# Forecasting VaR in WTI market by a dynamic extreme value approach.

1. IntroductionOver the last twenty years, the global economy has fluctuated strongly and been confronted with unforeseen risk. In the financial world, people have faced huge losses because of the bankruptcy or near bankruptcy of serval institutions first in the United States and then globally. These financial disasters have attracted the attention of academics, policy makers and economic agents at large. It has become clear to us that risk management especially with regard to market risk for financial institutions needs to be reassessed and the development of standard measures of the market risk is very necessary to the international economy. The VaR that was first used in late 1980s at J.P. Morgan has emerged as a standard and popular measure of market risk. VaR is simply a quantile of the loss or profit distribution (in this paper ,we only focus on the loss distribution) of a given asset portfolio over a prescribed holding period from a mathematical viewpoint. VaR suggests that how much people can lose with a given probability over a certain horizon.

The existing approaches for estimating VaR can be divided into three types: the nonparametric historical simulation (HS) method, parametric methods based on an econometric model for volatility dynamics and the assumption of conditional normality, and the extreme value theory based methods. The estimated loss distribution is simply given by the empirical distribution of past losses in the HS method. HS method avoids "ad-hoc-assumptions" on the form of the loss distribution and is easy to implement. However, in this method, the estimators of extreme quantiles are difficult to estimate accurately because extrapolation beyond past observations is impossible and extreme quantile estimates within sample tend to be very inefficient.

The methods based on econometric models such as GARCH-type models that assume conditional volatility which reflect the current volatility background. However, these methods suffer serious drawback that the assumption of conditional normality does not seem to hold for financial time series data. In order to overcome the drawbacks of normal distribution, GARCH models with skewed fat-tailed distributions are used to estimate VaR instead of GARCH models with normal distribution (such as Bali (2007),etc).

The estimation of return distributions of financial time series by using EVT is a topical issue and a lot of research for this method such as Embrechts et al. (1999), Longin (1996), Poon et al. (2004), etc. McNeil (1998) uses estimation techniques based on limited theorems for block maxima (BMM). McNeil uses a similar approach and shows how to correct for the clustering of extremal events caused by stochastic volatility. Danielsson and de Vries (1997) use a semi-parametric approach based on the Hill-estimator of the tail index. Embrechts et al. (1999) advocate the use of a parametric estimation technique which is based on a limit result for the excess-distribution over high thresholds. Recently, EVT has also been used to analyze extremes in financial markets due to the turmoils in various financial markets experienced around the globe. The tails of financial data series have been analyzed by Longin (1996), McNeil (1998), McNeil and Frey (2000), Neftci (2000), Gengay and Selguk (2004), Hill (2010), Kabundi and Muteba (2011), Allen et al. (2013), Singh et al. (2013), Degiannakis et al. (2013), Ghorbel and Trabelsi (2014), Dendramis et al. (2014), etc.

EVT-based methods take more attention on the tail of loss distribution and yield a more accurate estimator of VaR. However, most traditional EVT-based methods for quantile estimation yields VaR estimates which neglect the current volatility background. To overcome the disadvantages of each of the above methods and estimate VaR more efficient, McNeil and Frey (2000) propose an approach based on above mentioned three kinds of approaches. This approach present a two step dynamic VaR estimating method which uses EVT with GARCH model. In consideration of the current market volatility background, a GARCH model is used to model for the raw return series in first step. The GARCH residuals are closer to independent and identically distributed (i.i.d) than the raw return series, but usually GARCH residuals show to exhibit fat tails. Then in the second step, they apply EVT to the GARCH residuals. As such, both time- varying volatility and fat-tailed return distributions are accommodated in the two step GARCH-EVT model. Because the GARCH-EVT is more superior than the traditional EVT models and the simple GARCH-type models with normal distributions in many cases, it is applied and referenced widely by some researchers (see Poon et al. (2004), Kuester et al. (2006), Karmakar (2013), Kumar and Maheswaran (2014), Yi et al. (2014), etc.).

