.

The RV estimator is computed as the sum of squared intraday returns. Under the assumption of nearly infinite sampling, the RV is an asymptotically consistent estimate of the latent integrated variance (Andersen et al., 2001a, 2001b; Andersen and Bollerslev, 1998; Barndorff-Nielsen, 2002; Barndorff-Nielsen and Shephard, 2002). In practice, however, the RV estimate is biased owing to market microstructure noise and intraday price jumps. At high sampling frequencies, the market microstructure effects, such as bid–ask bounce and price rounding, induce a bias in the RV. The market microstructure bias in the variance estimate gets progressively worse with an increase in the sampling frequency (Bandi and Russell, 2006; Bandi and Russell, 2008; Zhou, 1996). One way to mitigate such bias is to use a lower sampling frequency, typically 5 to 30 min. The 5-min RV, calculated as the sum of squared 5-min returns, is a commonly used realized estimator. Unfortunately, sampling at a lower frequency leads to loss of a large number of price observations and reduces the efficiency of the RV estimate. Moreover, the assumption of nearly infinite sampling may not be appropriate for lower sampling frequencies. Additionally, in the presence of intraday price jumps, the RV estimator is biased, as it measures the true integrated variance plus a contribution equal to the cumulative squared jumps (Andersen et al., 2012; Barndorff-Nielsen and Shephard, 2004a, 2004b).

Recent literature has proposed realized estimators that are robust to microstructure noise and price jumps, and it make efficient use of the intraday data. The realized kernel (RK) estimator of Barndorff-Nielsen et al. (2008) is robust to the microstructure noise, and it makes use of all the available data (tick-by-tick sampling). The realized bipower variance (BV) estimator of Barndorff-Nielsen and Shephard (2004a) is robust to bias induced by the intraday price jumps. Sharma and Sharma (2015) reported that these robust RV estimators provide more precise volatility forecasts as compared to the RV.

In this study, we forecast VaR and ES of five European stock markets using a novel approach of combining the extreme value theory (EVT) framework with the Realized GARCH model. The EVT relies on the distribution of extreme returns rather than the conditional moments of the entire return distribution. Because the tail of the return distribution is characterized by rare and extreme outcomes, it generally does not have the same form as the parent distribution. By modelling the tail behaviour independently of the parent distribution, the EVT approach accommodates more generalized distribution for modelling the extreme returns. It is also more suited for modelling the stylized facts such as “fat tails” in the empirical return distribution. Empirical evidence suggests that the EVT models provide better VaR forecasts than the stand-alone GARCH-type models (Bali, 2003; Dimitrakopoulos et al., 2010; Ergen, 2015; Gençay et al., 2003; Gençay and Selçuk, 2004; Ho et al., 2000; Karmakar and Paul, 2016; Karmakar and Shukla, 2015; Yi et al., 2014; Paul and Sharma, 2017).

The EVT assumes that the observations are independent and identically distributed (iid). Pagan (1996) noted that the financial return series are non-iid, as they exhibit serial correlation and a stochastic volatility structure (GARCH effects). Wagner and Marsh (2005) demonstrated that in the presence of such GARCH effects, the quantile estimates may be affected by bias. They recommend an unconditional modelling technique which adjusts for the GARCH effects in the original return series. An alternative approach is to model the conditional moments by using the two-stage conditional EVT model proposed by McNeil and Frey (2000). In the first stage of the conditional EVT model, the GARCH effects of the return series are removed with a suitable GARCH model. The standardized residuals of the GARCH model thus obtained are approximately iid. In the second stage, the EVT is used to model the tail of the standardized residuals series obtained from the first stage.

The two-stage conditional EVT model is widely used for VaR modelling; however, it does have certain drawbacks. First, the two-stage estimation procedure increases the estimation error in the VaR model, as it uses the standardized GARCH residuals, instead of the true innovations, for estimating the EVT model. Second, the distribution of tail itself may change over time or it may be affected by structural changes (Galbraith and Zernov, 2004; Quintos et al., 2001). More advanced models, such as the marked point process model of Chavez-Demoulin et al. (2005) and the autoregressive conditional tail (ARCT) model of Wagner (2005), attempt to address these drawbacks by using a time-varying specification to model the tail behaviour. In addition, these models can be estimated on the original return series without estimating any intermediate times-series model to account for the non-iid behaviour of financial returns.

Despite these limitations, the conditional EVT model performs well in empirical comparisons of VaR forecasting models. With a data set of the US short-term interest, Bali and Neftci (2003) found that the GARCH-EVT model provides more precise estimates of VaR as compared to the stand-alone GARCH model. Several other studies have found that the conditional EVT model provides more accurate forecasts of VaR than the traditional VaR models (Byström, 2004, 2005; Cotter, 2007; Fernandez, 2005; Ghorbel and Trabelsi, 2008; Maghyereh and Al-Zoubi, 2008; Marimoutou et al., 2009; Santos and Alves, 2014). All of these studies have applied the conditional EVT approach with a standard GARCH model to estimate VaR.

The standard GARCH models rely on a noisy estimate of conditional variance (squared daily returns), and generally have poor out-of-sample performance. To overcome this limitation of the standard GARCH models, we use the Realized GARCH model for removing the GARCH effects from the return series.

This study investigates the effect of the choice of the realized estimator on the forecasting performance of the Realized GARCH-EVT model, in the context of VaR and ES forecasting. It also compares the out-of-sample performance of the various Realized GARCH-EVT models with that of the standard GARCH-EVT and the asymmetric EGARCH-EVT models. We generate one-step-ahead VaR and ES forecasts for five European stock indices using seven forecasting models: the standard GARCH-EVT model, the asymmetric EGARCH-EVT model and the Realized GARCH-EVT model implemented with five realized estimators – the RV estimator based on 5-min returns, the subsampled RV (SRV) based on 5-min returns with 1-min subsampling, the realized BV estimator based on 5-min returns, the subsampled BV (SBV) based on 5-min returns with 1-min subsampling and the RK estimator.

The work of Watanabe (2012) is the closest to our study. Watanabe (2012) reported that the Realized GARCH model provides superior quantile forecasts as compared to the EGARCH model based on daily returns. This study extends his analysis in a number of ways. First, it explores the forecasting ability of the Realized GARCH model within the conditional EVT framework. As discussed earlier, the conditional EVT model generally provides better VaR forecasts than the stand-alone GARCH models. Therefore, investigating the performance of the Realized GARCH model within the two-stage conditional EVT is of particular interest. Second, to the best of our knowledge, this is the first study that compares the performance of different realized estimators within the Realized GARCH-EVT framework. Third, this study has a reasonably long sample period in the context of high-frequency data-based forecasting studies. More importantly, the data set has cross-sectional dimension with multiple European stock indices. In comparison, the earlier studies by Watanabe (2012) and Hansen et al. (2012) were based only on the US market data. Comparing different forecasting models based on a restricted data set can be problematic as a certain model may outperform other models simply because of chance rather than because of its superior forecasting ability (Sharma and Vipul, 2015, 2016a). Our results also confirm that the forecasting performance of the various models can be sensitive to the choice of the data set. We found that the Realized GARCH-EVT model generally provides a more accurate forecast than the standard GARCH-EVT and asymmetric EGARCH-EVT models. In terms of the out-of-sample forecast comparisons of the various Realized GARCH-EVT models, we find that the robust realized estimators do not provide forecasting gains over the classical 5-min RV estimator. This result is consistent with that of Watanabe (2012), who reported no significant improvement in quantile forecasts while substituting RV with a robust realized estimator like RK.

