9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
TESTING GARCH HEDGE EFFECTIVENESS
under CONSIDERATION of
TRANSACTION COST in ENERGY MARKETS
KORAY SAYILI, Queen’s University †
Queen’s School of Business, Goodes Hall, Queen’s University, Room 405, Kingston, Ontario,
Canada, K7L 3N6, Tel: 613 533 2303, Fax: 613 533 2622, e-mail: ksayili@business.queensu.ca.
†
Author is grateful to Peter Sephton and Janelle Mann for their invaluable comments, critiques, and suggestions.
Any possible mistake in this paper and its liabilities solely pertain to the author.
October 16-17, 2009
Cambridge University, UK
1
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
TESTING GARCH HEDGE EFFECTIVENESS under CONSIDERATION
of TRANSACTION COST in ENERGY MARKETS
KORAY SAYILI, Queen’s University
ABSTRACT
This paper investigates the hedging performance of OLS relative to the appropriate GARCH
model for two energy resource contracts traded on the New York Commodity Exchange
(NYMEX) between January 2nd, 2002 and November 6th, 2008. The results for the out-of-sample
period (mainly 2008 data) show that for crude oil contracts, both OLS and GARCH BEKK
methods of hedging reduce the portfolio variance significantly with the latter outperforming the
former. Fractional integration and possibility of asymmetric variance was detected in the natural
gas spot and futures prices so the performance of ARFIMA-EGARCH model was tested. The
ARFIMA-EGARCH hedging method reduces portfolio variance more than the OLS hedging for
the natural gas portfolio similar to the GARCH BEKK outperforms the OLS method for crude
oil. For both energy portfolios, the inclusion of transaction costs in a fictional investment
scenario would not change the superiority of the GARCH hedging method to OLS as long as
several conditions are satisfied.
Keywords: Fractional integration, asymmetric volatility, constant and time dependent hedging,
portfolio variance reduction.
October 16-17, 2009
Cambridge University, UK
2
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
1. INTRODUCTION
With the increasing rate of globalisation, the financial markets have been facing steadily and
rapidly increasing transactions every day. These transactions, either in the form of selling or
buying, drastically raise not only the volume of transactions but also, in many cases, the
volatility which should be taken into account.
There are several methods to hedge existing positions either as portfolio variance minimization
for a given constant return or as portfolio return maximisation for a given constant variance that
have been tested by academics and used by market participants. One of the most widely-used
methods in this branch of econometric research is Generalized Auto Regressive Conditional
Heteroskedasticity (henceforth GARCH). Another widely-used method is Ordinary Least
Squares (henceforth OLS) regression estimation. The literature about the performance
comparison of these two models is quite rich for many products including commodities,
currencies, market indices, energy sources.
The basic motivation of this research project and the first reason for selecting the energy industry
is the popularity of the energy prices and high volatility in recent years. Also, mentioned by
Sephton (1993), oil and currency markets are interesting candidates for GARCH application in
time varying optimal hedge ratio.
October 16-17, 2009
Cambridge University, UK
3
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
The second reason is that very few countries abound plentiful crude oil and natural gas reserves
and they supply these crucial natural resources to the rest of the world. As announced by the U.S.
Energy Information Administration1, approximately three quarters of the world’s crude oil and
natural gas reserves are in Middle East and Central Asia regions. Due to political unrest,
international disputes, wars, strikes, rebel attacks on pipelines, and natural disasters such as
hurricanes, the supply of both crude oil and natural gas can be halted and/or impeded which
gives rise to high speculative price movements which can significantly affect the portfolio
strategy of market participants.
A third reason is that the change in price of oil and natural gas quickly impacts the price of many
goods and services since oil and natural gas are major inputs both for industry, transportation,
and household heating. In other words, the change in oil and natural gas prices starts a price
reaction in many goods and services we buy and provide every day.
The last reason is that there are numerous studies comparing OLS and GARCH hedging
performance, however, most of them either ignore or mention the existence of transaction costs
superficially. Few articles include equations that quantify and measure the effect of transaction
costs. This paper attempts to fill this gap in literature.
October 16-17, 2009
Cambridge University, UK
4
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
The rest of this research paper is organised as follows: The importance of volatility in crude oil
and natural gas and the selected models are introduced in Section 2. In Section 3 OHR is defined
and derived for dollar returns. Section 4 introduces the data used in this project and cardinal
features of data. The statistical methodology alongside the preliminary test results are given in
Section 5. Section 6 provides the details of hedging and empirical findings of both OLS and
implemented GARCH models. Finally, Section 7 summarises and provides possible extensions
of this research paper.
2. ECONOMIC IMPORTANCE OF ENERGY SOURCES & SELECTED
MODELS
In our modern world, all economies are dependent on some crucial natural resources like crude
oil and natural gas for several reasons. These resources serve as the main input of production in
many industries, and are required for transportation, heating and generating electricity. However,
statistical indicators show that the consumption of these scarce resources surged exponentially
after the industrial revolution. According to Shafiee & Topal (2009), these fossil based nonrenewable energy sources will be depleted in less than four decades if the current trend of price
and consumption continues.