The paper aims to study the tail-related risk measures including static and dynamic risk measures by extreme value theory (EVT) in the WTI market. Firstly, the study estimates the static VaR both by the classic variance-covariance method based on the assumption that the loss distribution follows normal distribution and the POT model based on the extreme value theory which pays more attention to the fact that the loss distribution is fat-tailed. Secondly, respect to the important properties of loss distribution of most financial data that it has time-varying volatilities and fat-tailed distribution, we employ a two stage approach of EVT which was originally proposed by McNeil and Frey (2000) to estimate dynamic value at risk. At the same time, in order to describe a stylized fact of the volatility of asset returns whose volatility response to a large positive return is considerably smaller than that of a negative return of the same magnitude, we fit the standard GJR-GARCH model proposed by Glosten et al. (1993) instead of the normal distribution GARCH (see Bollersler (1986)) model at first. Then the peak over threshold method (POT) is applied to the residual extracted from the GJR-GARCH model for a suitable threshold to estimate VaR. Finally, the results of backtesting of historical daily log-return series indicate that the procedure gives better estimates than the methods which ignore the fat tails of the innovations or the stochastic nature of the volatility.

The rest of this paper is organized as follows. In Section 2, the models we consider are shortly discussed. Section 3 presents the empirical procedure and results about the West Texas Intermediate market daily log-return series by using the models in Section 2. In addition, Section 4 presents some conclusions about the research.

2. The models

2.1. Static VaR - based on POT model

There are two main kinds of models that researchers apply EVT to estimate the value at risk of a portfolio: the first is the block maxima model (BMM) approach based on the generalized extreme value distribution (GEV), while the second is the peak over threshold approach based on the generalized Pareto distribution (GPD). The BMM is a traditional method for fitting a block of maxima (extreme events) in a time series of independent and identically distributed observations to GEV using different statistical methods. But POT is considered more efficient and powerful in modeling for limited data as it approximats the exceedances of all large observations which exceed over a given threshold in a data set to GPD.

In practice, the BMM has a major defect that it is very wasteful of data. Therefore, it has been largely superseded by the POT, which we use all data that are extreme in the sense that they exceed a particular designated high level. In this paper, we adopt the POT to identity the extreme observations that exceed a high threshold and estimate the static VaR of our empirical data. Now we focus on the specific details of the POT and some properties of the models.

The main distributional model for exceedances over thresholds is the GPD whose definition is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

Where [beta] > 0, and x [greater than or equal to] 0 when [xi] [greater than or equal to] 0 and 0 [less than or equal to] x [less than or equal to] [beta]/[zeta] when [xi] < 0 .The parameters [xi] and [beta] are referred to, respectively, as the shape and scale parameters.

The GPD in the extreme value theory has played an important role in modeling the excess distribution over a high threshold. This concept is defined along with the mean excess function, which will also play an important role in the theory in returns. The following is the definition of excess distribution over threshold u and mean excess function.

Let X be random variable with distribution function F, the excess distribution over the threshold u has distribution function

[F.sub.u](x) = P(X - u [less than or equal to] x | X > u) = F(x + u) - F(u)/1 - F(u) (2)

for 0 [less than or equal to] x < [x.sub.F] - u, where [x.sub.F] [less than or equal to] [infinity] is the right endpoint of F .

The mean excess function of a random variable X with finite mean is defined by:

e(u) = E(X - u|X > u). (3)

The excess distribution function describes the distribution of the excess loss over the threshold u, given that u is exceeded. For a large class of underlying distributions F, the excess distribution function [F.sub.u] can be approximated by GPD for some high threshold u (see Pickand (1975), Balkema and de Hann (1974)). In this paper we follow the common practice assumption that the exceedances can be modeled as stationary i.i.d processes.

Now the procedure of the POT is introduced. Given loss data [X.sub.1], ......, [X.sub.n] from F, let [N.sub.u] be the number of the data exceeding our threshold u, and relabel these data [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for convenience. In this paper, the method of maximum likelihood is applied to estimate the parameters of a GPD model by fitting this distribution to the [N.sub.u] excess losses.