The rest of the paper is structured as follows: Section 2 presents the theoretical framework of the EVT, and defines the various forecasting models. Section 3 describes the data set used in this analysis. Section 4 discusses the empirical results. Section 5 provides some concluding remarks.

2. Modelling and specifications

2.1 Extreme value theory

EVT is widely applied for statistical modelling of rare and extreme events. There are two popular methods for identifying extreme observations – the block maxima (BM) method and the peak-over-threshold (POT) method. In the BM method, the total period is separated into different non-overlapping blocks (or sub periods) of equal size, say months or years. From each such block, the maximum observation is identified as an extreme return. However, this approach is not suitable for financial time series. The empirical financial returns exhibit volatility clustering, and therefore, the extreme observations tend to be clustered together. As the BM method only considers single return per block, it may exclude several relevant observations. The POT method, on the other hand, includes all observations that exceed a certain threshold, u. Therefore, it selects all extreme observations as long as they exceed the threshold, irrespective of the fact that they are clustered. In this study, we use the POT method for detecting the extreme observations. The POT method can be formally defined as follows:

Let X = {x1, x2, …, xt} represent the series of log returns for any sample stock index. A return x is classified as an extreme observation (or exceedance) if it exceeds a particular threshold u, and the magnitude of such exceedance is y = x − u. The cumulative probability distribution of y, Fu(y), is defined as:

(1) Fu(y)=Pr(Xuy|X>u)

Because equation (1) denotes a conditional probability, it can be expressed as:

(2) Fu(y)=Pr(Xuy,X>u)Pr(X>u)=F(y+u)F(u)1F(u)=F(x)F(u)1F(u)
where, x = y + u is the exceedance.

Assuming that the returns series follows an iid process, Balkema and De Haan (1974) and Pickands (1975) established that for a sufficiently high threshold u, Fu(y) converges to the generalized Pareto distribution (GPD). The GPD is defined as:

(3) Gξψy=1-1+ξyψ-1/ξ,ifξ01-e-y/ψ,ifξ=0
where, ξ is the shape parameter and ψ is the scale parameter. These two GPD parameters, ξ and ψ, are estimated using the method of maximum likelihood. In other words, for a sufficiently high threshold u, Fu(y) ≈ Gξψ(y) (Embrechts et al., 1997). Therefore, by combining the equations (2) and (3), we can approximate the distribution of exceedances F(x) as:
(4) Fx=1-FuGξψy+Fu

F(u) can be empirically estimated as (nk)/n, where n denotes the total number of observations, and k denotes the number of exceedances. Substituting F(u) = (nk)/n in equation (4), and combining equations (3) and (4) we get:

(5) Fx=1-kn1+ξx-uψ-1/ξ

2.2 Value-at-Risk and expected shortfall

Let F(·) denote the cumulative distribution function. For a given probability q, VaR can be defined as the q-th quantile of the return distribution:

(6) VaRq=F1(1q)
where, F−1(·) is the quantile function defined as the inverse of the distribution function F (·). By inverting the distribution function F(x) given in equation (5), VaR can be estimated as:
(7) VaRq=xq=u+ψξ[(1qk/n)ξ1]

A major limitation of the VaR measure is that it only provides a specific quantile of the return distribution and ignores the information contained in the tail beyond that quantile. Moreover, Artzner et al. (1999) noted that the VaR measure lacks desirable properties such as sub-additivity, and they recommend the use of the ES measure. Unlike VaR, the ES measure uses all information in the tail distribution. The ES for random variable X at a given probability level q is calculated as:

(8) ESq=E(X|X>VaRq)

Following McNeil and Frey (2000), the ES is estimated as follows:

(9) ESq=VaRq1ξ+ψξu1ξ

2.3 Conditional EVT applied to VaR and ES

We use the two-stage conditional EVT approach proposed by McNeil and Frey (2000) to counter the problem of non-iid returns. In the first stage, we filter the original return series with the standard GARCH, asymmetric EGARCH and the Realized GARCH models. If the GARCH models are well specified, the standardized residuals (innovations) are free from serial correlation and conditional heteroskedasticity and hence are closer to iid. In the second stage, we apply EVT on these standardized residuals to model the tail distribution. Then, for a certain probability q, we calculate the VaR quantile (VaRq) and ES quantile (ESq) of the innovations using equations (7) and (9), respectively.

The conditional mean of the returns is modelled by the following ARMA (p1, q1) model:

(10) rt=μt+εt,εt=htZt
where, μt=a0+i=1p1airti+j=1q1bjεtj; a0, ai and bj are the parameters; rti represents the lagged returns; and ht represents the conditional variance of εt. Zt=εt/ht are the standardized residuals. The conditional variance ht is modelled by the standard GARCH, asymmetric EGARCH and Realized GARCH models. εt denotes the residuals which are assumed to follow a skewed Student t (skst) distribution. The distributional assumption of the innovation process has a significant influence on estimates of VaR and ES. We estimate the GARCH models with skst innovations, which capture the empirical characteristics of financial asset returns: skewness (asymmetry) and excess kurtosis (fat tails). Several studies note that using skst innovations, instead of the usual Gaussian innovations, can significantly improve the VaR forecasting performance of GARCH-type models (Giot, 2005; Giot and Laurent, 2003a, 2003b, 2004; Kuester et al., 2006; Lambert and Laurent, 2001; Watanabe, 2012).

2.3.1 GARCH (p2, q2) model.

The generalized autoregressive conditional heteroskedasticity GARCH (p2, q2) model (Bollerslev, 1986) is defined as:

(11) ht=ω+i=1p2αiεti2+j=1q2βjhtj

2.3.2 EGARCH (p3,q3) model.

Nelson (1991) proposed the following EGARCH model to allow for leverage effects:

(12) loght=ω+i=1p3αiεti+γi|εti|hti+j=1q3βjloghtj

Note that the left-hand side is the log of the conditional variance. This implies that the asymmetric effect is exponential, rather than quadratic, and that forecasts of the conditional variance are guaranteed to be nonnegative. Here a positive εti contributes (αi+γi)|εti|/hti to the log of the conditional volatility, whereas a negative εti contributes (αi+γi)|εti|/hti. The parameter αi captures the sign effect and γi the size effect. Again, we expect αi to be negative in real applications.

2.3.3 Realized GARCH (p4, q4) model.

The Realized GARCH model (Hansen et al., 2012) incorporates different realized measures of volatility within the GARCH framework. The unique feature of this model is the measurement equation which models the RV process xt. The conditional variance equation and the measurement equation are defined as:

(13) loght=ω+i=1p4αilogxti+j=1q4βjloghtj
(14) logxt=η+ϕloght+τ1Zt+τ2(Zt21)+ζt,ζti.i.d.(0,ζt)

The parameters of all the above-described GARCH models are estimated using the maximum likelihood method

The optimum lag lengths of ARMA-GARCH specifications have been identified by the following procedure. First, ARMA (0,0)-GARCH (1,1) models are estimated and then they are augmented with additional AR, MA, ARCH and GARCH terms when necessary to eliminate autocorrelation in the standardized and squared standardized residuals, respectively.

