Value At Risk Estimation Of Energy Commodity

Published Date: 02 Nov 2017

aResearch Laboratory for Economy, Management and Quantitative Finance (LaREMFiQ), IHEC - University of Sousse

*E-mail address: [email protected], [email protected], [email protected]

ABSTRACT

In this paper, we evaluate Value-at-Risk (VaR) for crude oil market. We adopt three long-memory - models including,, and to forecast crude oil volatility by capturing some volatility stylized fact such as long-range memory, heteroscedasticity, asymmetry and fat-tails. Then we consider Extreme Value Theory which concentrates on the tail distribution rather than the entire distribution. is considered as a potentially framework for the separate treatment of tails of distributions which allows for asymmetry. Our results show that the model with extreme value theory performs better in predicting the one-day-ahead VaR. Our findings confirm that taking into account long-range memory, asymmetry and fat tails in the behavior of time series combined with filtering process such as are important in improving risk management assessments and hedging strategies.

Key words: Extreme Value Theory, long- range- memory, Value-at-Risk, oil price volatility,

Introduction

The world crude oil prices have raised dramatically during the past decade, consequently the oil market have become very volatile and risky. Moreover, volatile oil prices may lead to price variability of other energy commodities and can have wide-spread impacts on the international economy. Therefore, forecasts of oil price volatility are important for both academicians and market participants. Many forecasting approaches and risk measurement tools have been proposed in the existing literature, in order to provide financial institutions, risk mangers and market participant with appropriate and easy technical approaches to measure financial and energy markets risk. In this paper we are interested on commodity assets, especially on forecasting oil prices risk. In fact, a large body of empirical studies shows that oil price fluctuations have considerable effects on economic activity. Hamilton (1983) argues that oil price shocks are responsible, at least partly, for every U.S recession after the second world-war. Sadorsky (1999) find that oil price volatility shocks have asymmetric effects on the economy and find evidence of the importance of oil price movements in explaining movements in stock returns. Consequently, it’s important to model these oil price fluctuations and implement an accurate tool for energy price risk management. In this context, Value-at-Risk (VaR), originally proposed by J.P. Morgan in 1994, has become a popular risk measures in the financial industry (Duffie and Pan (1997) and Engle and Manganelli (2004)). The concept of VaR is defined as a potential amount of loss on a portfolio with a given probability over a certain fixed time horizon. In fact, VaR reduces the risk associated with any portfolio to just one number, the loss occurred given a certain probability. Since the diffusion of the Risk Metrics system (RM), an academic debate has been emerged among academicians and practitioners about the appropriate approach to calculate VaR. Different approaches have been proposed in the existing literature and may be classified into three families. First, the non-parametric historical simulation (HS) approaches. Second, the parametric models approaches based on an econometric model for volatility dynamics under the normality assumption of the returns distribution. Third, the extreme value theory approach which models only the tails of the return distribution. Since VaR estimations are only related to the tails of a probability distribution, techniques from may be particularly effective. Extreme value theory has been applied in different area where extreme losses may appear, in hydrology (Davidson and Smith, 1990; Katz and al., 2002). In insurance (McNeil, 1997; RootzNen and tajvidi, 1997) and in finance (Longin, 1996; Danielsson and De varies, 1997; McNeil, 1998; Embrechts and al., 1999, McNeil and Frey, 2000; Gençay and Selçuk, 2004…). However, none of the previous study has reflected the current volatility background. In order to overcome the drawbacks of these methods, McNeil and Frey (2000) proposed a combined approach that reflects two stylized facts exhibited by most financial returns series, namely stochastic volatility and fat-tailedness of conditional returns distribution. In this context, the use of extreme value theory in oil market to implement a risk measure constitutes an important issue. Different empirical studies have investigated the predictive performance of approach to measure risk forecasts on the oil market despite the significant need and interest to manage energy price risks (Krehbiel and Adkins, 2006; Cabedo and Moya, 2003; Costello et al., 2008, Huang et al., 2008, Fan et al., 2008, Marimoutou and al., 2009). To the best of our knowledge, in the empirical literature, the above mentioned studies have not considered the eventual long-range memory in the crude oil volatility when using the theory of extremes to quantify market risks. They were focusing on heteroscedasticity, fat tails and normality in the empirical distribution of returns time series. Degiannakis (2004) have analyzed the forecasting performance of different risk models in order to estimate the one-day ahead realized volatility and the daily VaR. He concludes that the Fractional Integrated with skewed-student-t conditionally distributed innovations is more appropriate to take into account the major stylized facts of equity price behavior. Tang and Shieh (2006) have investigated the long-memory proprieties of three stock index futures markets. They conclude that the hyperbolic with skewed-student-t distribution performs better. Similar conclusions have been confirmed by Kang and Yoon (2007). These studies are concerned only with developed financial assets. There are only few studies focused on energy markets. Aloui and Mabrouk (2010) have estimated the Value-at-Risk for three long memory models, and with different error’s distribution assumptions for some major crude oil and gas commodities. They conclude that models taking into account asymmetries in the volatility specifications and fractional integration in the volatility process perform better in the VaR’s prediction. The focus of this paper is to further explore the usefulness of in predicting extreme risks in oil market. To this end, we adopt some long-memory approach -type models to forecast energy prices volatility by capturing some volatility stylized fact such as long-range memory, heteroscedasticity, asymmetry and fat-tails. Then we consider approach which concentrates on the tail distribution rather than the entire distribution. For his finality, conditional approaches are used to forecast VaR. This paper differs from the existing literature in at least two points. Firstly, for our knowledge, the long-memory type models have not been implemented with to forecast energy market risk. Secondly, compared with same previous studies, we have considered both the right and left tail to evaluate the VaR’s accuracy for predicting oil market risk. This paper is organized as follow. Section 2 represents a brief review of extreme value theory and exposes the forecasting models of long-range memory of energy market volatility. Data and preliminary analysis are provided in section 3. Section 4 provides our empirical findings and section 5 concludes the paper.

