1. Introduction
Energy, metal and agricultural commodities are very relevant in everyday life for a number of reasons. To start with, they play a crucial role in the global economy, since changes in their prices (and their volatility) have an impact on consumers, producers and the general economic activity. This is particularly true for oil, given that its rising prices generate not only effects on the demand and supply side of the economy, but also on financial markets. For instance,
Sadorsky (
1999) showed that shocks to oil prices tend to depress real stock returns. More recently,
Algieri and Leccadito (
2017) documented contagion effects from commodity markets to the rest of the economy. Specifically, they demonstrated that adverse shocks hitting one or more commodity markets tend to spread to the entire economic system as a consequence of the financialisation and integration processes of commodity markets.
Similarly, the intensified price interlinkages between energy and agricultural prices have raised concerns about deeper volatility connections and likely adverse effects on economic actors. More volatile agricultural prices can lead to sub-optimal productions, make poor consumers in countries dependent on grains more vulnerable to price swings, with the consequent increase in social and political tensions (
Mensi et al. 2017;
Tadesse et al. 2014).
Another reason for the importance of energy and non-energy commodities stems from their increasing role in portfolio strategies over the last two decades (
Bhardwaj et al. 2016;
Tang and Xiong 2012). Indeed, commodities are perceived as an alternative instrument to equities and fixed income securities, due to their low correlation with those two asset classes, and, as such, offer an interesting diversification opportunity and, in some cases, represent a ‘safe haven’. For instance, gold and oil have emerged as an important hedging tool to diversify market risk and precious metals have been widely used in risk management strategies. However, the stronger price volatility interlinkages across commodities could weaken the diversification benefits of holding commodities in a portfolio (
Kumar et al. 2019).
For the aforementioned reasons, it is crucial to identify appropriate statistical models to quantify and measure the risks associated with commodities. Over the recent years, Value-at-Risk (VaR) and Expected Shortfall (ES) have been adopted in the literature as risk measures especially for the examination of the risks linked to equities, interest rates and foreign exchanges. Commodity price risk evaluation has received less attention compared to other asset classes.
In our study, we do not examine tail dependence across commodity pairs, but assess commodity market risk for single commodity over a short-term time horizon by considering the history of commodity returns and enclosing the typical features of their time series, such as fat tails. In this respect, we focus on purely statistical models to describe the evolution of commodity returns without relying on economic models that aim to forecast commodity prices considering supply and demand imbalances (
Giot and Laurent 2003). Our approach allows the characterisation of risks for single commodity useful to disentangle specific commodity risk patterns.
More in detail, to capture the behaviour of the tails for the commodity return distributions, we adopt the Extreme Value Theory (EVT) and, similar to
Taylor and Yu (
2016), combine it with models that assume tail probabilities evolving over time as an auto-regressive process (the Conditional Auto-Regressive Logit models (CARL) models). According to the Extreme Value Theory, the tail of a distribution can be, at least approximately, described by the so-called Generalised Pareto Distribution (GPD) (
Pickands 1975). We follow the Peaks-Over-Threshold (POT) method and fit the GPD to losses larger than a given threshold. The POT method allows us to identify the magnitude of threshold exceedances. Then, we consider a time-varying scale parameter of the GPD which follows an auto-regressive process in order to obtain a Time Varying POT (TVPOT). Finally, tail probabilities entering the TVPOT method are made time varying by means of CARL models. The latter methods explain tail probabilities (constructed as twice the logit of probabilities) as a function of past extreme events, past returns and lagged values of some explanatory variables. CARL family models were introduced by
Taylor and Yu (
2016) as an improvement of older models including those by
Rydberg and Shephard (
2003) and
Anatolyev and Gospodinov (
2010). In the context of energy markets, CARL models have been shown to have a a better performance in forecasting tail probabilities than alternative models such as Historical Simulation, GARCH and Quantile-Augmented Volatility models (
Algieri and Leccadito 2019).
In our analysis, we use the TVPOT methodology to forecast Value-at-Risk (VaR) and Expected Shortfall (ES) for four commodities of different type, namely WTI crude oil, natural gas, gold and corn, and backtest the latter measure using recently proposed testing procedures.
Despite not being coherent (a risk measure is said to be coherent if it satisfies a number of desirable mathematical properties (see
Artzner et al. 1999)), Value-at-Risk, defined as a quantile of the profit and loss distribution, is the most popular risk measure among practitioners and regulators due to its simplicity (for instance, the capital requirement prescribed by Solvency II is the 0.5% VaR of the annual loss distribution). VaR, however, is insensitive to the severity of tail losses, by its definition of a quantile. Recent regulatory accords have thus shifted emphasis to Expected Shortfall, which is defined as the expected value of the profit and loss given that VaR has been exceeded. Basel III indeed prescribes 10-day 2.5%-ES as the risk measure on which capital requirements are to be based. Furthermore, ES has been proven to be a coherent risk measure (
Acerbi and Tasche 2002).
Thus, in our study, we assess the accuracy of ES predictions for the proposed models using the multinomial tests introduced by
Kratz et al. (
2018). The latter category of testing procedures, while designed to backtest ES, are based on the idea that ES is successfully backtested if a number of VaRs at properly chosen coverage probabilities has been simultaneously successfully backtested and, hence, they only require the sequences of VaR violations (the so-called hit sequences) at the different coverage thresholds.
The remainder of the paper is organised as follows.
Section 2 presents the POT methodology and CARL models.
Section 3 gives the results of the empirical application on WTI crude oil, natural gas, gold and corn futures.
Section 4 concludes.
3. Empirical Application
Following
Taylor and Yu (
2016), we combine the methods presented in
Section 2.1 and
Section 2.2 to predict VaR and ES risk measures. In particular, given an estimation window of length
T, we consider out-of-sample VaR forecasts for time
using Equation (
3) with the estimated parameters
based on the window and the forecast for the tail probability
The latter is obtained using the estimated parameter vector
, again based on the window of
T observations. The accuracy of the prediction is established via the recent tests of
Kratz et al. (
2018).
3.1. Data Description
To model tail risks and predict the time varying probability that commodity returns would exceed a given threshold, we compute daily log returns for WTI crude oil, gold, natural gas and corn by considering their first generic futures contracts extracted from Bloomberg (tickers CL1, GC1, HO1, and C1, respectively). The investigation covers the period 2 December 2003 to 27 October 2017, for a total of 3500 daily observations. Time-series plots for futures prices of the four commodities are given in
Figure 1. It is interesting to notice that, while corn and energy commodity prices registered severe falls during the global financial crisis of 2008, gold was the only commodity to act in a counter-cyclical manner. This behaviour testifies the hedging feature of this precious metal.
Daily returns for each commodity
i are computed as log differences in futures price (
):
Table 1 reports the descriptive statistics for daily futures returns. From the table, it is clear that the commodity with highest volatility is natural gas. Indeed, the standard deviations suggest that this energy commodity records the highest dispersion around the mean in comparison to other commodities, while gold shows the lowest volatility, thus confirming its characteristic of safe-haven. In addition, the average returns are higher for gold and WTI crude oil, while are negative for natural gas. The largest daily loss on commodity futures is 26.86 per cent for corn, followed by natural gas, crude oil and gold.
Natural gas displays the highest positive skewness, which implies frequent small losses and a few extreme gains compared to the other commodities. Gold and corn returns are negatively skewed, meaning frequent small gains and a few extreme losses. All series are not distributed as a bell-shaped Gaussian, since the Jarque–Bera test displays a p-values of zero and there is an excess kurtosis especially for corn. The results of the Augmented Dickey–Fuller unit root test suggest that all return are stationary given that the null hypothesis of unit root is always rejected.
Table 2 presents the correlation coefficients of the four commodity returns. It can be noticed that the correlation across commodities is always positive and it never overcomes 25 per cent. This means that there are not significant relationships among commodities, and they can be used to diversify investment portfolios.
In our analysis, commodity log returns represent our dependent variable and commodity implied volatility is used as explanatory variable in some variants of the adopted CARL models.
In particular, the implied volatility is calculated from prices of at-the-money (call) options expiring at least twenty business days from the observation date. This metrics is considered a forward-looking indicator for future return volatility. We consider the inverse of the implied volatility in line with
Algieri and Leccadito (
2019).
To assess how accurately our predictive model performs, we build out-of sample predictions. Specifically, we recursively estimate the parameters using a rolling window of observations to obtain a new one-period-ahead forecast. We use 2500 rolling windows each of length 1000. This implies that the first prediction is made for the end of November 2007.
Figure 2 plots the (negative of the) returns over an evaluation period of about 10 years. We select the threshold
Q to the 85% unconditional quantile of the losses
. This choice guarantees that the exceedance probability is larger than
for each in-sample period with an exceedance. In particular,
Figure 2 shows that the estimated measures of risk probability tend to move in the same fashion of returns. The VaR measures lay always below the ES measures by construction.
3.2. ES Testing Procedures
To evaluate the accuracy of the considered models to predict ES, we implement the multinomial test of
Kratz et al. (
2018). The testing procedure is based on a formula that approximates the
-ES as the mean of a number of quantiles associated to equally spaced probability levels, with the largest one equal to
. Explicitly:
The idea is therefore that
is successfully backtested if the VaRs with levels
with
are deemed reliable since they have been simultaneously backtested. Assume
K critical thresholds so that
and
; we have, by convention,
,
,
. Violations are defined for
via the exception indicator series (
denotes the indicator function taking on value 1 if
A is true and 0 otherwise.)
Let
be the total number of VaR violations in period
at the different thresholds. Denote by
the probability of falling in between the VaR associated to
and
, with
. Assuming a number of observations equal to
T, the number of observations falling between
and
are defined as
. If both the unconditional coverage hypothesis—i.e., that
,
, for all
t—and the independence hypothesis—i.e.,
is independent of
for
—hold, then the random vector
follows the multinomial distribution,
.