As mentioned previously, the prices of these natural resources fluctuate significantly in some
periods. Considering their crucial importance for the modern economic system, the price shocks
October 16-17, 2009
Cambridge University, UK
5
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
may cause serious distortions and deterioration in economic indices. Many studies2 have found
evidence supporting the existence of negative effects of oil price shocks (mostly in the direction
of oil price escalation followed by opposite direction movement) on inflation, economic activity
measured with growth rates, and consumption. For example, according to Cunado & de Gracia
(2004), oil price shocks had important short-run effects on price indices and economic activities
in Asian economies. Lee & Ni (2002) tested the effect of oil price shocks on different industries
and concluded that both industries using oil as input and as output are seriously affected.
Moreover, the negative effect of oil price shocks not only is limited to direct input cost but also
impose a slowdown in many other industries and delays in durable goods sales.
Under these circumstances and conditions, risk-averse oil producers and oil importers want a
smooth and predictable price for these natural resources which gives occasion to the introduction
of derivative products for these energy sources.
There are numerous methods developed to determine the appropriate number of future contracts
needed to minimize the risk of a portfolio. These methods have a varying level of complexity,
ranging from simple OLS to sophisticated GARCH models supported with an Error Correction
Model (henceforth ECM) in case of significantly different return levels.
October 16-17, 2009
Cambridge University, UK
6
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
In the simple OLS regression method, one can run a regression where the dependent variable is
defined as the return in spot market and the explanatory variable is defined as the return in
futures market. In this case, the estimated coefficient for the futures return gives the optimal
hedge ratio (henceforth OHR). Even though the OLS method is simple and empirically
successful3, this method has a theoretical drawback: It assumes constant variance over time.4
On the other hand, it has been recognized for a long time that asset returns exhibit volatility
clusters, meaning low volatility trading days will be followed by tranquil days and vice versa
unless there is an occurrence of an unexpected incident which has a tremendous impact on
markets. Taking this empirical fact into account, Bollerslev (1986) developed a model called
GARCH to estimate the future volatility in light of past volatilities and error terms. The main
theoretical superiority of this method to constant variance methods like simple OLS is that
GARCH allows for variance to change over time, a more realistic scenario.
Among all GARCH models, GARCH BEKK was chosen to be implemented for all data series
that are not fractionally integrated in this research paper. The reason for this choice is quite
straightforward: First of all, GARCH BEKK guarantees non-negativity of conditional variancecovariance matrix (Engle & Kroner; 1995). Secondly, GARCH BEKK is widely-preferred in
academic studies conducted mainly due to the non-negativity characteristic and its relative
October 16-17, 2009
Cambridge University, UK
7
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
simplicity. Karolyi (1995), Gagnon, Lypny, & McCurdy (1998), and Tse (2000) are some
examples of articles implementing GARCH BEKK.
However, if data series one analyses is fractionally integrated (in which fractional integration
level, d, is defined as 0 < d < 1), it is more appropriate to implement a GARCH model that
embraces this aspect. Hence, the ARFIMA-EGARCH was chosen to be implemented in case of
fractionally integrated data series.
The final theoretical point that needs to be kept in mind is that no matter which GARCH model
is implemented, the cointegration between spot and futures returns must be controlled. This
ensures that the assumption of constant returns can be made and the focus can be put on
variance. If this is not the case, the GARCH model needs to be extended with ECM to capture
the return’s effect on the result.
3. OPTIMAL HEDGE RATIO (OHR)
In this section, the derivation of optimal hedge ratio5 (OHR) is presented. It is worth noting that,
for the sake of simplicity, scenarios for portfolios including more than one type of energy source
position as well as cross-hedging options are ignored.
Mathematically, the OHR can be derived as follows6 :
October 16-17, 2009
Cambridge University, UK
8
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Assuming that
St
represents the spot price of any energy resource (or commodity),
Ft
represents the futures price of the same energy resource (or commodity), and
h
stands for the minimum variance providing hedge ratio (or simply OHR).
The return on time t both in spot and futures markets is defined as:
ΔSt = St – St-1
ΔFt = Ft – Ft-1
(The return in spot market at time t)
(1)
(The return in futures market at time t)
(2)
and the return of the portfolio that uses OHR is defined as:
Retportfolio = ΔSt – (h x ΔFt)
(3)
The variance of such a portfolio can be calculated by:
Var (Retportfolio) = Var (ΔSt) + [(h)2 x Var(ΔFt)] - [2h x Cov(ΔSt , ΔFt)]
(4)
The first order condition (F.O.C.) of portfolio variance with respect to h is:
(5)
October 16-17, 2009
Cambridge University, UK
9
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
The portfolio variance is minimised when the FOC derived above is equal to zero, thus the
minimum portfolio variance is derived as follows:
(6)
(7)
(8)
The hedge ratio is provided by the minimum portfolio variance and it is simply the ratio of
covariance of spot and futures returns to variance of futures returns.
It is important to note that the minimum variance providing hedge ratio defined above is valid if
and only if one uses dollar returns. The majority of previous studies have applied this hedge ratio
rule to discrete time or continuous percentage returns (as in the form of logarithm) which might
cause fallacy in the findings. Terry (2005) noticed this common mistake in the OHR
determination and rigorously derived proper OHR equations for discrete and continuous
percentage return scenarios.7
To keep things simple and to work with a familiar OHR equation, dollar returns in both spot and
futures markets were selected for the characterisation and empirical analysis of this data.