Methodology

Measures of extreme risk

Consider a strictly stationary time series representing daily observations of the negative log-return on a financial asset price. The dynamic of is assumed to be governed by:

, (1)

Where the innovations are a strict white noise process, independent and identically distributed, with zero mean, unit variance and marginal distribution function. We assume that and are measurable with respect to the information about the return process available up to time

Let denote the marginal distribution of and, for a horizon let denote the predictive distribution of the return over the next days, given knowledge of returns up to and including day .

We are interested of estimating unconditional and conditional quantiles in the tails of negative log-returns in these distributions. We remind that for the unconditional quantile is a quantile for the marginal distribution denoted by

,

And a conditional quantile is a quantile of the predictive distribution for the return over the next days denoted by

,

We also consider the expected shortfall (ES), known to be a measure of risk for the tail of a distribution. The ES is a coherent measure of risk in the sense of Artzner, Delbaen, Elsner, and Heath (2000). The unconditional expected shortfall is defined as:

And the conditional shortfall to be

We are principally interested in quantiles and expected shortfalls for the 1-step predictive distribution. Thus we denote the quantiles respectively by and. Since

These measures simplify to

(2)

(3)

Where is the upper quantile of the marginal distribution of which by assumption does not depend on

To implement an estimation procedure for these measures we must choose a particular model for the dynamics of the conditional mean and volatility. Many different models for volatility dynamics has been proposed in the econometric literature including models from the family (Bollerslev et al., 1992), process (Muller et al., 1997) and stochastic volatility models (Sheppard, 1996). In this paper, we use the parsimonious but effective process for the volatility. Our paper differ from the others papers existing in the literature by introducing some long-memory -type models to forecast energy price volatility by capturing some volatility stylized facts such as asymmetry and fat tails in the energy price return innovations and to provide better VaR’s computations.

Modeling oil price volatility

For predictive purpose we fix a constant memory so that at the end of day our data consist of the last negative log-returns. We consider these to be a realization from a process. Hence, the conditional variance of the mean-adjusted series is given by

Where and .