We use five RV estimators to estimate the RV process xt from equation (14). These estimators are – the RV estimator, the subsampled version of the RV estimator, the BV estimator, the subsampled version of the BV estimator and the RK estimator. Next, we provide few notations for defining these realized estimators. Let {pi}i=0m represent the time-series of intraday prices. We standardize the notations by defining a function γh, d(w) as:

(15) γh,d(w)=i=1o(pid+hp(i1)d+h)(p(i+w)d+hp(i1+w)d+h)

The RV estimator is calculated as:

(16) RV=γ0,d(0)

d = 1 denotes the highest sampling frequency, i.e. γ0,1(0) represents the RV estimate using all intraday price points. The 5-min RV is calculated using d = m/l, where l is the number of 5-min intervals in the trading day. Sampling at a lower frequency reduces the microstructure bias; however, it induces inefficiency in the RV estimate due to the loss of a huge number of intraday observations. Therefore, we also calculate a subsampled RV estimate, which uses 5-min returns with 1-min subsampling. The calculation of this subsampled estimate is as follows. Suppose the RV estimate is computed using the prices of the time points 9:30, 9:35, 9:40 …, etc. Another RV estimate is similarly computed using the prices of the time points 9:31, 9:36, 9:41 …, etc. In this manner, five values of RV estimates are computed using five non-overlapping subsamples for each day. As the starting and the ending time points of these subsamples may differ from those of the trading session, these RV estimates may ignore a small number of observations. To adjust for this loss, the RV estimates are proportionally inflated. Then the subsampled RV estimate is calculated as the average of these five RV estimates.

Barndorff-Nielsen and Shephard (2004a, 2004b) proposed the BV estimator, which is robust to intraday price jumps. To avoid the microstructure effects, the BV estimator is computed using 5-min returns. Following Barndorff-Nielsen and Shephard (2004a, 2004b), the BV is computed as:

(17) BV=π2i=1m|pidp(i1)d||p(i+1)dpid|

For 5-min sampling frequency, we use d = m/l, where l is the total number of 5-min intervals in the trading day. We also compute an SBV estimate which uses 5-min returns with 1-min subsampling. Finally, we use the RK estimator of Barndorff-Nielsen et al. (2008). The RK estimator is robust to microstructure noise, and it utilizes all intraday observations. The RK is computed as:

(18) RK=γ0,1(0)+2h=1Hκ(h1H)γ0,1(h)
where, κ(x) is a kernel weight function. The optimal bandwidth parameter, H, is calculated using the procedure of Barndorff-Nielsen et al. (2009). Following Barndorff-Nielsen et al. (2011), we use the “non-flat-top” Parzen kernel function, as it ensures a positive variance estimate, while allowing for endogeneity or dependence in the microstructure noise process. The Parzen kernel function is defined as:
k(x)={ 16 x 2+6 x 3 0x1/22 ( 1x ) 3 0x1/20x>1

Because the RK estimator makes use of all intraday price observations, the subsampled version of this estimator is not required. Five Realized GARCH models are estimated using these five realized measures of volatility.

We generate one-step-ahead forecasts of conditional mean and conditional variance using the GARCH and the Realized GARCH models. The one-step-ahead conditional mean, μ^t+1 is forecasted as:

(19) μ^t+1=a^0+i=1p1a^irti+1+j=1q1b^jεtj+1

The one-step-ahead conditional variance, h^t+1, is forecasted using equations (11), (12) and (13) for the standard GARCH, the asymmetric EGARCH and the Realized GARCH models, respectively. Next, we apply the EVT to the standardized residuals series ( Zt=εt/ht) and forecast VaR and ES using equations (7) and (9), respectively. The one-step-ahead conditional VaR, VaRqt+1, is forecasted as:

(20) VaRqt+1=μ^t+1+h^t+1VaRq

Similarly, the one-step-ahead conditional ES, ESqt+1, is forecasted as:

(21) ESqt+1=μ^t+1+h^t+1ESq

3. Data and preliminary analysis

The data set comprises daily data on returns and realized volatilities for five European stock indices. All data are sourced from the Oxford–Man Institute’s Realized Library (Heber et al., 2009). The sample period extends from 1 January 2003 to 8 October 2014. The sample stock indices and their descriptive statistics are listed in Table I.

Most of the daily returns distributions exhibit negative skewness and positive excess kurtosis that indicates the presence of fat tails. The Jarque–Bera tests reject the null hypothesis of normality at 1 per cent level of significance for all the sample indices. This validates our choice of the skst distribution, which allows for the negative skewness and the fat tails observed in the daily return distributions. The Ljung–Box tests have been performed on the daily returns and the squared daily returns series till 10 lags as adopted by earlier studies (In et al., 2001; Yang and Doong, 2004) The null hypothesis of this test is that the first ten autocorrelation are jointly zero. For all the sample indices, we find that both Q(10) and Q2(10) statistics are statistically significant. This indicates the presence of both linear and non-linear dependence in the daily returns series.

4. Empirical results

All forecasting models are estimated using an in-sample period that extends from 1 January 2003 to 31 December 2011. The out-of-sample period, 1 January 2012 to 8 October 2014, is used to compare the performance of the different forecasting models.

For each of the sample index, we generate one-step-ahead quantile forecasts with the standard GARCH-EVT model (G EVT), the asymmetric EGARCH-EVT model (EG EVT) and the five Realized GARCH models estimated using different realized estimators: RG EVTRV, RG EVTSRV, RG EVTBV, RG EVTSBV and RG EVTRK.

4.1 In-sample estimation

For all of the sample stock indices, we estimate the appropriate ARMA-GARCH model and then calculate the standardised residuals Zt as:

(22) Zt=(rtμt)/ht

The model estimation results are provided in Table II. To test the serial correlation and heteroskedasticity in the original returns series, we use the Ljung–Box tests on the daily returns series and the squared daily returns series. The results are provided in Column 3 of Table III. The Ljung–Box statistics Q(10) and Q2(10) suggest that both daily returns and the daily squared returns series exhibit significant autocorrelation. Because the EVT assumes iid observations, it cannot be directly applied to the original returns series.

We remove these effects by filtering the original return series by the standard GARCH, the asymmetric EGARCH and the Realized GARCH models. The standardized residuals obtained from the GARCH models are tested again for serial correlation and heteroskedasticity. The results are provided in Columns 4 to 9 of Table III. We find that regardless of the choice of the GARCH filter, the standardized residuals and the squared standardized residuals series do not exhibit significant serial dependence. For all the GARCH models, the Q(10) and Q2(10) statistics are found to be insignificant for all the sample stock indices. Therefore, the standardized residuals do not demonstrate linear and non-linear serial dependence, and consequently, they are suitable for the implementation of EVT.

To choose the threshold u for the POT method, we use the plot of the mean excess function (MEF). The MEF is the sum of exceedances over the chosen threshold u, divided by the total number of observations which exceed the threshold u:

(23) MEF(u)=i=1n(Xiu)i=1nI(Xi>u)
where n is the total number of observations and I(⋅) is an indicator function which equals 1 if Xi > u, and zero, otherwise.

For the GPD, the MEF plot becomes linear at the threshold level u. We construct the MEF plot using different values of the threshold (u) and select that threshold value for which the plot is sufficiently linear. We report the chosen thresholds (u) and the number of exceedances (k) in Table IV. Out of the total number of in-sample observations (n), the percentage of such exceedances (k/n) varies from 8.10 to 12.17 per cent, which is consistent with the findings of McNeil and Frey (2000). Table IV also reports the estimated GPD parameters – shape (ξ) and scale (ψ) – under the selected thresholds. The ξ parameter is positive and significant for majority of the cases, indicating that the distributions of standardized residuals are characterized by fat left-tails. Using equations (7) and (9), we estimate the unconditional VaRq and ESq at three confidence intervals: 95, 99 and 99.5 per cent.