4. DATA DESCRIPTION
October 16-17, 2009
Cambridge University, UK
10
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
This research paper analyses two energy resources: crude oil and natural gas. The abbreviations
used for each series follows this simple logic: the first letter “C” and “F” stand for “Cash” and
“Futures” respectively and the last two letters are “CO” and “NG” for crude oil and natural gas
respectively.
The spot and futures price series for both energy resources were excerpted from New York
Mercantile Exchange (NYMEX) contracts which existed in the Commodity Research Bureau
(CRB) InfoTech CD database. The New York Mercantile Exchange (NYMEX) was selected due
to the depth of the existing contracts and data availability.
The time period covered in the empirical analysis includes all dates between January 2, 2002 and
ends on November 6, 2008. The basic reason for this selection is that this period includes both a
tranquil sub-period in early 2000s and highly volatile sub-period in the most recent years.
Prior to empirical estimation, the data was cleansed. Specifically, days such that either one or
both of the spot / futures markets were closed were eliminated from the dataset. Moreover, after
the graphical analysis of price and dollar return series, the extreme outliers in the data were
double checked and possible sources of these unexpected movements were clarified. A
continuous futures price was established by rolling over the contract 15 days before expiry.
October 16-17, 2009
Cambridge University, UK
11
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Two different econometrics softwares, RATS 7.1 and OxMetrics 5, and a spreadsheet, Microsoft
Office Excel 2007, were employed to analyse data both numerically and graphically.
The visual examination of dollar return series8 supports the hypothesis put forward by Bollerslev
(1986) that financial markets in real life show volatility clustering. For instance, approximately
the first 1500 observations (covers the period from January 2002 till December 2007) in both
cash and futures crude oil dollar returns are fairly stable and exhibit little fluctuations around
their means. On the contrary, the last 200 observations in both series predicate high volatility
which can easily be distinguished from the previous tranquil period. Despite the fact that
graphical examination of series acknowledges the existence of volatility clusters, this doesn’t
guarantee each and every low volatility day will be followed by low volatility day and each and
every high volatility day will be followed by high volatility day. Exceptions to this process
include sudden shocks (i.e. a natural disaster that demolishes an oil refinery) and arbitrary
reasons (i.e. speculation). For example, the late February 2003 price surge and plunge in the spot
natural gas market is an example of such an exception.
5. METHODOLOGY & PRELIMINARY ANALYSIS
Since the data is in the form of a time-series, some basic tests need to be completed to see
whether the series have specific characteristic in order to avoid misleading results. First, it needs
to be determined if the data series are stationary or have a unit root. There are several tests that
October 16-17, 2009
Cambridge University, UK
12
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
can be employed, all of which have different strengths and weaknesses. In order not to be
confused or misled by the weaknesses of any test on this matter, three widely-used tests which
provide complimentary information will be implemented.
The first test which was developed by Dickey & Fuller (1979) and later modified and extended
by Said & Dickey (1984), Augmented Dickey Fuller Test (henceforth ADF Test) investigates the
presence of a unit root. The null hypothesis of ADF test is that the series is I(1) and the
alternative is the series is I(0).
The second test which also tests for a unit root in the null hypothesis was introduced by Phillips
& Perron (1988). The Phillips-Perron Test (henceforth PP Test) is a nonparametric unit root test.
This differs from the ADF Test in the sense that they account for autocorrelation in the error
terms. This engenders the necessity for a modification in the Dickey Fuller statistics while the
ADF test extends the standard Dickey Fuller Test by adding additional lags.9
The third and final test was developed by Kwiatkowski et al (1992). The distinctive feature of
this test, hereafter will be referred to as the KPSS Test, is that it tests the null hypothesis of
stationarity, I(0), against the alternative of integrated series, I(1). Under normal circumstances,
one rejecting the null hypothesis of the ADF and PP Tests expects not to reject the null
hypothesis of KPSS Test and vice versa.
Nevertheless, this expectation does not result for all time series because all three tests have the
common weakness of assuming the integration level under the null and alternative hypothesis is
October 16-17, 2009
Cambridge University, UK
13
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
an integer. When the level of integration, d, is not integer, the results of these tests might
contradict each other. In such cases, a fractional integration test can overcome this complication.
Running all three tests for the four price series and running ADF Test for the four dollar return
series10 indicate that all series have consistent unit root and stationarity results except the spot
price series of natural gas (CNG). As mentioned above, in fractionally integrated series, unit root
and stationarity test results can contradict as in the CNG series.
The method developed by Geweke & Porter-Hudak (henceforth GPH) is used to determine the
level of integration in CNG. According to Geweke & Porter-Hudak (1983), when series are
fractionally integrated, the results given by integer unit root tests can be misleading.
Following the finding of Silvapulle & Moosa (1999), which revealed the concurrent reaction to
new information by spot and futures markets, it is beneficial to check the integration level of
natural gas contracts in futures markets (FNG) since there can be a bi-directional causality
between spot and futures prices.11
After checking the fractional integration in both price series, it has been detected that the GPH
test results12 are close to, but less than one. Thus, we can say that the spot and futures price series
October 16-17, 2009
Cambridge University, UK
14
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
for natural gas are fractionally integrated. The level of integration for both series fall between 0.5
and 1, indicating the series are non-stationary but still mean-reverting.