The long memory concept

According to Ding and Granger (1996), a series is said to have a long-memory if it displays a slowly declining autocorrelation function (ACF) and an infinite spectrum at zero frequency. Specifically, the series is said to be a stationary long- memory process if the ACF, behaves as,

(4)

Where and c is some positive constant. The left hand side and the right hand side in Eq. (4) tends to1 as The ACF in Eq.(1)displays a very slow rate of decay to zero as goes to infinity and . This slow rate of decay can be contrasted with processes which have an exponential rate of decay, and satisfy the following bound, and consequently . A process that satisfies this condition is termed short-memory. Equivalently, long-memory can be defined as a spectrum that goes to infinity at the origin. This is, .

A simple example of long-memory is the fractionally integrated noise process, , which is , where is the lag operator, and This model includes the traditional extremes of a stationary process, and a nonstationary process. The fractional difference operator is well defined for a fractional and the ACF of this process displays a hyperbolic decay consistent with Eq.( 4). A model that incorporates the fractional differencing operator is a natural starting point to capture long-memory. This is the underlying idea of the and class of processes. In practice we must resort to estimating the ACF with usual sample quantities

(5)

A second approach to measure the degree of long-memory has been to use semi-parametric methods. This allows one to review the specific parametric form, which is miss-specified and could lead to an inconsistent estimate of the long memory parameter. In this paper, we consider the most two frequently used estimators of long memory parameter d. The first is the Geweke and Porter- Hudak (GPH) (1983) estimator, based on a log-periodogram regression. Suppose is the dataset and define the periodogram for the first m ordinates as

(6)

Where , is chosen positive integer. The estimate of can then be derived from linear regression of on a constant and the variable which gives

(7)

Robinson (1995a) provides formal proofs of consistency and asymptotic normality for the Gauss case with . The asymptotic standard error is . The bandwidth parameter must converge infinitely with the sample size, but at a slower rate than . Clearly, a larger choice of m reduces the asymptotic standard error, but the bias may increase. The bandwidth parameter was set to in Geweke and Porter-Hudak (1983). While Hurvich et al. (1998) show the optimal rate to be ). Velasco (1999) has shown that consistency extends to and asymptotic normality to . The other popular semi-parametric estimator is due to Robinson (1995b). Essentially, this estimator is based on the log-periodogram and solves:

(8)

The estimator is asymptotically more efficient that the GPH estimator and consistency and asymptotic normality of are available under weaker assumptions than for the Gaussian case.

The fractional integrated model

Baillie et al. (1996) have extended the traditional model by considering an eventual fractional integration model. They suggested the mode which is able to distinguish between short memory and long memory in the conditional variance behavior. Formally, the process is defined as follows:

(9)

Or

Where is the lag operator. , et is an infinite summation which, in practice, has to be truncated. According to Baillie et al. (1996), should be truncated at 1000 lags. is the fractional differencing operator. It can be defined as follows:

Where (10)

The fractional integrated asymmetric power model

Tse (1998) have extended the model in order to take into account asymmetry and the long-memory feature in the process of the conditional variance. He has introduced the function of the process. The can be written as follows;

(11)

where are the model parameters. The process can consider for some stylized facts on volatility of financial and commodity prices. More specifically, (1) if then volatility exhibits the long-memory property; (2) if negative shocks have more impact on volatility than positive shocks and inversely; (3) is the power term in the volatility structure. It should be specified by the data; (4) the process also nests the process when and . Consequently, the process is superior to the because it takes into account asymmetry and long memory in the conditional variance behavior.

The hyperbolic

The model (Davidson, 2004) is obtained by extending the conditional variance of the model by introducing weights in the difference operator. The conditional variance of the model is expressed as follows:

(12)

The and stable cases correspond to =1 and 0, respectively. We should note that the main advantage of this model is to take jointly into account volatility clustering, long-range memory and leptokurtosis in the time series behavior. However, this model is unable to consider for asymmetry in the return distribution.

Extreme Value Theory and the peaks-over-threshold model

In this section, we briefly describe how we obtain the quantile by applying techniques to the distribution of -models filtered innovations. We fix a high threshold u and we assume that excess residuals over this threshold have a generalized Pareto distribution (GPD) with tail index,

(13)

Where and the support is when and when . The choice of this distribution follows from a limit result in EVT. Consider a general distribution function F and the corresponding excess distribution above the threshold defined by:

(14)

For wher is the (finite or infinite) right endpoint of F. Balkema and de Haan (1974) and Pickinds (1975) showed for a large class of distributions F that is possible to find a positive measurable function such that

This result was shown by Balkema and da Haan (1974) and Pickands (1975). This result holds for continuous distributions used in statistics.

