Value at Risk (VaR)

^{Jorion (2001)} defines Value at Risk as the maximum expected loss in a given time horizon for a given confidence level.

Definition. Given X, a continuous random variable defined over the sample space Ω, which represents the change in the value of the asset (yield). Assuming that X : Ω → ℝ is defined over a fixed probability space (Ω, A, P), then the Value at Risk of X at level 1-q is defined as the minimum of the upper dimensions for a confidence interval of (1-q) %, such that:

This definition indicates that it is possible to obtain the VaR given the function of cumulative distribution of asset yields:

where FX-1(·) is the inverse of the function of cumulative distribution of asset yields over a period. In other words, VaR is the q-quantile of F _X. Therefore, the essence of the VaR calculations is the estimation of the lower quantiles of the cumulative distribution function of asset returns, which in practice, is unknown.

VaR estimation methods suggest different ways of constructing this function. The most common are: the parametric method based on the assumption that a parametric distribution characterizes changes in the value of the portfolio; the historical simulation where it is not necessary to assume a specific distribution of yields, and instead the predictions of its behavior are inferred using the historical behavior of the data; and the Monte Carlo simulation in which approximations of the expected yield behavior of a portfolio or financial instrument are obtained through simulations that generate random trajectories of the yields of the portfolio or financial instrument, considering certain initial assumptions on the volatilities and correlations of the risk factors.

In this work, the VaR estimation is carried out through Monte Carlo simulation, under the stable hypothesis, generalized asymmetric Student t, and normal Student t.

α-stable distribution and Generalized Asymmetric Student t-distribution

In risk management, it is essential to find a distribution that appropriately describes the financial data. Commonly financial returns are not adequately described by a normal distribution, as empirical results show that financial data are generally asymmetric and have heavy tails (^{Fama, 1965}; ^{Bollerslev, 1986}; ^{Bali and Theodossiou, 2007}; ^{Champagnat et al., 2013}). However, this characteristic is not exclusive to financial assets; the energy yield series also share these characteristics (^{Giot and Laurent, 2003}; ^{Hang et al., 2007}; ^{Fan et al., 2008}).

Currently, most studies estimating the volatility of oil yields do so under the Gaussian hypothesis (^{Sadeghu and Shavvalpour, 2006}). The existing literature regarding the estimation of volatility in the energy sector applying alternative distributions to the normal distribution is very scarce (^{Giot and Laurent, 2003}; ^{Hang et al., 2007}; ^{Fan et al., 2008}; ^{Hung et al., 2008}; ^{Marimoutou et al., 2009}; ^{Aloui and Mabrouk, 2010}; ^{Cheng and Hung, 2011}; ^{Youssef et al., 2015}).

This work presents two families of distributions: stable distribution and generalized asymmetric Student t-distribution (GST), which allow describing the characteristics of asymmetry and yield kurtosis in the oil sector.

Generalized asymmetric Student t-distribution

There are several parameterizations of the generalized asymmetric Student t-distribution proposed in previous researches. Due to its simplicity, this study follows the parameterization proposed by ^{Hansen (1994)}, which suggests an alternative parametric approach to model the conditional density function of the normalized error. It consists of selecting a distribution that depends on a vector of parameters of few dimensions and allowing this vector to vary according to the conditional variables.

Definition. The generalized asymmetric Student t-distribution (GST) is the generalization of the Student t-distribution, which considers asymmetry. The probability density function of the standard GST distribution is defined as:

^{Hansen (1994)} demonstrates that this is an appropriate density function with a mean of 0 and a variance of 1. Parameter η controls the thickness of the tail, and λ controls asymmetry. When η→∞, the distribution is reduced to asymmetrical normal distribution; when λ=0, it is reduced to Student t-distribution.

If a random Z variable follows a standard GST distribution with parameters η and λ, it is denoted as Z◻GST(η, λ). If the random variable Z follows a non-standard GST distribution with mean μ and variance σ², it is denoted as Z◻GST(μ, σ, η, λ).

α-stable distribution

The family of α-stable or simply stable distributions is a class of probability distributions that allow asymmetry and heavy tails. In 1920, French mathematician Paul Lévy characterized this class of distributions in his study of sums of identically distributed independent terms. Stable distributions do not have explicit analytical expressions for either the probability density function (PDF) or the cumulative distribution function (CDF); however, their characteristic function (CF) describes them.