We now jump to an entirely different scenario for the unit root and stationarity tests. In addition
to assuming an integer order of integration, the ADF, PP, and KPSS Tests have one more
common weakness; they all assume that no structural break occurs during the period. If there is a
structural break and it is ignored, some undesirable consequences may arise. Lee & Strazizich
(2004) determine that in the presence of a structural break, endogenous break unit root tests
suffer from size distortions. This leads to higher rates of rejection. Lee, Huang & Shin (1997)
study the effect of a structural break on stationarity tests and they conclude that size distortion
problems in stationarity tests correspond to power loss in unit root tests.
Considering the high probability of a structural break in the series, it is essential to control the
unit root for a structural break. To achieve this, a unit root test with one structural break,
Minimum LM Unit Root Test is employed.
The null hypothesis of unit root with one structural break was only rejected for the natural gas
spot price series (CNG) at 10% critical value.13 For the other three price series, the null
hypothesis of a unit root could not be rejected. These findings are consistent with the no
structural break unit root tests implemented.14
October 16-17, 2009
Cambridge University, UK
15
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
After determining that the dollar return series are stationary, there is one more important concept
that needs to be deliberated: cointegration.
The efficient market hypothesis supports the existence of cointegration between spot and futures
market prices. There are numerous studies that endorse the validity of this relationship. Lien
(1996) touched on the vitality of cointegration for futures hedging and argued that either
intentionally or mistakenly omitting cointegration in the modelling of the OHR may result in
significantly different hedge ratios and performance.
The preferred method to test for cointegration15 was developed by Engle & Granger (1987).
According to Engle Granger Cointegration Test (henceforth EG Test), if two I(1) series are run a
simple OLS regression and shown that error terms in the OLS regression model are I(0), that
designates correlation between those two series. This paper uses the augmented-EG Test
(henceforth AEG Test). It differs from standard EG Test because it includes additional lags. The
number of lags in the cointegration model are determined by the minimum number of lags given
by Akaike Information Criterion (AIC) in ADF Test.16 As the final step, the AEG Test results
were compared with the approximate critical values provided by James MacKinnon (1991) to
reject or not to reject the null hypothesis of no cointegration.
October 16-17, 2009
Cambridge University, UK
16
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
The AEG Test indicated that the test statistics for CCO-FCO are much lower than the
approximate MacKinnon critical values for 1%, 5%, and 10%. Thus, we do not need to write an
EC model and add it to our GARCH model based OHR estimation. Bessler & Covey (1991),
Baillie & Myers (1991) and Sephton (1993) are some of the previous studies that found similar
results.17
As it is mentioned above, the ADF Test on residuals might not show evidence supporting
cointegration, however, this may stem from the fact that structural breaks in the time series were
ignored. As hypothesized, tested, and proven for the public and private savings rate relationship
by Mandal & Payne (2007), allowance for structural break in the cointegration analysis might
reverse the conclusion in favour of cointegration. In order not to fall into this fallacy, the
Gregory-Hansen Cointegration Test (1996) is employed. The Gregory-Hansen Cointegration
Test allows for endogenous structural break(s).
As is the case with AEG Test, the minimum t-statistics for each option in Gregory-Hansen
Cointegration Test is lower than the critical values at 1%, 5%, and 10%. This signifies that both
price series are not cointegrated.
Since both AEG and Gregory-Hansen cointegration tests cannot reject the null of no
cointegration, the necessity for constructing an ECM is not crucial in the determination of
October 16-17, 2009
Cambridge University, UK
17
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
hedging ratios thus the focus can be on variance minimization with the assumption of constant
return for both portfolios.
6. EMPIRICAL ANALYSIS & FINDINGS
This section of the research paper determines the conventional OLS OHR and GARCH OHR for
the portfolios. The hedge ratios will be compared with each other as well as with the unhedged
portfolio. Transaction costs will be taken into account in the performance measurement.
For all hedging analysis in this paper, the first 1500 observations are considered to be part of the
in-sample period and the remaining data are considered to be part of out-of-sample period.18
The analysis begins with the OLS OHR estimation. As shown in section 3, the optimal hedge
ratio for a portfolio using dollar returns is the ratio of the covariance between spot and futures
market dollar returns to the variance of futures market return. This formulisation is identical to
the coefficient description of an independent variable when estimated using the OLS method,
thence running a regression using dollar return in spot market as the dependent variable and
dollar return in futures market as the explanatory variable provides the OHR as the coefficient of
dollar returns in futures market.
Mathematically,
(9)
October 16-17, 2009
Cambridge University, UK
18
9th Global Conference on Business & Economics
Where
ISBN : 978-0-9742114-2-7
is the spot market dollar return at time t,
It is worthwhile to note one interesting characteristic of crude oil OHR estimation19. Its
coefficient is not significantly different from one at a 5% level of significance. This result
indicates that the usage of one-to-one hedging (naive hedge ratio) is applicable to our crude oil
data series.