Kratz et al. (
2018) proposed several procedures to test the above hypotheses. The Pearson type of test is defined as:
and under the null it has a chi-square distribution with
K degrees of freedom. A correction to the above test is given by the Nass test defined as
where
The null distribution of the Nass test statistic is chi-square with
degrees of freedom. Finally,
Kratz et al. (
2018) considered the Likelihood-Ratio Test (LRT) defined as:
which, under the null, is chi-squared distributed with
K degrees of freedom.
Results of the above testing procedures for
and
are reported in
Table 3 and
Table 4, respectively. In detail, the tables show the estimated p-values for the Indicator and Absolute CARL models and their augmented version with the implied volatility used as explanatory variable. For a comparison with alternative models,
Table 3 and
Table 4 further report the p-values obtained using a Normal GARCH (1,1), the Historical Simulation (HS), the Filtered Historical Simulation (FHS) methods (the FHS method we consider involves modeling the variance) and the Quantile-Augmented Volatility (QAV) model (see
Han 2016 and Appendix A of
Algieri and Leccadito 2019 for details on QAV models) proposed by
Han (
2016).
According to all the procedures, the best model to forecast 5%-ES is the augmented CARL-Abs-X model for crude oil, the augmented CARL-Ind-X models for gold and natural gas and the simple CARL-Ind model for corn. Conventional econometric models based on GARCH-methods are less capable to model and predict extreme commodity returns showing relevant time-varying volatility. This finding is consistent with most of the previous conclusions that commodity markets tend to suffer from extreme shocks (
Ji and Fan 2012;
Pindyck and Rotemberg 1990), and that there is some heterogeneity between oil and gas markets that indicates that the two commodities are not perfect substitutes (
Ji et al. 2018) and, therefore, different prediction models can be necessary to forecast their tail risks. In addition, the results confirm that implied volatility linked to products can be considered a important variable to improve forecasting models and thereby facilitate risk management (
Algieri and Leccadito 2017;
Mensi et al. 2017).
The threshold
Q plays a crucial role in the TVPOT-CARL models (and EVT in general). Indeed, there is a trade-off in the choice of
Q. If the threshold is too large, then only few observations are left in the tail and it becomes difficult to accurately estimate the parameters of the GPD. However, choosing
Q too small on the one hand increases the number of exceedances and, hence, the number of observations used when estimating the GPD parameters, but on the other hand it implies that the focus is no longer (only) on the tail of the distribution and the GPD may not be used to describe exceedances. As a robustness check, we consider alternative values for
Q. In particular,
Table 5 and
Table 6 give the results of the testing procedures to assess the accuracy of
-ES predicted using the TVPOT-CARL models when
Q is equal to the 83% and 87% unconditional quantile of the losses, respectively. Overall, the findings presented in
Table 5 and
Table 6 confirm the outcomes registered for the 85% quantile.
In brief, the adoption of CARL with the POT methods could be considered an important technical tool to regularly monitor and minimise the market exposure to price risks and, at the same time, build optimal strategies that maximise investors’ profitability given a certain acceptable amount of risk.
4. Conclusions
Commodity markets are often characterised by high uncertainty and pronounced price volatility. These dynamics and the consequent hazy market conditions make necessary an examination of the most appropriate methodologies and tools to predict extreme price risks. To this purpose, we considered a novel class of models to measure and forecast tail events for WTI crude oil, natural gas, gold and corn over the years 2007–2017.
In particular, we used the Time Varying Peaks-Over-Threshold (TVPOT) method combined with a set of Conditional Auto-Regressive Logit (CARL) models and compared them with alternative benchmark models. After predicting the Value-at-Risk and Expected Shortfall risk measures specific to each commodity, we backtested 5%-ES to establish which model is accurate in measuring market risk.
We found that conventional econometric models based on GARCH, HS, FHS and QAV methods are less able to characterise the dynamics of commodity returns and forecast tail events and, hence, Expected Shortfall. Conversely, the TVPOT methodology in combination with CARL models and their augmented versions, which include time-varying implied volatility, generally allows more accurate risk predictions. The results further highlight that different commodities appear to work better with different variants of CARL-models. In particular, better forecasts are obtained using the augmented CARL-Abs-X model for crude oil, the augmented CARL-Ind-X models for gold and natural gas and the simple CARL-Ind model for corn. Some heterogeneity between oil and gas markets would confirm that the two commodities are not perfect substitutes and, therefore, different prediction models are necessary to forecast their tail risks.
The adoption of CARL within the POT framework could be useful for portfolio managers, risk arbitrageurs and traders who closely monitor commodity market trends and need to have a better knowledge on possible changes in prices for their investments and risk management decisions. At the same time, CARL and POT models would be useful to policy makers to predict in advance risks and therefore protect the poor from the severe impact of high food and energy prices. This is because the poorest segments of society are more exposed to increases in commodity prices given the large share especially of food and in their overall consumption basket.