According to the above discussion, we can obtain, for x > u,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Which provides a formula for calculating tail probabilities if F(u) is given. This formula (5) may be inverted to obtain a high quantile of the underlying distribution, which is interpreted as value at risk. For [alpha] [greater than or equal to] F(u), we yields (see Embrechts et al. (1997)):

[VaR.sub.[alpha]] = [q.sub.[alpha]](F) = u + [beta]/[xi] [((1 + [alpha]/[bar.F](u)).sup.-[xi]] (5)

We estimate these quantities by replacing [xi] and [beta] their estimations with maximum likelihood method. And the simple empirical estimator [N.sub.u]/n is taken to an estimate of [bar.F](u). So we can gain an estimation of the value at risk in the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

We compare the estimation of the value at risk by POT with that by traditional variancecovariance method. Following Duffie and Pan (1997), the variance-covariance based value at risk at time t is computed as:

[VaR.sub.[alpha]] = [Z.sub.[alpha]] [[sigma].sub.p] [square root of ([DELTA]t)] (7)

where [Z.sub.[alpha]], [[sigma].sub.p] and [DELTA]t are the standardized normal variable, the portfolio volatility and the time horizon, respectively. We use this formula to generate estimations of the value at risk based on the variance-covariance method and compare them with the estimation based on the peak over threshold method. Finally, we also demonstrate the efficiency of these two approaches by backtesting.

2.2. Dynamic VaR--based on GJR-GARCH and EVT

EVT can not only be used to calculate static VaR but also can be combined with auto-correlated time series models to forecast VaR. McNeil and Frey (2000), proposed a two step dynamic VaR forecasting method based on EVT. Considered the conditional heteroscedasticity of most financial data, they applied GARCH model to model for return sequences in the first step. In the second step, the POT model based on EVT is applied to the GARCH residuals which are closer to iid than the original returns. We now present our model which referenced the approach proposed by McNeil and Frey (2000) in the following. This approach is denoted as GJR-GARCH-EVT in this study.

Here we first use the ARMA(1,1)-GJR-GARCH model with the mean return modeled as an ARMA(1,1) process and the conditional variance of the returns as a GARCH (1,1) model:

[r.sub.t] = U + [[alpha].sub.1][r.sub.t-1] + [[epsilon].sub.t] + [[beta].sub.1][[epsilon].sub.t-1] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where [[epsilon].sub.t]|[[OMEGA].sub.t] [right arrow] skewed t-distribution with mean = 0, variance = [[sigma].sup.2.sub.t], and [[OMEGA].sub.t], is the information set of all information at time t.

As the return series in this paper are asymmetrical, skewed, fat-tailed and peaked around the mode, it is appropriate and accurate to use the skewed t-distribution with GJRGARCH model instead of the normal distribution with GARCH model when estimating VaR. It is clearly demonstrated that the skewed fat-tailed distributions with GARCH provide very accurate and robust estimates of the actual VaR thresholds and perform equally well as the more specialized extreme value distribution in many papers (see Bali (2007), Karmakar (2013), Allen et al. (2013), etc).

GJR-GARCH model is chosen to remove the time series volatility clustering, which may be inconsistent with given assumption that exceedances of financial return series are i.i.d. Here we estimate these parameters of the GJR-GARCH model by maximum likelihood method. Then standard residuals are closer to i.i.d and are researched by the POT model.

For a 1-day horizon, an estimate of the conditional VaR is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where the sequences {[Z.sub.t] = [r.sub.t] - [u.sub.t]/[sigma]} are standard residuals series and V[??]R[([Z.sub.t]).sub.[alpha]] is given in Equation (10), [[??].sub.t+1] and [[??].sup.[sigma].sub.t] are respectively the one step forecast for the conditional mean and variance in time t +1.

3. Data and empirical results

This empirical study focuses on modeling for WTI daily log-return series. The sample data period in this paper is from 2000.1.4-2014.10.30, which provides us approximate 15 years of data with 3724 daily log-returns.

As we consider only the right tail of the generalized extreme value, which represents losses for an investor with a long positions in the abovementioned West Texas Intermediate market. We compute the distribution of maximal losses for the above investors before. This distribution is obtained as follows. Denote pt the closing price at time f in the WTI market, and we construct the negative return as [r.sub.t] = [-100.sup.*] log([p.sub.t]/[p.sub.t-1]), where t = 1, 2, ..., T .