As mentioned in the introduction, the OLS OHR estimation assumes a constant variance over
time, which is not the case in real life. To account for this, two GARCH hedging models are
implemented. For the crude oil series, the bivariate GARCH BEKK (1, 2) model has been
selected to determine the OHR. For the natural gas series which is fractionally integrated, the
ARFIMA (0, d, 0) EGARCH (1, 2) has been selected to determine the OHR. These models fit
the data and are widely-used in the literature.
Following Engle & Kroner (1995), the GARCH BEKK model is derived as follows:
October 16-17, 2009
Cambridge University, UK
19
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
First, by defining the vector of residuals as εt and the information set at time t-1 as ωt-1, we can
assume
In order words, this means that the residuals are distributed
conditionally normal with mean of zero and variance (covariance) matrix Ht.
Under these assumptions, the conditional variance-covariance matrix for GARCH BEKK (p, q),
can be written as
(10)
In case of BEKK GARCH (1, 1) this can also be written as20
(11)
The most important feature of the GARCH BEKK specification is the non-negativity of
conditional variance-covariance matrix. This is guaranteed by the pairing of each matrix (A, B,
and C) with their transpose.21
Now that the BEKK GARCH model has been derived, p and q must be selected in such a way
that the portfolio variance is minimised. After checking the dollar return data individually for all
series, some unexpectedly high and opposite movements in spot and futures series were
encountered. It was assumed that these incidents, e.g. late March 2003 price surge as a result of
October 16-17, 2009
Cambridge University, UK
20
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
the U.S. bombardment of Iraq, cannot be known a priori therefore the model was constructed
without dummy variables; however, if one can predict such incidents, a dummy variable can be
added to conditional mean equation. Several different p and q combinations were estimated and
it has been determined that the minimum variance providing p and q combination is BEKK
GARCH (1, 2) for crude oil.22 A combination higher than (1, 1) is rarely found in the literature;
however, Engle (2001) cited that a higher order combination may befit for larger datasets since it
gives the advantage of both fast and slow decay of information.23
The choice of p and q for the fractionally integrated series (natural gas) is even more important
due to the fact that price series in spot and futures markets are both fractionally integrated so
implication of any model cannot provide meaningful results. A specific model that considers the
fractional integration characteristic must be chosen. For this reason, ARFIMA (0, d, 0) EGARCH (p, q) is decided to be implemented.
The Autoregressive Fractional Integration Moving Average (henceforth ARFIMA) model
developed by Granger and Joyeux (1980), is used to analyse the fractionally integrated series.
The model states that ARFIMA (p, d, q) is valid when time series,
and
in this case, exhibit
the following characteristic:
(12)
October 16-17, 2009
Cambridge University, UK
21
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
(13)
where
L is the lag operator,
=
,
(14)
(15)
0 < d < 1.
The results of the GPH test indicate that
and
are both within the range (0, 1).
To keep things simple and neat, the ARFIMA (0, d, 0) model was implemented alongside the
appropriate EGARCH (p, q) specification.
Developed by Nelson (1991), EGARCH (exponential GARCH)’s distinctive feature arises from
its allowance for asymmetry in
developed based on
function. On the other hand, GARCH (p, q) has been
’s magnitude with no regard for the sign.24
construction of a GARCH model that allows for asymmetry in the
The rationale behind
function is that unexpected
happenings in different directions (in signs) may not have same impact on future volatility, thus
sign as well as the magnitude of the
October 16-17, 2009
Cambridge University, UK
must be taken into account.
22
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Asymmetric volatility in the oil futures markets has been mentioned by Switzer & El-Khoury
(2007). They succeeded to improve the hedging performance by taking asymmetry into account.
The formulation of EGARCH (p, q) is as follows:
(16)
In the EGARCH (p, q) equation, the coefficients
and
are not restricted to guarantee the
non-negativity of conditional variance as opposed to GARCH (p, q).25 This non-negativity is
provided by the use of absolute value and natural logarithm in the equation.
As in the case of crude oil, the most appropriate (p, q) combination was determined
heuristically.26 After several trials, the EGARCH (1, 2) was found to converge in both spot and
futures dollar returns, reaching an interesting final hedging result.27
After determining the GARCH coefficients for both energy sources’ returns, the daily hedge
ratios were calculated using the equation 8.28
October 16-17, 2009
Cambridge University, UK
23
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Before computing the variance reduction, it is worthwhile to briefly explain why the crude oil
GARCH hedge ratio has such an enormous upward/downward movement around late September
2008. The basic reason for such events is that even though the spot and futures markets have the
same sign and similar percentage returns, there were days in which returns in spot and futures
markets were quite different from each other. For such days, since GARCH is using the previous
day’s data, big deviations from the previous days occur. On the other hand, most of these
enormous deviations prevailed for only a very short time period because the market participants
realised this enormous deviation between the spot and futures market and the following day’s
neutralised returns, again with different signs but in the opposite direction for both markets. The
sudden surge in GARCH hedge ratio for the crude oil portfolio was the result of the different
return magnitudes in the spot and futures market. This was a result of the announcement about
the massive bailout plan in the U.S. with the weakening U.S. Dollar on September 23rd, 2008.