Finally, the one-step-ahead conditional VaRqt+1 and ESqt+1 forecasts are computed using equations (20) and (21), respectively.

4.2 Out-of-sample forecast evaluation

4.2.1 Out-of-sample performance of the VaR models.

We evaluate the VaR forecasts using binomial tests and likelihood ratio tests. To compare the out-of-sample performance of the different VaR forecasting models, we calculate an empirical violation ratio in the following manner. For the day t + 1, if the actual daily return (rt+1) exceeds the forecasted VaR ( VaRqt+1), we consider it as a violation. The empirical violation ratio is the total number of such violations divided by the total number of one-step-ahead forecasts (f). The expected violation ratio is calculated as 1 − q, i.e. for the 95 per cent level of confidence, we expect only that 5 per cent of the returns would exceed the forecasted VaR. If a particular model provides precise VaR forecasts, the empirical violation ratio should not be significantly different from the expected violation ratio. We test this hypothesis using the binomial test. A negative (positive) and significant binomial test statistic indicates that the empirical violation ratio is significantly lower (higher) than the expected violation ratio. The VaR forecasts are calculated and compared at three tail quantiles: q = 0.95, 0.99 and 0.995.

Table V displays the binomial test statistics and their level of significance. Overall, we make 15 forecasts with each model (5 stock indices × 3 quantiles). Out of the 15 cases, we reject the null hypothesis four times in case of the G EVT model and two times in case of the EG EVT model. Regardless of the choice of the realized estimator, the VaR forecasting ability of the Realized GARCH-EVT models is significantly better than that of the standard GARCH EVT models. For all the five Realized GARCH-EVT models, the null hypothesis is not rejected for any of the sample index.

Next, we apply the likelihood ratio-based testing framework of Christoffersen (1998) to evaluate the performance of the VaR forecasting models. We use three statistical tests: the test for unconditional coverage, the test for independence and the test for conditional coverage. If the VaR is calculated at a 95 per cent confidence level, on an average, one would expect five exceedances out of every hundred observations. The unconditional coverage test verifies whether the number of such exceedances is in line with the specified confidence level. The test for independence verifies whether there is clustering of exceedances over time. For example, say a particular model forecasting 95 per cent VaR results in exactly 5 per cent exceedances. In this case, the model will be considered perfect by the unconditional coverage test. However, if the exceedances are clustered together, and we spot an exceedance today, the conditional probability of observing another exceedance tomorrow would be higher than the unconditional probability (5 per cent). Therefore, to ensure that both the conditional and unconditional probabilities of observing exceedances are consistent with the confidence level of the VaR forecast, it is important that the exceedances should be independent and not clustered over time. The independence test excludes the VaR models which lead to clustered exceedances. The conditional coverage test is a joint test of the unconditional coverage and the test of independence of the exceedances. One limitation of Christoffersen’s (1998) VaR backtesting framework is that it tests for the dependence between exceedances only for the first lag. Chen et al. (2012) extend the independence and conditional coverage tests by considering multiple lags. In line with their study, we implement these tests with four lags. For detailed descriptions of these tests, we refer the reader to Chen et al. (2012). For alternative methods of evaluating VaR forecasts, the readers may refer to Nieto and Ruiz (2016) and Gerlach et al. (2016).

Table VI reports the results of the tests for unconditional coverage. The null hypothesis of this test is that the actual proportion of exceedances is equal to the expected proportion of exceedances at a certain confidence level q. Out of the 15 cases, the null hypothesis is rejected four times for the G EVT model and twice for the EG EVT model. In contrast, it is never rejected for any of the Realized GARCH-EVT models. Therefore, in terms of the unconditional coverage, the Realized GARCH-EVT models fare considerably better than the G EVT and EG EVT models. However, among the Realized GARCH-EVT models, there is no particular RV measure which dominates the others. For example, the values of the LR statistics show that at 5 per cent confidence level, the RG EVTRV model performs the best for the AEX index, RG EVTSRV is the best model for the DAX and CAC40 indices, RG EVTBV outperforms for the SMI index and RG EVTRK is the best performer for the FTSE and the CAC40 indices.

Table VII presents the results for the tests of independence. The null hypothesis of this test is that the exceedances do not cluster over time. In all the cases, we fail to reject the null hypothesis. These results indicate that for all the forecasting models, the exceedances are generally independent. Table VIII presents the results for test of conditional coverage. The null hypothesis of this test is a joint hypothesis that the observed and expected proportions of exceedances are equal, and all such exceedances are independent of each other. The results are consistent with our earlier findings. The forecasting performances of the Realized-GARCH EVT models are considerably better than those of the G EVT and the EG EVT models. The null hypothesis is rejected in four instances for the G EVT model and twice for the EG EVT model. But, it is never rejected for any of the Realized GARCH-EVT models.

As earlier, the magnitudes of the LR statistics suggest that the robust RV estimators are not preferable to the conventional RV estimator. This finding is consistent with that of Watanabe (2012), who found that substituting the RK estimator for the RV estimator in the Realized GARCH model does not lead to forecasting gains. The Realized GARCH specification can adjust for bias induced by the microstructure noise and non-trading hours. Consequently, we observe no forecasting gains by using a microstructure noise robust estimator, such as the RK estimator, in the Realized GARCH model.

4.2.2 Out-of-sample performance of the ES models.

Backtesting VaR is relatively straightforward, as the predicted VaR quantile can be compared with the actual returns in the out-of-sample period. However, it is significantly harder to backtest ES, as it involves the comparison of a forecast of an expectation with some estimate of that expectation. Moreover, many parametric statistical tests for ES, such as those by Berkowitz (2001) and Kerkhof and Melenberg (2004), rely on large samples for convergence to limiting distributions. This is inherently problematic, as the realized ES is invariably based on a small sample of exceedances (Wong, 2008). The difficulty in backtesting is further compounded by the fact that the ES measure is not elicitable (Gneiting, 2011; Weber, 2006). Gneiting (2011) showed that elicitability provides a natural way of comparing different forecasting methods, which led to an argument that it may be impossible to backtest ES. We refer the reader to Gneiting and Katzfuss (2014) for a detailed discussion on the property of elicitability and how it can be used to it to compare different forecasting methods.

Acerbi and Székely (2014) note that exploiting the property of elicitability is not the only way of comparing forecasting models. For instance, although VaR is elicitable, most VaR backtests are based on counting the number of exceedances and do not rely on elicitability. In addition, whereas the property of elicitability can be used for making relative comparisons between different forecasting models, it cannot be used for the statistical validation of an individual model. The authors suggest three non-parametric statistical tests to backtest ES. The main advantage of these tests is that they are model-free and they do not assume any limiting distribution for the test statistic. In this analysis, we use the second test of Acerbi and Székely (2014), which is recommended by the authors as a stand-alone test for ES. Other notable approaches to backtest ES include the works of Emmer et al. (2015) and Du and Escanciano (2016). Emmer et al. (2015) proposed an empirical approach which involves substituting ES by a set of four quantiles, which are then tested using conventional VaR backtests. Du and Escanciano (2016) proposed statistical backtests for ES which are analogous to VaR backtests of Berkowitz et al. (2011), Christoffersen (1998) and Kupiec (1995).