3.1. Description of data

The basic statistics are given in Table 1 and the time series plot of daily negative logreturns of the WTI market from the indices during 2000.1.1-2014.10 in Figure 1.

The kurtosis in Table 1 is greater than three for return series which suggests that the distribution of return in the WTI market is fat-tailed, and the skewness suggests this distribution is biased. The value of Jarque-Bera statistic for the series indicates that the returns do not follow a normal distribution.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Figure 1 shows that the time series plots of daily returns of the indices during 2000.12014.10. The extreme behaviour of the return series during the sample data period is showed by the above graphs.

3.2. Static VaR estimation

3.2.1. The VaR based on the POT approach and the variance-covariance approach

In this Subsection, We implement the POT approach and the traditional variancecovariance approach to model the extreme risk for the WTI market and estimate the VaR of the above return series. At first, as mentioned earlier, we employ the POT method by using GPD for tail estimation of the raw return series in the following.

The first step in this modeling is to select the right threshold for identifying the relevant tail region. Several techniques are successful for threshold determination. Here the sample mean excess function (MEF) is utilized to choose an appropriate threshold and it is expressed by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

where [X.sub.1], [X.sub.2], ..., [X.sub.n] are sample data. So [e.sub.n](u) is an empirical estimator of the mean excess function e(u) which describes the expected overshoot of a threshold once an exceedance occurs. If the empirical MEF is a positively sloped straight line above a certain threshold u, it is an indication that the data follows the GPD with a positive shape parameter [xi]. The MEF of returns can be applied to choose a suitable thresholds in POT model.

Figure 2 gives a mean excess plot for daily negative log-returns in WTI market, and from the MEF the threshold can be chosen as u = 1.7. Using this threshold, we estimate the parameters of the POT method by the maximum likelihood method and forecast the VaR with different quantiles based on the formula (6). The estimators of parameters are given in Table 2.

[FIGURE 3 OMITTED]

The POT method requires a suitable threshold u , values above which are fitted to GPD. To determine how the GPD fits the tails of the return distribution, we plot the empirical distribution of exceedances along with the cumulative distribution simulated by a GPD and compare the results visually in Figure 3. It proposes the empirical excess distribution function follows the trace of a corresponding GPD closely and implies that the threshold in this POT model is appropriate.

Next we use the same daily return data set to quantify static VaR based on the traditional variance-covariance approach. And the values of VaR based on the POT approach and the traditional variance-covariance approach are presented in Table 2.

3.2.2 Backtesting of static VaR

One approach used to evaluate the performance of the value at risk techniques is the two sided binomial test. The theory we used to proceed our backtesting is that the expected number of breaches m which the actual loss exceeds the forecasted the value at risk is n(1 - [alpha]) if the value at risk model is actual. In other word, if the model is effective, the number of exceedances (which is denoted by m) follows a binomial distribution whose expectation is n(1 - [alpha]) and variance is n[alpha](1 - [alpha]). We use the test statistic

z = m - n(1 - [alpha])/[square root of (n[alpha](1 - [alpha]))] (12)

Whose approximation distribution is normal distribution.

On the one hand, if the number of exceedances is much larger than n(1 - [alpha]), the model is considered to underestimate risk. On the other hand, if the number of breaches is too smaller than expected violations, the model is viewed to overestimate the risk. The values of the test statistic z and the numbers of breaches for the two sided binomial test are given in Table 4.

From Table 4, we can find that the method based on the POT model is accepted in this paper at every confidence level ([alpha] = 0.99,0.975,0.95). From another point of view, the method based on traditional variance-covariance with the assumption of normal distribution is rejected at confidence level [alpha] = 0.99. The abovementioned conclusion is consistent with the result proposed by Kabundi and Muteba (2011) that the peak over threshold method is effective and accurate in high quantiles.