In our case we assume that the tail of the underlying distribution begins at the threshold, with the random variables of exceeding observations. For a sample for total size the random proportion of extremes is then . If we assume that the the excesses over the threshold are i.d.d with exact GPD distribution, the parameters ξ and β are estimated by maximum likelihood. Smith (1987) has shown that maximum likelihood estimates and of the GPD parameters ξ and β are consistent and asymptotically normal as , provided . Even under an approximate GPD distribution, parameters estimates and are unbiased and asymptotically normal, provided a sufficent rate of convergence. Under the assumption of dependent data, the GPD-based tail estimator is still asymptotically valid, but provides much less stable results compared to the case. Embrechts and al (1997) provides a related example involving an process.

The following equality holds for points in the tail of

(15)

If we estimates the first term, , using the random proportion of the data on the tail , and if we estimate the term , where is defined in eq (, by approximating the excess distribution with a GPD fitted by maximum likelihood, we get the tail estimator

,

For , let represent the ordred residuals. If we fix the number of the data in tha tail to be , this gives us a random threshold at the order statistic. The GPD with parameters ξ and β is fitted to the data, the excess amounts over the threshold for all residuals exceeding the threshold. The form of the tail estimator for is then

(16)

For we can invert this tail formula to get

the quantile of the data distribution (17)

In order to estimate risk measures, VaR for crude oil market, our main interest is on extreme value theory based models: we consider only the conditional GPD approach.

The peak over threshold: conditional GPD approach

Different approaches have proposed in the literature to estimate risk measures. The unconditional GPD represent the advantage is that it focuses attention directly on the tail of the distribution. However it doesn’t recognize the fact that returns are none. The econometric models of volatility dynamics such us -process under different innovation’s distributions yield VaR estimates which reflects the current volatility background. The weakness of this modeling approach is that focuses in the modeling of the hole conditional return distribution as time-varying, and not only on the part we are interested in, the tail. This approach may fail to estimate accurately risk measures like VaR and ES. In order to overcome drawbacks of each of the above methods, McNeil and Frey (2000) proposed to combine ideas from these two approaches. The advantage of this - combination lies in its ability to capture conditional heteroscedasticity in the time series through the framework. While, simultaneously, modeling the extreme tails shape through the method. Bali and Neftci (2003) apply the model to U.S short-term interest rates and show that the models yields best estimates of VaR than that obtained from a student-t distributed with models. Bystrom (2004), find similar conclusions indicating that performs better than the parametric models in forecasting VaR different international stock markets.

In the context of energy market, Bystrom (2005) apply to NordPool hourly electricity returns. Krehbiel and Adkins (2005) employ the some approach to the NYMEX energy complex. Marimoutou and al., (2009) apply this methodology to estimate risk measures in oil market. They find similar results confirming that conditional model produces more accurate estimates of extreme tails.

The combined approach, denoted conditional GPD, may be presented in the following three steps:

-Step 1: fit a -type model to the return data by the quasi-maximum likelihood. Estimate and from the fitted model and extract the residuals.

-Step 2: consider the standardized residuals computed in step 1 to be a realization of a white noise process, and estimate the tail of the innovations using the extreme value theory and then compute the quantiles of the innovations.

-step 3: construct VaR from parameters estimated in step 1 and 2.

Statistical accuracy of model-based VaR estimations

In order to back-test the accuracy for the estimated VaRs, we computed the empirical failure rates. By definition, the failure rate is the number of times returns (in absolute values) exceed the forecasted VaR. If the model is correctly specified, the failure rate should be equal to the specified VaR’s level. In this study, the backtesting VaR is based on the Kupiec’s (1995) and Engel and Manganelli (2004) for unconditional and conditional coverage tests.