Definition. An α-stable random variable X is commonly described by its characteristic function (CF), which is defined as:

where, sgn(t)= {1 si t > 0,0 si t = 0, -1 si t < 0}, is the stability index or characteristic exponent that reflects the size of the tails of the distribution; -1 ≤ β ≤ 1 is the asymmetry parameter that indicates the symmetry of the distribution; γ ≥ 0 is a scale parameter also called dispersion; and δ ∈ R is the position parameter.

If a random Z variable follows a stable distribution it is denoted as Z◻S(α, β, γ, δ); thus a standard α-stable random variable is denoted as Z◻S(α, β, 1, 0).

GARCH models

Volatility is the primary variable on which economic and financial pricing and hedging models are developed, so making estimates with the correct specifications of conditional distribution is crucial to improving their efficiency. This work implements GARCH(1,1) under the stable hypothesis, GST, and normal to describe the volatility of oil yields. ^{Sadorsky (2006)} states that the GARCH model (1.1) has an excellent performance in estimating volatility in the oil sector.

Stable GARCH(1,1) model

The stable distribution was used to describe the empirical characteristics of oil yields, such as heavy tails, asymmetry, and volatility clusters. Moreover, it was used to describe the innovation process in the GARCH(1,1) model. Because in the case of stable distribution not all moments are defined, the TS-GARCH model proposed by ^{Taylor (1986)} and ^{Schwert (1989)} was used, in which yields are modeled as follows:

where R _t is the series of yields of the action in time t; δ _t and γ _t are the position and z _t conditional dispersion parameters, respectively; and are standardized stable random variables identical and independently distributed, z _{t ~ S(α, β, 1, 0) 1 < α < 2.}

The parameters are estimated by maximum likelihood, using the STABLE program described in ^{Nolan (1997)}.

GARCH(1,1)-GST model

Similarly, the GST distribution was used on the GARCH model to describe the stylized facts characteristic of the oil yield series. The model is below:

To estimate the parameters of the GARCH model, the maximum likelihood estimation (MLE) method was used, where the probability density function of z _t was approximated using the approach proposed by ^{Hansen (1994)}.

Normal GARCH(1,1) model

In this model, share yields are modeled as follows:

Evaluation of the VaR performance

In this research, the evaluation of VaR performance is done in terms of its probability of empirical coverage, using ^{Kupiec’s statistic (1995)}. This statistic is an unconditional test that counts the number of VaR violations over the entire period, estimating whether the expected proportion of violations is equal to the level of significance α.

Kupiec’s statistical test for large samples is distributed as a Chi-square with a degree of freedom and is given by:

where T represents the size of the sample, n is the number of violations, and p=n/T is the percentage of violations. The null hypothesis, H ₀ : n/T = α, is rejected with a significance level of 1% if the value of Kupiec’s statistical test exceeds or is equal to the critical value of a Chi-square distribution with a degree of freedom, i.e., LRUC ≥ 6.635.

Description of data and statistical analysis

This paper uses the daily closing prices of West Texas Intermediate (WTI), North Sea (Brent), and the Mexican export oil blend (MME), the sample covers the period from January 2, 2013, to December 29, 2017, obtaining a total of 1275 observations and the reference currency is the U.S. dollar. The price series were obtained from Bloomberg in the case of the Mexican oil mixture, and from the Energy Information Administration (EIA) website in the case of WTI and Brent.

These three oil mixtures were selected because, in the case of Brent, its price is the benchmark in European markets; WTI is the most accepted world reference price of a barrel of oil; and MME is the benchmark in the Mexican market.

The sample period was selected for two reasons: first, to analyze the VaR performance under the different distribution functions considered in this work in periods of high volatility, and second, because there is no reference to the VaR measurement in this specific period for this type of crude oil under the stability hypothesis. Additionally, there was a severe plunge in crude oil prices during this period, which experienced the third largest semi-annual depreciation in the last 24 years. The World Bank (Global Economic Prospects January 2015) identified four reasons for the 2014-2015 fall in oil prices by pointing to the first three factors as dominant: 1) oversupply at a time of weakening demand, 2) change in OPEC objectives, 3) reduced concern regarding geopolitical supply disruptions, and 4) appreciation of the US dollar.