3.3. Dynamic VaR estimation

3.3.1. The VaR based on the GJR-GARCH-EVT model

In this Subsection, followed the model in Subsection 2.2, we first use a GJR- GARCH model to model for the raw returns. In consideration of the AIC, here an ARMA (1,1) model is applied as the mean equation. Table 5 presents the estimated parameters of the mean and volatility equations of the GJR-GARCH model with skewed t- distributed innovations applied to daily return series. Both the constant term and the ARMA(1,1) coefficient in the mean equation are found to be significant. Similarly, the parameters in the volatility equation: the constant, the ARCH(1) coefficient and the GARCH (1) coefficient, are all found to be significant. At the same time, the parameter which is to measure asymmetry of the series is not zero significantly and it illustrates that the GJRGRACH model instead of GARCH model in the study is necessary.

Next we study the standardized residuals of our model followed subsection 2.2 and some results are given in Table 6.Table 6 presents some diagnostic statistics of raw returns and standardized residuals. The significant value of Ljung-Box Q(16) statistic indicates that raw returns are correlated and hence are not i.i.d as required by EVT, but the standardized residuals are closer to i.i.d as their Q(16) statistic is not significant. Thus the filtering procedure advocated by McNeil and Frey (2000) has been effective in producing i.i. d residuals on which EVT can be implemented. The Q2(16) statistic of standardized residuals also suggests that the GJR-GARCH model is well specified. However, the skewness, excess kurtosis and Jarque-Bera statistics in the table demonstrate that neither the return series nor the standardized residual series are normally distributed. All these findings motivate the second stage in Subsection 2.2 EVT implementation, where the standardized residuals is fat- tailed and skewed is explicitly modeled.

On the other view, we plot correlograms for the raw returns and the standardized residuals in Figure 4 and it suggests the raw data are clearly not i.i.d where the standardized residuals are closer to iid, then we can applied EVT to the standardized residuals. At the same time, the adequacy of the GJR-GARCH modeling can be verified by this fact.

As discussed in Subection 2.2, EVT can also be used in a dynamic framework to forecast VaR. In this part of the empirical analysis we use a 2724 day log return moving window to evaluate the daily VaR for our data sample. The advantage of the moving window technique is that it allows us to capture dynamic time-varying characteristics of the data in different time periods. The data period here is approximately 17.5 years which gives us 1000 daily predictions. We choose a higher 10% quantile as threshold u to fit the standardized residuals from the GJR-GARCH model to GPD followed many researchers (see McNeil and Frey (2000), Allen et al. (2013), etc).

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Figure 5 shows the efficacy of estimated dynamic VaR based on the GJR-GARCH-EVT model. In this figure we plots the returns; superimposed on this figure is the dynamic VaR estimates at different confidence levels. It appears from the figure that the VaR changes quickly to reflect market fluctuations. The VaR also increases in periods of extreme volatility, while the values of VaR appear to be stable for a normal market condition. The fast changing position of VaR in response to market condition is tenable here. In the abovementioned model, the auto regression and asymmetry in returns, conditional volatility clustering, the heavy tails and skewed of the standardized residuals are all described and accommodated. So these features ensure that VaR estimates from the GJR-GARCH-EVT model at any given time can reflect the most recent and relevant information.

3.3.2. Backtesting of dynamic VaR

In this subsection, in order to verify that the GJR-GARCH-EVT model is efficient and precise, we exercise backtesting on the dynamic VaR estimator not only by the violations introduced in Subsection 3.2.2 but also by the LR statics proposed by Kupiec (1995) which is used widely. The results are presented in Table 7.

The results in Table 7 reveal that the actual violations are very approximate to the expected violations at every confidence level and the results manifest that the model forecasts the VaR preferably.

4. Conclusions

As the volatility in the financial market increases, implementing an effective risk management system becomes more and more important. In risk management, the VaR methodology as a measure of market risk is widely accepted and used in both institutions and regulators. The high volatility of the WTI market demands the implementation of effective risk management. EVT is a successful and valid theory to estimate the effects of extreme events in fluctuating markets where extreme values may appear based on sound statistical methodology.