The Kupiec’s (1995) (LR UC ) test

The main idea of the Kupiec’s (1995) is to estimate of the probability of observing a loss greater than the VaR’s amount. In order to test the accuracy and to evaluate the performance of the model-based VaR estimates, Kupiec (1995) provided a likelihood ratio test () for testing whether the failure rate of the model is statistically equal to the expected one (unconditional coverage).

Consider that is the number of exceptions in the simple size T. Then

(18)

Follows a binomial distribution, . If is the expected exception frequency (i.e. the expected ratio of violations), then the hypothesis for testing whether the failure rate of the model is equal to the expected one is expressed as follows: is the prescribed VaR level. Thus the appropriate likelihood ratio statistic in the presence of the null hypothesis is given by

(19)

Under the null hypothesis, has a as an asymptical distribution. Consequently, a preferred model of VaR prediction should display the property that the unconditional coverage measured by should equals the desired coverage level

Dynamic quantile test

Engel and Manganelli (2004) proposed a conditional coverage test by using a linear regression model based on the process of hit function:

Where { } is a centered process in the target probability. The dynamic of the hit function is modeled as:

(20)

Where is an process with mean of zero and is a function of past exceedances and of variable .

Under the hypothesis that the VaR estimation can deliver accurate VaR estimates and also the occurrence of consecutive exceedances is uncorrelated, the regressors should have no explanatory power. Hence the dynamic quantile test is defined as:

It’s easy to show that the dynamic quantile test statistic, in association with the Wald statistic, is:

(21)

Where denotes the covariates matrix in equation (19), and .

Fig.1. crude oil spot closing prices

Data description and preliminary analysis

data description

In this study we tend to estimate risk measures for energy market. For this aim, we consider closing daily spot prices of the Cushing West Intermediate crude oil (WTI). The closing price of energy commodity is provided by the Energy Information Administration (EIA). For the selected serie, the data covers the period (Mars, 2003-Mai 2011), totalling more than 2000 observations. The continuously compounded daily returns are computed as follows:

(22) Where and are the return in percent and the energy commodity closing price on day, respectivly. This data set is challenging to model as is characterized by large price increases and decreases that reflect a substantial rise in the volatility of real oil price (see Fattouh, 2005). We should mention that for the serie, under study, the dataset is subdivided into two subsets. The last 600 daily returns are reserved for the out- of-sample analysis. The first subset is used for the in-sample analysis.

Fig.2. crude oil daily returns

Some preliminary results

Descriptive statistics, unit root, stationarity and long range memory tests are reported in Table 1. From the panel A, we can see that energy commodity returns are skewed toward the left and they did not correspond to the normal distribution assumption. According to the Jarque-Béra (1980) test statistic, we can surely reject the null hypothesis of Gaussian distribution. Using the Ljung-Box statistic of order 10 based on the squared returns, we can also reject the hypothesis of white noise and assert that time serie is autocorrelated.

Descriptive graphs (daily price levels, daily returns, normal probability plots of daily returns and histogram of daily returns against normal distribution) for crude oil are reported in figs 1-3. WTI crude oil returns were extremely volatile around the 2007-2008 periods, which led to a succession of extremely large positive and negative returns within a very short time horizon. Volatility clustering is manifestly apparent for crude oil returns revealing the presence of heteroscedasticity. Moreover, the normal probability plots and the histograms against normal distribution graphs show that daily returns of WTI exhibit asymmetry and fat tails. Finally, before fitting serie, we employ some tests in order to check for the presence of unit roots and to test stationarity. In panel B we represent the results of the augmented Dickey-Fuller (1979) (ADF) and the Philips Peron (1988) (PP) unit roots tests and the Kwiatkowski, Phillips, Schmidt and Shin (1992) (KPSS) stationarity test. The ADF and PP tests undoubtedly reject hypothesis of unit root for the time serie studied. So, we can conclude that energy commodity price returns is governed by an process which have no long-range memory. Furthermore, the KPSS test’s results reveal that we cannot reject the stationarity null hypothesis at a 1% significant level for all the energy returns time serie.

Fig.3. Normal probability plots and histogram X normal distribution for crude oil daily returns.

table 1:

summary statistics, unit root, stationary test and long memory tests for daily log returns

 

 

 

 

 

WTI

 

panel A: descriptive statistics

 

 

 

 

mean

 

 

 

0,0608

 

maximum

 

 

 

16,4137

 

minimum

 

 

 

-12,8267

 

S.D

 

 

 

2,5519

 

skweness

 

 

 

-0.0076

 

(0.8879)

kurtosis (excess)

 

 

 

4.3764**

 

(0.0000)

J-B test

 

 

 

1612,7058**

 

(0.0000)

 

 

 

981,759**

 

 

 

 

 

(0.0000)

 

panel B: unit root and stationary tests

 

 

 

 

ADF test

 

 

 

-24,4858**

 

(0.0000)

PP test

 

 

 

-45,71223**

 

(0.0000)

KPSS test

 

 

 

0,08229**

 

 

 

 

 

(0.0463) 

 

Panel C: long-memory test statistics

 

 

 

 

return

 

 

 

 

 

Lo's R/S test

 

 

 

1,44791

 

(0.94761) 

squared return

 

 

 

 

Lo's R/S test

 

 

 

4,9465 **

 

(0.0000)

absolute return

 

 

 

 

 

Lo's R/S test

 

 

 

5,162**

 

(0.0000)

Note: S.D. is the standard deviation of WTI returns, the descriptive statistics for cash daily returns are expressed on percentage, J-B test is the Jarque- Béra (1980) normality test statistic, Q2(10) is the Ljung-Box Q-statistics of order 10 on the squared returns. ADF is the augmented Dickuy Fuller (1979) unit-root test statistics, PP is the Phillip-Peron (1988) unit-root test statistics, KPSS is the Kwiatkowski, Phillips, Schmidt and Shin (1992) stationary test statistic. P-values are given into brackets. ***, ** denotes significance at 1% and 5% level respectively.

Testing long memory in crude oil volatility

In this paper, we consider two proxies of the daily volatility squared return and absolute returns. To test long memory we use two different long-memory tests: Lo’s (1991) test and two semi-parametric estimators of long memory parameter, the log-periodogram regression (GPH) of Geweke and Porter-Hudak (1983) and the Gaussian semi-parametric (GSP) of Robinson (1995a). Empirical results are given in table 2. From these results we can conclude that crude oil returns does not display long-rang memory in their mean’s returns equation. Furthermore, Lo’s test (R/S) does not reject the null hypothesis of no long-range memory.

Table 2: GPH, GSP and R/V tests results

GPH (1983)

 

GSP (1995a) Robinson

 

Lo’s (1991) test

Squared return

 

Squared return

Squared return

 

0.6104

0.2221

 

3.0465

0.4923

0.3703

 

1.8241

0.2429

0.7425

 

1.2387

Absolute return

 

Absolute return

Absolute return

 

0.6889

0.2411

 

3.1924

0.5762

0.3978

 

1.8950

0.2186

 

0.7608

 

1.2783

Notes: and are, respectively, log return, squared log return and absolute log return. denotes the bandwidth for the Geweke and Porter-Hudak’s(1983) and the GSP Robinson (1995a) tests.

In fact we fail to reject the long-range memory in the return volatility of crude oil at the 5% significance level since the evaluated statistic are over the critical value. Moreover the GPH and GSP tests statistics rejects the hypothesis of long-range memory for a significance level of 5% for squared and absolute returns. Thus, crude oil volatility seems to be will described by a fractionally integrated process. Finally, we can suppose that clustering volatility, fat tails, asymmetry and long-range memory characteristics could be captured by an accurate model describing the dynamic behavior of energy commodity.

Empirical results

Estimating -type models

Tables 3 represent the estimation results of the , and models. Results in panel (table 3) reveal that models are able to capture the long-range memory phenomenon for crude oil return volatility. Therefore all the long-memory parameters strongly reject the null hypothesis at a 1% significance level. Turning to the goodness-of-fit tests, our results indicate that we cannot reject the null hypothesis of correct model specification since the Box-Pierce test statistics computed with 10 lags for both standardized residuals and squared standardized residuals show no serial correlation and no remaining effect.

Table 3: estimation results and diagnostic tests for, and models

 

Panel A: estimation results

 

 

 

 

 

0.1348***

0.1259***

0.0903*

 

 

(2.9930)

(2.7140)

(1.8980)

 

0.1653***

0.0512***

0.1909***

 

(2.8150)

(2.6870)

(3.8550)

 

0.3836***

1.0466***

0.4025***

 

(6.1020)

(23.09)

(6.6400)

 

0.7123***

0.9638***

0.7081***

 

(12.6600)

(80.0400)

(14.12)

 

0.4689***

0.0512

0.4516***

 

(7.7830)

(1.1810)

(8.809)

 

-0.010***

 

 

 

(-2.4420)

 

 

 

0.3418***

 

 

(4.5920)

 

 

1.5346***

 

 

(8.484)

log-likelihood

 

-4489.9400

-4482.05

-4482.13

Panel B: diagnostic tests

 

 

 

 

4.2178**

3.3510**

5.5851**

 

(0.8370)

(0.9104)

(0.6936)

 

0.43279**

0.33818**

0.57115**

 

(0.9311)

(0.9708)

(0.8386)

AIC

 

4.4770

4.4702

4.4712

SIC

 

4.4910

4.4869

4.4908

Notes: (1) is the value of the maximized log-likelihood, t-values are reported in brackets. and are the box-Pierce statistics for remaining serial correlation for respectively standardized and squared standardized returns with p-values in brackets. AIC and SIC are the Akaike (1974) and Shibata information criterion respectively. ***, **, * denotes significantly at the 10%, 5% and 1% respectively.

Table (3) reports the estimation results of the model. The hyperbolic parameters are not significantly different from zero for crude oil price returns indicating that the components are covariance stationary. For the goodness-of-fit test, the diagnostic results reveal that the model is suitable to depict the heteroscedasticity exhibited in the time series since the Box-Pierce and the effect tests do not reject the null hypothesis of a correct model specification.

Concerning the estimation results (see table 3), crude oil display strong evidence of volatility asymmetry since the () parameters is statistically significant. The coefficient of asymmetric response of volatility to news () is significant and positive. As in the previous models, the diagnostic results show the power of the model to take into account the major stylized facts of time series prices behavior.

Overall comparing different type models, en terms of the AIC and SIC information criterion, we can conclude that the model out performs the and models for the serie studied. This model is more appropriate to capture both long-range memory and clustering volatility in the time series behavior. On the other hand, it’s shown that residual series is found to be free from autocorrelation. We can conclude that the filtering process successfully removes the time series dynamics from the return series and obtain an series free from any time series dynamics. Therefore, we can apply successfully methods to the residual series. Obviously, in what follow we choose the approach to compute the one-day-ahead VaRs for both energy market. In our study we choose the as a benchmark approach.

Table 4: in-sample VaR estimations calculated by the appraoch

Quantile

 

Kupiec test

Engle & Manganelli test

Short position

 

p-value

 

p-value

0.95

 

 

0.2285

0.6327

 

2.2667

0.8936

0.975

 

 

0.1674

0.6825

 

6.103

0.4118

0.99

 

 

2.8293

0.0926

 

4.3069

0.6352

0.995

 

 

1.9793

0.1595

 

3.1254

0.793

0.9975

 

 

0.9876

0.3203

 

1.5469

0.9563

long position

 

 

 

 

 

 

0.05

 

 

0.0566

0.8119

 

6.5874

0.3606

0.025

 

 

0.3457

0.5565

 

4.9080

0.5556

0.01

 

 

1.1874

0.2758

 

1.8900

0.9295

0.005

 

 

0.0342

0.8531

 

0.2046

0.9998

0.0025

 

 

1.1434

0.2849

 

0.9704

0.9867

Table 5: out-of-sample VaR estimation calculated by the approach

Quantile

 

Kupiec test

Engle & Manganelli test

short position

 

p-value

 

p-value

0,95

 

 

1,1152

0,2909

 

4,8590

0,5620

0,975

 

 

0,4141

0,5198

 

13,3620

0,0376

0,99

 

 

1,1539

0,2827

 

15,6280

0,0158

0,995

 

 

0,9702

0,3246

 

1,3920

0,9663

0,9975

 

 

0,2117

0,6454

 

0,2462

0,9997

Long position

 

 

 

 

 

 

0,05

 

 

0,0662

0,7969

 

15,3930

0,0174

0,025

 

 

0,0081

0,9282

 

4,9481

0,5504

0,01

 

 

0,0127

0,9102

 

6,2204

0,3989

0,005

 

 

0,9702

0,3246

 

83,7310

0,0000

0,0025

 

 

0,0073

0,9320

 

155,8300

0,0000

Fig 4. WTI return and its VaRs at the 99% confidence level.