Logarithmic yields were estimated as follows: Rt=100ln⁡PtPt-1 where Pt is the price in day t. Figure 1 shows the graph of the behavior of the daily closing oil prices and their respective logarithmic yields. In this figure, it is possible to observe that the series of logarithmic yields show heteroscedasticity and volatility clusters.

Source: own elaboration

Figure 1 Daily prices and logarithmic yields 

Table 1 presents the descriptive statistics of the yields. Their normality is contrasted, and the unit root, stationarity; and ARCH effects tests are presented. It is possible to observe that the means of the three yields are small, negative, and show similar values. Conversely, the corresponding standard deviation is high in comparison to the mean, and the kurtosis indicates a leptokurtic behavior of the series. The statisticians ^{Jarque-Bera (1980)}, ^{Shapiro-Wilks (1965)}, and Anderson-Darling reject the hypothesis of normality in the series, and the unit root tests ADF (^{Dickey-Fuller, 1979}) and PP (^{Phillips-Perron, 1988}) reject the unit root hypothesis for the time series studied, meaning that the series are stationary. Furthermore, the results of the KPSS test (^{Kwiatkowski, Phillips, Schmidt, and Shin, 1992}) reveal that it is not possible to reject the null hypothesis of stationarity with a deterministic trend at a significant level of 1%, i.e., all series present stationarity in trend. Finally, the ARCH-LM test rejects the hypothesis of the non-existence of ARCH effects (^{Engle, 1982}), i.e., the series of studied yields show ARCH effects.

Goodness of fit tests

In order to compare the goodness of fit of the alternative distributions considered in this document the Kolmogorov-Smirnov (KS) test is considered, the null hypothesis of which is H0: The data analyzed follow the distribution indicated. The null hypothesis is not rejected if the p-value exceeds or is equal to the chosen level of significance.

Table 2 shows the contrast statistic values of the KS test and its respective p-value, which indicate a non-rejection of the null hypothesis with a level of significance of α=.01, except for the MME series in the case of the GST distribution.

Estimation of the parameters of the probability distributions

The parameters of the alternative distributions considered were estimated by the maximum likelihood estimation (MLE) method. Table 3 shows the values obtained.

Estimation of the GARCH models

To describe the empirical characteristics of oil yields such as heavy tails, asymmetry, and volatility clusters, the GARCH(1,1) model was implemented under the stable, GST, and normal hypothesis. MLE estimated the parameters of the GARCH models, which are significant at 1%. Table 4 displays these estimates.

Table 5 displays the values of the ARCH effects test on the standardized residuals. The results show the absence of conditional heteroscedasticity in the series of standardized residuals for each of the respective GARCH models.

It is concluded from Tables 4 and 5 that the GARCH models analyzed adequately describe the clusters of conditional volatility, an empirical characteristic of oil yields, in addition to the fact that parameters a ₁ and b ₁ satisfy the condition of seasonality and indicate a high degree of persistence of volatility in yields.

VaR estimations

As previously mentioned, in the VaR estimation, the choice of the appropriate distribution of the innovation process is crucial since it directly impacts the quality of the estimation of the quantiles required to estimate the risk. Besides, the assumption on the distribution in the GARCH model is also fundamental in the VaR forecast, since the required likelihood functions to estimate the parameters are constructed based on this assumption and the future distribution of the risk is determined conditioned to the predicted volatility.

This document estimates one-day VaR by considering three alternative distributions in the innovation process: stable, generalized asymmetric Student t, and normal. Table 6 shows that the estimates of VaR-stable are higher than the estimates of VaR-normal and VaR-GST. It is also important to note that the estimates of VaR-normal at 95% confidence are higher than those of VaR-GST, whereas at 99% confidence these estimations show an opposite behavior.

Evaluation of VaR performance

The evaluation of VaR predictive performance under GARCH-stable, GARCH-GST, and GARCH-normal out-of-sample models is performed using historical data from the last year of the sample to predict the current VaR. Table 7 shows the results of Kupiec’s statistical test, whose null hypothesis, H ₀ : n/T = α, is rejected with a level of significance of α=1% if the value of the statistic exceeds or is equal to the critical value of a Chi-square distribution with a degree of freedom, that is, LRUC ≥ 6.635. Values in bold indicate the model with the best performance for estimating VaR according to Kupiec’s statistical test.

Table 7 presents the following results: