An Analysis of Value at Risk Methods for U.S. Energy Futures Robert Trevor Samuel III * 2 October 2012 † Abstract We estimate Value-at-Risk (VaR) statistics using parametric, non-parametric, and Extreme Value Theory (EVT) based techniques on the logarithmic price changes for continuous futures prices of Crude Oil, Natural Gas and Heating Oil from the New York Mercantile Exchange (NYMEX). Our results illustrate that the VaR confidence level, α, matters along with the amount of data used (’window size’) but overall finds poor results for all five methods of VaR tested with some positive results for specific parameterizations of a few methods. * Master’s Candidate, Clemson University. Correspondence: [email protected]† revised; first edition: 7 September 2012; second edition: 28 September 2012
21
Embed
An Analysis of Value at Risk Methods for U.S. Energy Futures
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
An Analysis of Value at Risk Methods for U.S. Energy
Futures
Robert Trevor Samuel III∗
2 October 2012†
Abstract
We estimate Value-at-Risk (VaR) statistics using parametric, non-parametric, and
Extreme Value Theory (EVT) based techniques on the logarithmic price changes for
continuous futures prices of Crude Oil, Natural Gas and Heating Oil from the New
York Mercantile Exchange (NYMEX). Our results illustrate that the VaR confidence
level, α, matters along with the amount of data used (’window size’) but overall finds
poor results for all five methods of VaR tested with some positive results for specific
parameterizations of a few methods.
∗Master’s Candidate, Clemson University. Correspondence: [email protected]†revised; first edition: 7 September 2012; second edition: 28 September 2012
1 Introduction
Until Markowitz (1952) people rarely considered risk when making investment or portfolio
allocation decisions. Investments were made in isolation and the portfolio allocation decision
was merely to arbitrarily specify portfolio weights for the disparate investments. Markowitz
in his seminal work demonstrated that there existed an optimal boundary when looking
at the aggregate portfolio’s expected return versus its expected risk. Any combination of
expected returns and risk that was not on this optimal boundary was sub-optimal. The
objective then became one of defining risk and solving for the optimal combination using
linear optimization technique(s).
Markowitz used the standard deviation of returns as his measure of risk but alluded to the
fact that there may be better measurements of risk. In Markowitz (1959) he recommended
the use of semi-variance which only uses deviations below the mean return. The assumption
is that long-only investors are only concerned with downside deviations from the mean. In
fact, they would favor right-skewed distributions and large deviations above the mean; and
since standard deviation does not distinguish between upside and downside deviations it
therefore may not be an accurate reflection of risk. Fishburn (1977) advocated the usage of
differing forms of risk/return utility depending on where an observation occurred within a
distribution of returns. In addition, he questioned the a priori assumption that semi-variance
is the best model and showed that there is a general class of models, the α− t models, that
are dominant and align with an investors risk/return utility as articulated within the Von
Neumann & Morgenstern framework. Nawrocki (1991) extended upon Fishburn (1977) and
investigated the performance of lower partial moment (LPM) estimators of risk. In their
study they can not say that LPM is superior to traditional covariance analysis as articulated
by Markowitz but can say that LPMs are part of the second-degree stochastic-dominance
efficient set.
During the same time period as these authors others were rephrasing the question by
asking whether it was the distribution below the mean that mattered or maybe that extreme
1
values are what matters in regards to risk. Davison & Smith (1990) provided a review of
models using Extreme Value Theory (EVT) by analyzing the limit distribution of extreme
values as first proposed by Fisher & Tippett (1928). They found that the Generalized Ex-
treme Value (GEV) distribution provided an excellent framework for analyzing the extreme
values of a distribution. Others began to use these models with financial data and found that
the GEV distribution had strong explanatory power when looking at extreme logarithmic
prices changes. Specifically related to the analysis in this paper Edwards & Netfci (1988)
and Longin (1999) looked at GEV models in regards to logarithmic price changes with com-
modity futures. Their analyses was related to counter-party risk and the optimal margin
level but they demonstrated that EVT provided an appropriate framework for looking at
extreme prices changes in the futures markets. More recently Gabaix et al (2006) found em-
pirical evidence to support power law distributions, of the same family as GEV, as defining
distributions for price returns of stocks. However, although GEV distributions may provide
strong explanatory power for extreme price changes what is needed is a general framework
for looking at risk and for that we turn to Value at Risk.
2 Value at Risk
Value at Risk (VaR) is concerned with quantifying the largest expected loss over a spec-
ified time period for a specified level of confidence. Formally, let rt = ln(pt/pt−1) be the
logarithmic change in price at time t then VaR is defined as
Pr(rt ≤ V aRt(α)) = α (1)
where the objective becomes finding some F where F−1(α) = V aRt(α). Jorion (1996)
proposed, amongst others, to simply use the sample standard deviation and the standard
2
Normal CDF, Φ. In that context VaR becomes
V aRt(α) = Φ−1(α)σ + µ (2)
where σ is the standard deviation and µ is the mean associated with rt. Jorion (1997)
addressed some of the issues of determining VaR such as the assumption of normality in
regards to financial returns data and the estimation error associated when using sample
quantile methods. In addition he cautioned against the dependence of defining risk with
a single estimator even though the financial industry was rapidly embracing VaR out of
necessity and regulation (I.e. Basel banking accords). Lastly he suggested the usage of kernel
density estimation when the financial returns data is ’suspected to be strongly nonnormal.’
More recently others have advocated using EVT so as to estimate F within the VaR
framework. Neftci (2000) found that using EVT yielded much better out-of-sample results
versus traditional VaR estimates when examining interest rate and foreign exchange data.
Their methodology, which is similar to what we will propose, is to count the number of
observations that exceed a VaR estimate at time t for a specified period of time. Over
the two year period of 1997-1998 they find that across all data sets that EVT-based VaR
methods have a proportion of exceedences that is closer to the stated level of VaR than
compared to standard VaR as defined in (2). Gencay & Selcuk (2004) examine EVT-based
VaR methods in conjunction with emerging markets stock market indices and found that
EVT-based methods offer better estimation for out-of-sample VaR. Specifically due to the
heavy-tailed distributions in emerging markets, because of their associated financial crises,
EVT-based methods are better at estimating VaR especially for lower levels of α.
Krehbiel & Adkins (2005) look at EVT-based methods for VaR dealing with commodity
futures on the NYMEX. They find the best success with conditional-dependence EVT meth-
ods versus Exponentially Weighted Moving Average (EWMA) and Autoregressive-General
Autoregressive Conditional Hetereoskedascity [AR(1)-GARCH(1,1)] methods for the time
3
period analyzed. However, they acknowledge that noise can adversely impact the results
and that the selection of the threshold parameter for Peaks Over Threshold (POT) EVT
methods requires more research. Iglesias (2012) studied EVT-based VaR methods with ex-
change rates and finds that they offer strong explanatory power but there exists varying
results in regards to the type of EVT method used: for some exchange rates EVT-based
methods that take into account the presence of GARCH effects in the data offer better
results.
3 Data
We look at daily logarithmic price changes in three continuous contract1 commodity futures
listed on the New York Mercantile Exchange (NYMEX): Crude Oil (CL), Natural Gas (NG)
and Heating Oil (HO)2. All data used is provided by Norgate Investor Services3 and Table
1 contains descriptive statistics for the three data series analyzed. All series are decidedly
non-normal with all series failing the Jarque-Bera test’s null hypothesis of normal skew and
kurtosis using standard confidence levels. In addition Natural Gas is the only series with
both a negative mean logarithmic return and positive skew. Figures 1, 2 and 3 display the
disparate log price changes and it is discernible the heavy-tailed nature of the series. Natural
Gas shows an increase in dispersion in the latter part of the series which is a function of the
deregulation of the markets in the United States. Other obvious periods of variability would
be the First Gulf War in 1991, global financial crisis of 2007-2009 and the ’Arab Spring’ of
2011-2012 for Crude Oil and Heating Oil.
Yang (1978) first proposed the usage of the Mean Excess (ME) function and Davison
& Smith (1990) used a ME plot to visually determine whether the data conforms to a
1A continuous contract is a construct performed by aggregating multiple time series sequencestogether but then removing gaps that occur due to the fact that commodity future contractswith differing maturities will trade at different price levels. An overview can be found at:http://www.premiumdata.net/support/futurescontinuous.php.
2It should be noted that all futures analyzed have daily price limits such that on certain unspecified daysprices may reach their daily limits which in turn truncates the data.
3http://www.premiumdata.net/
4
Generalized Pareto Distribution (GPD). Given an independent and identically distributed
data sample then the ME function is defined as
M(µ) =
∑ni=1(Xi − µ)I[Xi > µ]∑n
i=1 I[Xi > µ], µ ≥ 0 (3)
where µ is a specified threshold value. These function values can be plotted against a range
of µ to determine an appropriate threshold level and whether a series is suited for EVT
analysis (see Ghosh $ Resnick (2010) for an overview of ME plots). Figures 4, 5 and 6 show
ME plots for the respective commodity futures and were generated using the evir package
in R. In all of the plots we can clearly see a linear trend as the threshold values become
more negative which is an indication of a distribution that fits within the EVT framework.
Hill (1975) offered a non-parametric approach to GEV distributions with his Hill estimator.
Define the Hill estimator as
ξ =1
k
k∑i=j
lnXj,n − lnXk,n (4)
where the data are ordered such that X1,n ≥ X2,n ≥ X3,n, . . . ,≥ Xn,n then α = 1
ξis called
the tail index statistic. Again using evir package in R we create plots of α for varying
order statistics, and their corresponding values, using the negative value for each element of
a series: this is done since by definition the Hill estimator deals with maxima and for the
purposes of our analysis we are only considering negative extremals which means we use the
negative of the returns, r′t = −rt, for our analysis. Figures 7, 8 and 9 show the respective
Hill plots for each series. In each we can clearly see that the standard error of the estimate
is a function of the order statistic selected with the confidence intervals narrowing as the