In this paper, on one hand, we pay attention to the POT method based on EVT which provides a statistical model for fitting extreme values above a threshold. In the empirical study, we demonstrates that EVT can be used to forecast VaR more accurate than the traditional variance-covariance approach. It is because that EVT provides more comprehensive results as it can model the extremes in a distribution which are less captured by models that based on the assumption of normality in the data. On the other hand, this study exhibits how to apply EVT to the daily return series and forecast the VaR which is used to measure the tail-related risk. The EVT based traditional models are usually applied to measure static market risk and they often neglect the stochastic volatility and conditional heteroscedasticity of most financial data. To overcome this drawback, we refer to McNeil and Frey (2000) and combine the GJR-GARCHEVT model. In this paper, as the return series are asymmetrical, skewed, fat-tailed, autocorrelation and stochastic volatility, we first model the skewed t- distribution with GJR-GARCH model for the raw returns. Then the standardized residuals from the GJRGARCH are verified that they are similar to i.i.d, so that EVT models can be applied to them. However, the residuals are not normally distributed, so it is more appropriate that we use EVT based models than normal distribution based models to analyze the standardized residuals. Application in the study captures the heavy-tailed behavior in daily returns and the asymmetric characteristics in distributions.

Recebido/Submission: 13/06/2016

Aceitacao/Acceptance: 23/09/2016

Acknowledgments

The paper is supported by the Humanity and Social Science of Ministry of Education Planning Foundation of China (10YJA880197).

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Xueyan Pan (1,2), Guanghui Cai (2) *

* Guanghui Cai, cghzju@163.com

(1) School of Mathematics and Computer Science, Anhui Normal University, 241000, Wuhu, China

(2) School of Statistic and Mathematics, Zhejiang Gongshang University, 310018, Hangzhou, China

Table 1--Summary statistics mean standard skew Kurt max JB statistic -0.01329248 1.052867 0.2485699 5.047643 7.422868 3998.6 Table 2--Parameter estimates for the POT model in WTI market Total in-sample observation T 3724 threshold u =1.7 Number of exceedances k 167 % of exceedances in-sample k/T 4.48 GPD shape parameter [xi] 0.2051246(0.11312398) GPD scale parameter [beta] 0.7235175(0.09849234) Table 3--Static VaR estimates [alpha] = 0.99 [alpha] = 0.975 [alpha] = 0.95 VaR in Equation 2.971366 2.149145 1.6222134 (6) VaR in Equation 2.449335 2.063582 1.731812 (7) Table 4--Results-backtesting of static VaR in WTI market Confidence level Expected Violation Violations Value of z violations based on based on based on Equation Equation Equation (6) (7) (6) [alpha] = 0.99 38 38 69 0.1251674 [alpha] = 0.975 94 89 99 -0.430335 [alpha] = 0.95 187 183 153 -0.2406015 Confidence level Value of z based on Equation (7) [alpha] = 0.99 5.230679 [alpha] = 0.975 0.6192626 [alpha] = 0.95 -2.496241 Table 5--Parameter estimates for the GJR-GARCH mode Parameter Estimate Std. Error P-value u -0.015000 0.012971 0.247489 [[alpha].sub.1] -0.036692 0.016518 0.026324 [omega] 0.004702 0.001552 0.002452 p 0041289 0.004448 0.000000 q 0.952442 0.003845 0.000000 [lambda] -0.202634 0.070535 0.004068 skew 1.091199 0.025446 0.000000 shape 7.039731 0.735744 0.000000 Table 6--Diagnostic statistics of raw returns and standardized residuals in WTI market Statistics Skew Kurt J-B Q(16) Q2(16) statistic Raw returns 0.2485699 5.047643 3998.6 49.6647 536.0456 (0.0000) (0.0000) (0.0000) Standardized 0.3429893 2.537795 1074.756 10.0352 21.8893 residuals (0.0000) (0.8648) (0.1468) Table 7--Results-backtesting (dynamic VaR) Expected Violations LR statistic Value of z violations in GJR- GRCH-EVT model [alpha] = 0.99 38 45 1-531458 1.278025 [alpha] = 0.975 89 98 0.2601014 0.5143029 [alpha] = 0.95 183 181 0.1542307 -0.3909774