In-sample VaR estimations

In this sub-section, we estimate the one-day-ahead VaRs via the and approaches. As shown in table 4, the backtesting VaR results reveal that the approach performs very well for the one day time horizon. More specifically, for the short trading position, the p-values associated to the Kupiec’s (1995) and Engel and Manganelli (2002) tests show that we are unable to reject the null hypothesis for all the significance levels. Our empirical results show quite similar findings for the long trading position. With reference to the as a benchmark approach we perceive that the and models combined with distribution provides very similar results. Over all, we can conclude that model combined with extreme value theory does very well in predicting critical loss for crude oil market. Our findings (see table 4) reveals that models considering for some stylized facts such that volatility clustering, long range memory and leptokurtosis in the time series behavior enhances the VaR predicting for both high and low confidence levels and for the short and the long trading position. Figure 4 Reports more details about the backtesting exercise to the conditional GPD approach. In fact, figure 4 display a part of the backtest for WTI returns. We have plotted, in this figure, the returns, superimposed on this plot is the 99% VaR estimates for both long and short position, to show the violation of the approach. It’s clear that the model responds rapidly to the high volatility around the 2008-2009 petroleum choc. In this thousand-observation window conditional cannot be rejected for either tails at standard confidence levels.

Out-of-sample VaR estimations

In this sub-section, we are interested to evaluate the performance of the selected model combined with extreme value theory by computing the out-of-sample forecasts. The models are re-estimated every 50 observations in the out-of-sample period. The out-of-sample VaR’s predictions are provided in table 5.

The obtained results (see table 5) reveal that the performs well for the out-of-samlpe forecasts for energy market and under all the confidence levels. Unless, the out-of-samlpe results are not as good as the in-samlpe predictions. This finding corresponds to the statement that type models produce accurate in-sample estimates but less satisfactory out-of-sample forecasts (Bollerslev etal., 1992; Figlewski, 1997; Poon and Granger, 2003). In facts, for both long and short trading position, the unconditional coverage’s test p-values show that we cannot reject the null hypothesis for all confidence levels, the out-of-sample results are similar to those of the in-sample VaR analysis. The conditional coverage’s test results show that we can reject the null hypothesis for the short trading position under the 97.5% and the 99% confidence levels, and under the 0.05%, 0.005% and 0.0025% confidence levels for long trading position. In addition, we remark that the in-samlpe p-values are greater than those of the out-of-sample forecasts. Furthermore, the method yields accurate VaR quantification which account for volatility dynamics. In facts, this model performs well for the crude oil market and for various confidence levels considered. The current findings also show that conditional GPD model performs well, especially for markets where the distributions of returns exhibits extreme moments.

Conclusion

As the volatility in the energy market increases, it’s extremely important to implement an effective risk management system against market risk. In this context, VaR has become the most popular tool to measure risk for institutions and regulators. In addition extreme value theory has been successfully applied in many fields where extreme values can appear. In this paper we introduce an extension of the McNeil and Frey (2000) approach based on the method. We extend the model to the fractional integrated models to take into account major stylized facts in to the price return volatilities of energy market. Our findings reveal that energy market is characterized by asymmetry, fat-tail and long range memory. In addition, oil market volatility exhibit strong evidence of long-range memory, which can be well captured by the model. In terms of predictive accuracy and referring to several statistical criteria, our results show that the outperform the and the models for the energy market. The model combined with extreme value theory performs better in predicting the one-day-ahead VaRs. The in-sample results are marginally better than those of the out-of-sample period. Overall, our findings confirm that taking into account long-range memory clustering volatility and fat tails in the behavior of time series, combined with filtering process such as are important in improving risk management assessments and hedging strategies.