Next, we provide some notations to describe the Acerbi and Székely (2014) test. Let Xt denote the realized returns in the out-of-sample period where t = 1,2, …, T. Ft is the distribution of the realized returns and Pt is the predictive distribution of the returns based on the forecasting model. Ft[q] and Pt[q] denote the tail distributions below the quantile q. The test statistic, Z(X), is defined:

(24) Z(X)=t=1TXtItTqESqt

The hypotheses for this test are:

H0:Ptq=Ftq,t
H1:ES^qtESqt,foralltand>forsomet
VaR^qtVaRqt,forallt
where VaR^qt and ES^qt are the sample VaR and ES from the realized returns, respectively. It is the indicator function that equals 1 when Xt<VaRqt, and 0 otherwise. Note that under the null hypothesis, H0, the expected value of Z(X) is zero. To compute the statistical significance, Acerbi and Székely (2014) simulated the model predictive distribution Pt under H0.

We generate one-step-ahead ES forecasts at three tail quantiles: q = 0.95, 0.99 and 0.995. The Acerbi and Székely (2014) test is used to compare the forecasting performance of the various models and results are presented in Table IX. We generate 5,000 simulated series, Xti, from the distribution Pt, where i = 1,2, […],5,000. For each simulated series Xti, we compute the statistic Zi=Z(Xti) using equation (24). The p-value for the test is calculated as pval=i=15000(Zi<Z(X))/5000. For each forecasting model, we make 15 set of ES forecasts (5 stock indices × 3 quantiles). The null hypothesis is rejected seven times for the G EVT model and five times for the EG EVT model, which suggests that, at least in some cases, these models may lead to under-estimation of ES. On the other hand, there is no instance of rejection in case of any of the Realized GARCH-EVT models. Therefore, in terms of ES forecasting, the Realized GARCH-EVT models perform better than the G EVT and EG EVT models. The comparison of different Realized GARCH-EVT models indicates that there is no specific RV measure that outperforms all others.

The Acerbi and Székely (2014) test is useful for the absolute validation of a forecasting model. We also compute the Embrechts et al. (2005) measure which is widely used for relative ranking of ES forecasting models (González-Pedraz et al., 2014; Karmakar and Paul, 2016; Lin et al., 2014; Takahashi et al., 2016; Watanabe, 2012). The standard backtesting measure for the ES estimates is:

(24) E1=1ctτϕt
where, ϕt=rt+1-ESqt+1, c is the number of intervals for which a violation of VaR occurs, and τ is the set of such intervals. An obvious limitation of this measure is that it depends on the estimated VaR quantile, which may deviate from the empirical quantile of the financial returns series. To correct this, it is combined with another measure, where the empirical α- quantile of ϕ is used in place of the VaR estimates:
(25) E2=1dtθϕt
where, d is the number of periods for which ϕt is less than the empirical quantile and θ is the set of intervals for which it happens.

The Embrechts et al. (2005) measure is given by:

(26) E=(|E1|+|E2|)/2
where a lower value of E indicates a better forecast of the ES. For a detailed discussion of the properties of this measure, we refer the reader to Embrechts et al. (2005).

Table X reports the backtesting results for the ES forecasts. For each sample index, the lowest value of the measure (E) is marked with an “asterisk”, which indicates the best forecasting model. Of the 15 model comparisons (5 stock indices × 3 quantiles), the G EVT model is never ranked as the best model, the EG EVT model ranks as the best model only in one case, whereas the Realized GARCH-EVT models provide the best forecasts in 14 cases. As earlier, we find that no specific realized measure dominates all others, in terms of the forecasting ability of the Realized GARCH-EVT model. More specifically, the robust realized estimators, such as the RK and BV estimators, do not provide forecasting gains over the classical RV estimator.

5. Conclusion

We investigate the forecasting ability of the standard GARCH-EVT, the asymmetric EGARCH-EVT and the Realized GARCH-EVT models, in the context of quantile forecasts. With the US market data, Hansen et al. (2012) and Watanabe (2012) showed that the Realized GARCH model provides better out-of-sample performance than the standard GARCH models that are based on daily returns. We extend their analysis by implementing the two-stage conditional EVT approach of McNeil and Frey (2000) with a Realized GARCH filter. Additionally, we compare the performance of different realized estimators with the Realized GARCH-EVT framework. Our results suggest that the Realized GARCH-EVT model generally provides more precise VaR and ES forecasts than the GARCH-EVT and the EGARCH-EVT models. However, there is no specific realized estimator, which outperforms all other realized estimators within the Realized GARCH-EVT framework. Because the Realized GARCH model can adjust for the microstructure noise bias in the basic RV estimate, no incremental forecasting gains are derived by using a microstructure noise robust estimator such as the RK.

The main advantage of the Realized GARCH-EVT framework lies in its ability to model most of the stylized facts of the financial return series. The conditional mean and conditional variance equations capture serial correlation in returns and the volatility clustering effects. The measurement equation of Realized GARCH model includes a realized measure of volatility that provides a more accurate estimate of the daily volatility than the squared daily returns. The specification of the measurement equation is quite general, and it adjusts for asymmetric leverage effects and microstructure noise bias in the RV process. Finally, the EVT approach captures the typical tail behaviours, such as the fat tails of standardized residuals, commonly observed in the distributions of financial returns.

Quantile forecasts are useful for a variety of practical applications. The Basel III guidelines propose the minimum capital requirements for financial institutions based on the 97.5 per cent ES measure, which is calculated over a one-year stressed period. Accurate forecasts of the ES measure are desirable for financial institutions as underestimation of ES may lead to undercapitalization, whereas overestimation of ES limits their profitability by discouraging them from taking risks (Kinateder, 2016). Quantile forecasts are also used for setting risk limits for trading desks (Berkowitz et al., 2011), computing margin requirements (Cotter and Dowd, 2006), setting optimal investment allocation (Alexander and Baptista, 2004; Basak and Shapiro, 2001; Cuoco et al., 2008; Wang et al., 2010) and implementing insurance risk management (Dowd and Blake, 2006). Our results suggest that the implementation of the Realized GARCH-EVT model would yield forecasting gains over the traditional forecasting models for such applications.

Descriptive statistics for the daily returns series

Country UK Germany France Netherland Switzerland
Index FTSE 100 Index Deutsche Börse Index CAC 40 Index AEX Index Swiss Market Index
Ticker FTSE DAX CAC40 AEX SMI
N 2959 2986 3008 3008 2950
n 2259 2284 2301 2301 2259
f 700 702 707 707 691
Mean 0.04 0.09 0.03 0.02 0.05
SD 0.18 0.22 0.22 0.22 0.18
Skewness −0.16 0.10 −0.01 −0.14 −0.04
Kurtosis 8.74 7.55 6.18 7.95 7.65
JB statistic 9439.47** 7107.21** 4790.89** 7938.55** 7206.64**
Q(10) 39.16** 17.29** 36.38** 47.62** 49.54**
Q2(10) 1647.21** 1120.68** 1427.08** 1836.14** 2253.01**
Notes:

All metrics are calculated for the daily returns series, over the period 1 January 2003 to 8 October 2014. For each stock index, N, n and f are the number of daily observations in the total sample period, in the in-sample period and in the out-of-sample period, respectively. The mean and standard deviation are annualized. JB statistic is the Jarque–Bera test statistic. Q(10) and Q2(10) are the Ljung–Box Q statistic for the daily returns and the squared daily returns series; *and