With the possession of OLS and GARCH hedge ratios alongside the dollar returns in spot and
futures markets, one can easily compute the daily unhedged and hedged portfolio returns and
compare the portfolio variances. This paper not only compares the OLS and GARCH hedged
portfolio variances but also takes unhedged portfolio into account, by calculating the variance
reduction rate in the hedged portfolios. During this process the following equation was utilised:
(17)
The results obtained can be summarised as follows29:
October 16-17, 2009
Cambridge University, UK
24
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
For the crude oil portfolio, both OLS and GARCH hedging succeeded to reduce variance by 90%
when compared to the unhedged portfolio. For the natural gas portfolio, the variance reduction
for both OLS and ARFIMA EGARCH hedging had limited success, with less than 6% reduction
in variance when compared to the unhedged portfolio.
GARCH hedging strictly outperformed OLS hedging strategy for the out-of-sample period for
both energy sources’ portfolios individually.
The result for natural gas is consistent with the findings of Coakley, Dollery & Kellard (2007)
and Dark (2007). Both of these papers suggest that if the price series are fractionally integrated,
one should implement a GARCH model that allows for fractional integration and long memory
characteristics.
In the end, they both empirically proved that a GARCH hedging model
internalising fractional integration outperforms conventional OLS hedging. The result for crude
oil is also consistent with the findings of some previous studies. For example, Floros & Vougas
(2004) found that M-GARCH outperformed OLS hedging in the Greek Stock Index Futures
Market. Moon, Yu, and Hong (2009) tested different GARCH models’ variance reduction
effectiveness against conventional OLS method and came up with the conclusion that all
GARCH methods performed better in the out-of-sample period. Sephton (1993) had a similar
finding where GARCH hedging outperformed OLS hedging for commodities traded on
Winnipeg Commodity Exchange. However, neither result takes the transaction cost into account.
This can be misleading since a small improvement in variance reduction with GARCH hedging
may come with a huge transaction cost burden to the investor.
October 16-17, 2009
Cambridge University, UK
25
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
One of the most convenient ways to incorporate transaction costs into the hedging strategy is to
write a utility function that consists of both expected return and the improvement of the portfolio
variance. A utility function that meets these requirements which was previously used by Kroner
& Sultan (1993) is:
(18)
However, this way of measuring utility improvement can be misleading since it utilises the
absolute difference of variance. For instance, imagine two different commodities’ OLS and
GARCH hedged portfolios. Also assume the first commodity’s OLS and GARCH hedged
portfolio variances are 0.05 and 0.04 whilst the second commodity’s OLS and GARCH hedged
portfolio variances are 1.05 and 1.04. Using the Kroner & Sultan’s utility function, both
commodities have the same utility improvement rate since the absolute difference in both
commodities’ OLS and GARCH hedged portfolio variances are same. In other words, the
variance differences must be standardised in order to overcome this problem. To do this, the
difference in percentage variance improvement has been used instead of the absolute difference
in the utility function.
Furthermore, since variance negatively affects total utility in the function (with coefficient ),
any reduction in variance should positively affect the total utility. After modifying the utility
function taking these two factors into account, the utility function can be rewriten as follows:
(19)
October 16-17, 2009
Cambridge University, UK
26
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
As found previously, the average GARCH hedged portfolio returns were -0.00566 for natural gas
and 0.0165 for crude oil. Considering the variance of the same hedged portfolios, it can be
assumed that expected returns were zero for both portfolios. The second term on the RHS of the
equation is the product of the risk aversion coefficient, , and the variance reduction in GARCH
hedged portfolio compared to OLS hedged ratio,
.
The coefficient of risk aversion has been estimated by several studies before. For instance,
Grossman & Shiller (1981) came up with the coefficient of four, Kroner & Sultan (1993) decided
on four in lieu of this coefficient, Settlage & Preckel (2002) used the coefficient of five, and
Chou (1988) calculated it as 4.5. Considering these four studies, it was deemed adequate to use a
coefficient of 4.5 for degree of risk aversion.
Under these circumstances, the utility gain from GARCH hedging in comparison to OLS
hedging can be calculated as follows:
For crude oil:
For natural gas:
On the total transaction cost side, it is well-known that OLS hedging does not bring any extra
cost because of the fixed hedge ratio. On the other hand, the time dependent GARCH hedging
strategy requires daily portfolio adjustments, which brings an inevitable transaction cost burden
to the market participant. To deal with this concern, the percentage transaction cost of GARCH
hedging strategy is determined as follows:
October 16-17, 2009
Cambridge University, UK
27
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Portfolio size (spot position) =
(adjusted daily through entire period)
Transaction cost = n (constant over entire period)
One Futures Contract Price =
(varies over the period)
Hedge Ratio =
In this case, the percentage transaction cost per contract is:
Futures contracts required on day t become:
Thus the daily cost of portfolio adjustment (in percentage terms) on day t is30:
(20)
By generalising this equation to the entire period, the total transaction cost (in percentage terms)
can be reached:
(21)
where
k is the total number of days in the hedging period.
Under these circumstances, if TTC < 14.08% for crude oil (or TTC < 15.84% for natural gas),
this means GARCH hedging elicits net utility surplus even after accounting for the cost of
transactions.