**

indicate significance at the 5% and 1% levels, respectively

Model estimation results for the ARMA-GARCH models

Ticker Model fitted ω α1 β1 γ1 η φ τ1 τ2
FTSE ARMA(0,0)-G(1,1) 0.000 0.085** 0.909**
ARMA(0,0)-EG(1,1) −0.106* −0.124** 0.989** 0.095**
ARMA(0,0)-RGRV (1,1) 0.095 0.389** 0.596** −0.741** 0.986** −0.151** 0.076**
ARMA(0,0)-RGsRV(1,1) 0.136 0.411** 0.576** −0.799** 0.982** −0.152** 0.073**
ARMA(0,0)-RGBV(1,1) 0.141 0.401** 0.584** −0.848** 0.985** −0.159** 0.075**
ARMA(0,0)-RGSBV(1,1) 0.117 0.423** 0.558** −0.754** 0.994** −0.157** 0.071**
ARMA(0,0)-RGRK(1,1) 0.071 0.396** 0.585** −0.656* 0.996** −0.149** 0.075**
DAX ARMA(0,0)-G(1,1) 0.000 0.083* 0.910**
ARMA(0,0)-EG(1,1) −0.157* −0.128** 0.982** 0.122**
ARMA(0,0)-RGRV (1,1) −0.025 0.407** 0.572** −0.484 0.991** −0.142** 0.103**
ARMA(0,0)-RGsRV(1,1) 0.076 0.421** 0.567** −0.678** 0.974** −0.157** 0.086**
ARMA(0,0)-RGBV(1,1) −0.065 0.422** 0.546** −0.402 1.015** −0.169** 0.071**
ARMA(0,0)-RGSBV(1,1) 0.011 0.437** 0.537** −0.551* 1.001** −0.172** 0.068**
ARMA(0,0)-RGRK(1,1) 0.038 0.423** 0.562** −0.595* 0.977** −0.148** 0.096**
CAC40 ARMA(0,0)-G(1,1) 0.000 0.085 0.906**
ARMA(0,0)-EG(1,1) −0.176* −0.151** 0.980** 0.104**
ARMA(0,0)-RGRV (1,1) −0.028 0.399** 0.576** −0.467 1.003** −0.143** 0.084**
ARMA(0,0)-RGsRV(1,1) 0.005 0.410** 0.567** −0.524* 0.998** −0.146** 0.086**
ARMA(0,0)-RGBV(1,1) −0.069 0.409** 0.556** −0.389 1.022** −0.153** 0.081**
ARMA(0,0)-RGSBV(1,1) −0.026 0.428** 0.541** −0.465 1.014** −0.154** 0.081**
ARMA(0,0)-RGRK(1,1) −0.012 0.384** 0.592** −0.482 1.003** −0.141** −0.094**
AEX ARMA(0,0)-G(1,1) 0.000 0.090 0.903**
ARMA(0,0)-EG(1,1) −0.123* −0.125** 0.986** 0.109**
ARMA(0,0)-RGRV (1,1) 0.282* 0.429** 0.576** −1.145** 0.935** −0.161** 0.078**
ARMA(0,0)-RGsRV(1,1) 0.277* 0.440** 0.563** −1.109** 0.941** −0.164** 0.073**
ARMA(0,0)-RGBV(1,1) 0.258* 0.434** 0.563** −1.094** 0.950** −0.162** 0.077**
ARMA(0,0)-RGSBV(1,1) 0.283* 0.457** 0.542** −1.107** 0.949** −0.166** 0.072**
ARMA(0,0)-RGRK(1,1) 0.247* 0.423** 0.578** −1.064** 0.944** −0.160** 0.082**
SMI ARMA(0,0)-G(1,1) 0.000 0.104** 0.883**
ARMA(0,0)-EG(1,1) −0.208* −0.139** 0.978** 0.131**
ARMA(0,0)-RV (1,1) 0.052 0.403** 0.582** −0.631** 0.985** −0.126** 0.064**
ARMA(0,0)-SRV(1,1) 0.089 0.430** 0.555** −0.694** 0.982** −0.133** 0.062**
ARMA(0,0)-BV(1,1) 0.150 0.412** 0.579** −0.844** 0.971** −0.132** 0.059**
ARMA(0,0)-SBV(1,1) 0.115 0.443** 0.542** −0.743** 0.983** −0.136** 0.059**
ARMA(0,0)-RK(1,1) −0.033 0.429** 0.543** −0.411 1.012** −0.133** 0.058**
Notes:

This table reports the model estimation results for the ARMA-GARCH models. All models are estimated using the method of maximum likelihood, with an in-sample period of 1 January 2003 to 31 December 2011. The notations have been used as follows: G = GARCH; EG = EGARCH; RGRV = Realized GARCH with the realized variance estimator; RGSRV = Realized GARCH with the subsampled realized variance estimator; RGBV = Realized GARCH with the BV estimator; RGSBV = Realized GARCH with the subsampled BV estimator; RGRK = Realized GARCH with the RK estimator;

*

and

**

indicate significance at 5% and 1% levels, respectively

Serial correlation and heteroskedasticity in the original and filtered return series

Ticker Ljung–Box test Original return Standardized residuals
GARCH EGARCH RGRV RGSRV RGBV RGSBV RGRK
FTSE Q (10) 41.4** 2.19 2.41 1.09 1.78 1.32 1.74 1.07
Q2 (10) 1213.54** 7.37 6.52 4.76 4.22 4.71 4.53 7.43
DAX Q (10) 15.39 4.73 4.07 1.04 0.95 0.93 0.93 3.45
Q2 (10) 838.15** 10.83 7.07 4.46 3.94 3.43 3.68 5.13
CAC40 Q (10) 35.59** 4.94 7.16 2.61 2.74 2.65 2.62 5.90
Q2 (10) 1095.28** 5.95 9.53 2.42 2.85 3.08 3.60 2.34
AEX Q (10) 40.62** 8.06 4.27 5.44 5.07 5.59 5.27 6.87
Q2 (10) 1345.98** 6.60 2.62 1.48 1.75 1.44 2.12 1.59
SMI Q (10) 48.74** 4.85 7.19 2.13 2.32 1.96 2.21 2.07
Q2 (10) 1698.05** 2.12 7.26 4.37 4.06 3.81 4.18 4.04
Notes:

This table reports the results of Ljung–Box tests on the original and filtered returns series. Filtered returns are the standardized residuals obtained after fitting a specific GARCH model on the original returns series. The estimation period of the GARCH models extends from 1 January 2003 to 31 December 2011. Q(10) and Q2 (10) are the Ljung–Box Q statistic for the daily (filtered) returns and the squared daily (filtered) returns series; *and

**

indicate significance at 5% and 1% levels, respectively

Model estimation results for the conditional EVT models

Ticker n Models U k k/n (%) ξ ψ VaR ES
0.95 0.99 0.995 0.95 0.99 0.995
FTSE 2,259 G EVT 1.309 226 10.00 −0.134* 0.725** 1.789 2.745 3.098 2.372 3.215 3.526
EG EVT 1.339 219 9.69 0.144** 0.679** 1.767 2.654 2.977 2.307 3.082 3.364
RG EVTRV 1.051 226 10.00 0.117* 0.728** 1.577 2.976 3.664 2.471 4.056 4.836
RG EVTSRV 1.004 242 10.71 0.123* 0.714** 1.574 2.970 3.661 2.468 4.060 4.849
RG EVTBV 1.023 234 10.36 0.114 0.729** 1.577 2.977 3.664 2.471 4.053 4.829
RG EVTSBV 1.019 235 10.40 0.116* 0.727** 1.575 2.975 3.663 2.470 4.052 4.830
RG EVTRK 1.013 238 10.54 0.124* 0.715** 1.572 2.967 3.659 2.466 4.058 4.847
DAX 2,284 G EVT 1.473 185 8.10 0.019 0.621** 1.771 2.745 3.156 2.374 3.330 3.733
EG EVT 1.399 202 8.84 0.074 0.645** 1.759 2.696 3.067 2.334 3.207 3.552
RG EVTRV 1.171 210 9.19 0.170* 0.642** 1.583 2.900 3.587 2.440 4.026 4.854
RG EVTSRV 1.082 241 10.55 0.166* 0.632** 1.584 2.903 3.589 2.441 4.023 4.845
RG EVTBV 1.088 235 10.29 0.148* 0.654** 1.586 2.907 3.580 2.439 3.988 4.778
RG EVTSBV 1.110 228 9.98 0.146* 0.658** 1.588 2.909 3.581 2.441 3.987 4.774
RG EVTRK 1.204 196 8.58 0.141 0.679** 1.586 2.909 3.579 2.439 3.980 4.760
CAC40 2,301 G EVT 1.333 228 9.91 0.087* 0.667** 1.776 2.720 3.088 2.355 3.223 3.560
EG EVT 1.263 243 10.56 0.129** 0.686** 1.752 2.656 2.992 2.303 3.104 3.401
RG EVTRV 1.229 258 11.21 0.128** 0.682** 1.752 2.645 2.977 2.296 3.089 3.383
RG EVTSRV 1.246 255 11.08 0.129** 0.668** 1.752 2.629 2.954 2.287 3.064 3.352
RG EVTBV 1.273 248 10.78 0.127** 0.657** 1.754 2.622 2.945 2.283 3.054 3.340
RG EVTSBV 1.293 240 10.43 0.125** 0.653** 1.751 2.620 2.944 2.281 3.053 3.341
RG EVTRK 1.271 245 10.65 0.126** 0.667** 1.752 2.635 2.964 2.290 3.075 3.368
AEX 2,301 G EVT 1.273 248 10.78 0.123* 0.701** 1.787 2.717 3.064 2.354 3.182 3.491
EG EVT 1.224 257 11.17 0.137** 0.689** 1.748 2.640 2.968 2.291 3.076 3.364
RG EVTRV 1.290 230 10.00 0.152** 0.705** 1.753 2.658 2.984 2.303 3.089 3.372
RG EVTSRV 1.280 237 10.30 0.139** 0.688** 1.754 2.651 2.979 2.300 3.088 3.376
RG EVTBV 1.290 230 10.00 0.152** 0.705** 1.753 2.658 2.984 2.303 3.089 3.372
RG EVTSBV 1.257 249 10.82 0.122** 0.671** 1.750 2.641 2.974 2.294 3.088 3.385
RG EVTRK 1.213 264 11.47 0.123** 0.680** 1.750 2.647 2.981 2.297 3.095 3.394
SMI 2,259 G EVT 1.183 267 11.82 0.049 0.702** 1.774 2.815 2.239 2.416 3.407 3.811
EG EVT 1.197 262 11.60 0.068 0.668** 1.744 2.707 3.090 2.335 2.237 3.596
RG EVTRV 1.194 262 11.60 0.081* 0.687** 1.753 2.721 3.100 2.346 3.241 3.592
RG EVTSRV 1.197 268 11.86 0.073 0.668** 1.745 2.699 3.076 2.330 3.219 3.572
RG EVTBV 1.215 256 11.33 0.087* 0.687** 1.746 2.707 3.081 2.335 3.219 3.563
RG EVTSBV 1.188 265 11.73 0.081* 0.688** 1.743 2.715 3.096 2.339 3.238 3.590
RG EVTRK 1.158 275 12.17 0.088* 0.702** 1.747 2.723 3.102 2.344 3.241 3.589
Notes:

This table reports the model estimation results for the conditional EVT models. All models are estimated using the method of maximum likelihood, with an in-sample period of 1 January 2003 to 31 December 2011. n = total no. of observations; u = threshold level; k = no. of exceedances; k/n = percentage of exceedances;

*

and

**

indicate significance at 5% and 1% levels, respectively

Comparison of the VaR forecasting models using the binomial tests

Ticker G EVT EG EVT RG EVTRV RG EVTSRV RG EVTBV RG EVTSBV RG EVTRK
α = 5%
FTSE −0.520 −1.041 −0.694 −0.867 −0.867 −0.694 0.000
DAX −0.364 −1.230 0.849 0.502 0.675 0.675 1.195
CAC40 −0.751 −0.923 −0.578 −0.406 −0.751 −0.578 −0.406
AEX −0.233 −0.923 −0.578 −0.923 −0.751 −0.923 −0.751
SMI 0.253 0.079 −0.096 −0.445 0.079 −0.445 −0.096
α = 1%
FTSE −0.760 −0.380 0.380 0.760 0.380 1.519 1.519
DAX −0.387 0.372 0.751 0.372 0.751 1.130 0.751
CAC40 −2.026* −0.026 0.352 0.352 0.352 0.352 −0.026
AEX 2.107* 1.930* −0.026 0.730 0.730 0.730 1.485
SMI 0.799 1.181 1.181 1.564 1.181 1.564 1.564
α = 0.5%
FTSE −0.268 −0.268 −0.268 0.804 −0.268 0.268 1.340
DAX 1.332 1.332 0.797 0.797 0.262 0.797 0.797
CAC40 2.214* 2.314* 0.781 0.781 1.314 0.781 0.781
AEX 2.781** 0.781 0.248 0.781 0.248 1.314 1.314
SMI 0.294 0.833 0.833 0.833 1.373 0.833 0.833
No. of Rejections 4 2 0 0 0 0 0
Notes:

This table reports the binomial test statistics for VaR violation ratio under each competing model;

*

and

**

indicate significance at 5% and 1% levels, respectively

Comparison of the VaR forecasting models using the test of unconditional coverage

Ticker G EVT EG EVT RG EVTRV RG EVTSRV RG EVTBV RG EVTSBV RG EVTRK
α = 5%
FTSE 1.278 1.147 0.500 0.788 0.788 0.500 0.000
DAX 1.135 1.620 0.690 0.246 0.441 0.441 1.347
CAC40 0.587 0.896 0.343 0.168 0.587 0.345 0.168
AEX 1.055 0.896 0.345 0.896 0.589 0.896 0.587
SMI 1.063 1.006 0.009 0.203 0.006 0.203 0.009
α = 1%
FTSE 0.641 0.152 0.138 0.529 0.138 1.967 1.967
DAX 0.157 0.132 0.518 0.132 1.129 1.129 0.518
CAC40 4.001* 0.001 0.119 0.119 0.119 0.119 0.001
AEX 4.087* 4.490* 0.001 0.490 0.490 0.490 1.087
SMI 0.583 1.226 1.226 2.073 1.226 2.073 2.073
α = 0.5%
FTSE 0.075 0.075 0.075 0.570 0.075 0.069 1.477
DAX 1.463 1.463 0.561 0.561 0.561 0.561 0.561
CAC40 4.427* 4.427* 0.540 0.540 0.143 0.540 0.540
AEX 7.540** 0.540 0.059 0.540 0.059 1.427 0.540
SMI 0.082 0.610 0.610 0.610 1.543 0.610 0.610
No. of Rejections 4 2 0 0 0 0 0
Notes:

This table reports the LR statistics for the unconditional coverage tests for the VaR forecasts under different competing models. The test is asymptotically distributed as χ2 with 1 degree of freedom;