October 16-17, 2009
Cambridge University, UK
28
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Just to make the comparison more straightforward, consider the following fictional scenarios
which have been applied to both the crude oil and natural gas portfolios31:
Portfolio size (spot position) =
= $10M
Transaction cost32 = $5
-
Natural gas portfolio-specific information:
Futures Contract Price = Varies between $65,870 and $147,570 (per 10,000 million British
thermal units or simply mmBtu)
Hedge Ratio = Varies between 0.1227 and 0.7705 within the period.
k = 206 days.
-
Crude oil portfolio-specific information:
Futures Contract Price = Varies between $60,770 and $145,660 (per 1,000 barrels)
Hedge Ratio = Varies between 0.966644 and 2.490627 within the period.
k = 216 days.
Under these circumstances, the total transaction cost for the natural gas portfolio adjustment
occurs as follows:
October 16-17, 2009
Cambridge University, UK
29
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Since 15.84 > 6.04, it can be said that the GARCH hedged natural gas portfolio performed better
than the unhedged and OLS hedged portfolios after accounting for transaction costs.33
Under these circumstances, the total transaction cost for the crude oil portfolio adjustment occurs
as follows:
Since 14.08 > 5.02, it can be said that the GARCH hedged crude oil portfolio performed better
than the unhedged and OLS hedged portfolios after accounting for transaction costs.34
7. CONCLUSION & EXTENSION OF RESEARCH
Using futures contracts to hedge spot positions has been discussed for many years. There have
been several methods developed and tested to achieve the goal of portfolio variance
minimisation or portfolio return maximisation. These methods vary from one-to-one hedging to
ARCH and simple OLS to GARCH enriched with an error correction model.
October 16-17, 2009
Cambridge University, UK
30
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
In this paper the hedging effectiveness of constant (OLS) and time dependent (GARCH)
hedging strategies in two different energy products, crude oil and natural gas. Preliminary
analyses about the existence of a unit root steered the selection of the GARCH model. For the
crude oil series, the unit root and stationarity tests provided consistent results, thus the GARCH
BEKK (p, q) model was employed due to its assurance of conditional variance-covariance matrix
non-negativity. For the natural gas series, the unit root and stationarity tests contradicted each
other, thus the existence and degree of fractional integration was tested. Afterwards the
ARFIMA (0, d, 0) EGARCH (p, q) model was employed due to the fact that the series were
fractionally integrated (captured by ARFIMA) and the asymmetries in volatility (captured by
EGARCH).
The results are quite interesting. For crude oil series, both OLS hedging and GARCH hedging
strategies reduced portfolio variance by about 90% when compared to the unhedged portfolio
case. Moreover, the GARCH hedging outperformed OLS hedging by about 3.1%; however, this
result does not reflect portfolio adjustment costs which might wipe away the superior
performance.
The variance reduction of simple OLS and ARFIMA-EGARCH hedging strategies in the natural
gas portfolio succeeded limitedly, less than 6% for both strategies. The gripping side of this
portfolio hedging was that just like the GARCH hedging strategy for crude oil, the ARFIMAEGARCH hedging strategy also outperformed OLS hedging, around 3.5%.
October 16-17, 2009
Cambridge University, UK
31
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
To be able to confidently reveal that both GARCH hedging strategies are better than OLS
hedging for crude oil and natural gas portfolios in the covered period, the impact of transaction
cost for time dependent hedging strategy had to be internalised. The modified version of meanvariance utility function was compared to the percentage increase in utility with the transaction
cost (as a percentage of cash position). The fictional scenario demonstrated that both GARCH
hedging strategies still outperformed OLS hedging after the inclusion of transaction cost. Finally,
the break-even point providing changes in the fictional scenario, i.e. higher transaction cost,
higher starting cash position, or lower risk aversion coefficient were presented.
For future research, this study can be extended in a couple of ways. First different GARCH
models and time dependent (rolling-over) OLS methods can be applied to the data series. It
might be possible that the untested methods outperform the findings in this paper. Secondly, one
can carefully investigate the correlations among these energy sources’ contracts with others in
order to determine appropriate cross-hedging opportunities and their viability.
October 16-17, 2009
Cambridge University, UK
32
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
FOOTNOTES & ENDNOTES
[1] Energy Information Administration: Official Energy Statistics from the U.S Government.
(2009). World Proved Reserves of Oil and Natural Gas, accessed April 13, 2009, [available at
http://www.eia.doe.gov/emeu/international/reserves.html]
[2] For those interested in the effect of natural gas price changes on economic indicator please
refer to Kliesen, K. L. (2006). Rising Natural Gas Prices and Real Economic Activity. Federal
Reserve Bank of St. Louis Review, 88(6), 511-26.
[3] This consequence will be explained in detail in the following sections.
[4] Lien, D., Tse, Y. K., & Tsui, A. K. C. (2002). Evaluating the hedging performance of the
constant-correlation GARCH model. Applied Financial Economics, 12, 791-798.
[5] Throughout this paper, the terms optimal hedge ratio and minimum variance hedge ratio
denote the same thing and are used interchangeably.
[6] Simplified version of Johnson (1960) and Stein (1961).
[7] The derivation of those two scenarios will not be detailed in this paper because dollar returns
will be used in the following sections. For those who are interested in the correct specification
for discrete time (symbolized with
) and continuous time (symbolized with
percentage return hedge ratios are as follows:
and
October 16-17, 2009
Cambridge University, UK
33
)
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
[8] The price and return series’ figures can be found at the figures section of the paper.