*

and

**

indicate significance at 5% and 1% levels, respectively

Comparison of the VaR forecasting models using the test of independence

Ticker G EVT EG EVT RG EVTRV RG EVTSRV RG EVTBV RG EVTSBV RG EVTRK
α = 5%
FTSE 0.043 0.760 0.001 0.167 0.167 0.328 0.101
DAX 0.368 0.032 0.130 0.401 0.717 0.222 0.586
CAC40 1.003 0.012 0.653 0.399 0.247 0.104 0.397
AEX 0.186 1.367 0.691 0.483 1.022 0.483 1.022
SMI 0.302 0.059 1.795 1.648 1.328 1.648 1.795
α = 1% 
FTSE 0.293 1.779 0.758 0.536 0.758 1.945 0.144
DAX 0.422 0.755 0.960 0.755 0.960 1.189 0.960
CAC40 2.572 0.572 0.750 0.750 0.750 0.750 0.572
AEX 2.180 2.952 0.572 0.547 0.952 0.547 0.315
SMI 1.975 1.519 0.270 0.227 2.459 0.227 0.227
α = 0.5%
FTSE 0.105 0.105 0.105 0.293 0.105 1.070 0.423
DAX 0.422 0.422 0.292 0.292 0.186 0.292 0.292
CAC40 2.419 2.419 0.290 0.290 0.419 0.290 0.290
AEX 0.290 0.290 0.185 0.290 0.185 0.419 0.290
SMI 2.189 2.906 2.344 2.344 1.685 2.344 2.344
No. of Rejections 0 0 0 0 0 0 0
Notes:

This table reports LR statistics of independence tests for the VaR forecasts under different competing models. Following Chen et al. (2012), we have extended the test up to four lags but reported the results for the fourth lag. The test is asymptotically distributed as χ2 with 1 degree of freedom; *and **indicate significance at 5% and 1% levels, respectively

Comparison of the VaR forecasting models using the test of conditional coverage

Ticker G EVT EG EVT RG EVTRV RG EVTSRV RG EVTBV RG EVTSBV RG EVTRK
α = 5%
FTSE 1.321 1.907 0.501 0.956 0.956 0.827 0.101
DAX 1.503 1.652 0.820 0.647 1.158 0.663 1.933
CAC40 1.590 0.908 0.998 0.565 0.833 0.449 0.565
AEX 1.241 2.263 1.036 1.379 1.609 1.379 1.609
SMI 1.365 1.065 1.804 1.851 1.334 1.851 1.804
α = 1%
FTSE 0.934 1.930 0.895 1.065 0.896 3.912 2.111
DAX 0.579 0.888 1.478 0.888 1.477 2.318 1.477
CAC40 6.141* 0.573 0.868 0.868 0.868 0.868 0.573
AEX 6.267* 7.442* 0.573 1.037 1.442 1.037 1.402
SMI 2.558 2.745 1.496 2.300 3.686 2.300 2.300
α = 0.5%
FTSE 0.180 0.180 0.180 0.863 0.180 1.138 1.900
DAX 1.884 1.884 0.853 0.853 0.252 0.853 0.853
CAC40 6.573* 6.846* 0.830 0.830 1.846 0.830 0.830
AEX 7.830* 0.830 0.244 0.830 0.244 1.846 0.830
SMI 2.271 3.516 2.954 2.954 3.228 2.954 2.954
No. of Rejections 4 2 0 0 0 0 0
Notes:

This table reports the LR statistics of the conditional coverage tests for the VaR forecasts under different competing models. Following Chen et al. (2012), we have extended the test up to four lags but reported the results for the fourth lag. The test is asymptotically distributed as χ2 with 2 degrees of freedom;

*and **indicate significance at 5% and 1% levels, respectively

Comparison of the ES forecasting models using Acerbi and Székely (2014) measure

Ticker G EVT EG EVT RG EVTRV RG EVTSRV RG EVTBV RG EVTSBV RG EVTRK
α = 5%
FTSE 1.894 1.904 1.909 1.895 1.881 1.918 2.039
DAX 4.259** 4.259** 2.100 2.099 2.104 2.121 2.122
CAC40 1.931 1.925 1.959 1.985 1.936 1.958 1.984
AEX 3.984* 1.899 1.936 1.898 1.914 1.899 1.928
SMI 2.035 2.075 2.015 1.970 2.047 1.975 2.009
α = 1%
FTSE 1.684 2.859* 2.104 2.256 2.100 2.503 2.556
DAX 4.305** 4.356** 1.850 1.848 1.849 2.308 1.864
CAC40 2.152 2.203 2.187 2.189 2.201 2.180 2.061
AEX 2.310 2.325 1.989 2.247 2.141 2.155 2.368
SMI 3.259* 2.512 2.417 2.437 2.458 2.522 2.433
α = 0.5%
FTSE 3.780* 1.723 1.801 2.333 1.784 2.072 2.654
DAX 4.506** 4.599** 1.840 1.854 1.830 2.367 1.792
CAC40 2.859 2.940 2.465 2.472 2.741 2.454 2.462
AEX 3.323* 2.378 2.074 2.239 2.007 2.484 2.282
SMI 2.123 2.969* 2.386 2.389 2.688 2.404 2.383
No. of Rejections 7 5 0 0 0 0 0
Notes:

This table reports the backtesting results for the ES forecasts using the test suggested by Acerbi and Székely (2014);

*

and

**

indicate significance at 5% and 1% levels, respectively

Comparison of the ES forecasting models using the Embrechts et al. (2005) measure

Ticker G EVT EG EVT RG EVTRV RG EVTSRV RG EVTBV RG EVTSBV RG EVTRK
α = 5%
FTSE 0.612 0.255 0.231 0.459 0.213* 0.391 0.690
DAX 0.530 0.565 0.582 0.444* 0.491 0.544 0.703
CAC40 0.558 0.493* 0.567 0.591 0.574 0.601 0.605
AEX 0.669 0.432 0.367* 0.576 0.429 0.589 0.665
SMI 0.421 0.856 0.444 0.552 0.492 0.598 0.356*
α = 1%
FTSE 1.114 0.978 0.430 0.223* 0.480 0.412 0.494
DAX 1.469 1.534 0.767 0.579* 0.606 0.846 0.931
CAC40 4.410 3.943 1.555 1.469 1.808 1.284* 1.754
AEX 0.930 1.282 0.121* 0.337 0.422 0.230 0.503
SMI 0.956 0.896 0.743* 0.907 0.857 0.948 0.883
α = 0.5%
FTSE 2.032 2.343 0.334 0.418 0.379 0.487 0.314*
DAX 0.952 1.721 0.898 0.833 0.518* 0.918 0.676
CAC40 3.548 2.907 1.188 1.334 1.050 1.004* 1.208
AEX 0.891 1.279 1.055 0.950 1.063 1.259 0.817*
SMI 0.836 0.633 0.732 0.529 0.871 0.524* 0.611
Min Value Occurrence 0 1 3 3 2 3 3
Notes:

This table reports the Embrechts et al (2005) measure (scaled up by a factor of 1,000) for the ES forecasts;

*

indicates the minimum value of the measure among the models for a particular sample index at a certain confidence level

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Corresponding author

Prateek Sharma can be contacted at: prateeksharma1985@gmail.com

About the authors

Samit Paul is currently associated with the Indian Institute of Management, Calcutta, as an Assistant Professor in the area of Finance and Control. Currently, his research interest lies in the area of downside risk management, portfolio risk management and risk modelling.

Prateek Sharma is currently associated with the Indian Institute of Management, Udaipur, as an Assistant Professor in the area of Finance and Accounting. He pursues his research in the area of volatility and VaR modelling, alternative risk management and risk modelling.