[9] Verbeek, M. (2008). A Guide to Modern Econometrics (3rd edition). West Sussex: England: John
Wiley & Sons Ltd.
[10] The unit root and stationarity test results table can be found at the tables section.
[11] The other price series were also tested with GPH and the results are available to the author.
[12] The GPH test results table can be found at the tables section.
[13] The LM unit root test results table can be found at the tables section.
[14] For all unit root tests, the null hypothesis of a unit root was only rejected for CNG.
[15] Cointegration analysis was implemented on crude oil (CO) price series due to the fact that
both spot and futures market prices are I(1). On the other hand, cointegration for natural gas
(NG) series was not considered for two reasons: the level of integration is different for spot and
futures prices, and the order of integration for both spot and futures prices is less than one.
[16] Number of lags for the crude oil (CO) price series is five.
[17] There are also several studies supporting the existence of cointegration between spot and
futures prices.
[18] Only out-of-sample hedging performances are presented throughout the paper.
[19] The OLS results table can be found at the tables section.
October 16-17, 2009
Cambridge University, UK
34
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
[20] The reason to demonstrate GARCH (1, 1) instead of GARCH (p, q) stems from the fact that
GARCH (1, 1) is the simplest case of GARCH (p, q) and the logic behind the equation does not
change for higher orders.
[21] Karanasos, M., & Kim, J. (2005). The Inflation-Output Variability Relationship in the G3:
A Bivariate GARCH (BEKK) Approach. Risk Letters, 1 (2), 17-22.
[22] For crude oil series, ARCH (2) also performed as well as GARCH (1, 2) and the difference
was infinitesimal. Since this paper compares the performance of OLS and GARCH hedging, we
are only reporting GARCH (1, 2) results.
[23] The quantitative GARCH BEKK (1,2) results are available upon request from the author.
[24] Bollerslev, T., Chou, R. Y., & Kroner, K. F. (1992). ARCH Modeling in Finance: A review
of the theory and empirical evidence. Journal of Econometrics, 52, 5-59.
[25] Bollerslev, Chou, & Kroner, ibid.
[26] As expected the search was started with EGARCH (1, 1) but the spot dollar return series did
not converge so different p and q combinations were tried.
[27] The quantitative ARFIMA (0,d,0) EGARCH (1,2) results are available upon request from
the author.
[28] The descriptive statistics of GARCH and OLS hedge ratios table can be found at the tables
section. Moreover, the figures of hedge ratios for crude oil and natural gas are available at the
figures section.
[29] The variance reduction table can be found at the tables section.
October 16-17, 2009
Cambridge University, UK
35
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
[30] The reason for using absolute difference in hedge ratio stems from the fact that negative and
positive adjustments might cancel out each other which can be deceiving.
[31] Trading futures contracts in decimals is assumed to be possible.
[32] KIT Finance Europe’s brokerage fee schedule. Accessed April 11, 2009, [available at
http://www.kitfinance.ee/en/?p=15]
[33] The break-even analysis is as follows:
Ceteris paribus, the transaction cost (per contract) of $13.1,
Ceteris paribus, the spot position of $26.23M,
Ceteris paribus, a risk aversion coefficient of 1.715 would make the investors indifferent
between OLS and GARCH hedging options.
[34] The break-even analysis is as follows:
Ceteris paribus, the transaction cost (per contract) of $14,
Ceteris paribus, the spot position of $28M,
Ceteris paribus, a risk aversion coefficient of 1.607 would make the investors indifferent
between OLS and GARCH hedging options.
[35] The number of lags for the ADF Test was 13, in the default settings in RATS 7.1
[36] The number of lags for the KPSS Test was determined by the formula given in Kwiatkowski
et al. (1992),
1/4
. Results of another method of lag length determination, I4,
are available upon request from the author.
[37] To be consistent with the ADF Test, the number of lags used in the PP Test was also chosen
as 13. Different lag length results are available upon request from the author.
October 16-17, 2009
Cambridge University, UK
36
9th Global Conference on Business & Economics
October 16-17, 2009
Cambridge University, UK
ISBN : 978-0-9742114-2-7
37
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
REFERENCES

Baillie, R. T., Bollerslev, T., & Mikkelsen, H. O. (1996). Fractionally Integrated
Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics, 74
(1), 3–30.

Baillie, R. T., & Myers, R. J. (1991). Bivariate GARCH Estimation of the Optimal
Commodity Futures Hedge. Journal of Applied Econometrics, 6 (2), 109-124.

Bessler, D. A., & Covey, T. (1991). Cointegration: Some Results on U.S. Cattle Prices.
Journal of Futures Markets, 11 (4), 461–474.

Bollerslev, T. (1986). Generalized Autoregressive Conditional Heteroskedasticity.
Journal of Econometrics, 31 (3), 307-327.

Bollerslev, T., Chou, R. Y., & Kroner, K. F. (1992). ARCH Modeling in Finance: A
Review of the Theory and Empirical Evidence. Journal of Econometrics, 52 (1), 5-59.

Chou, R. Y. (1988). Volatility Persistence and Stock Valuations? Some Empirical
Evidence Using GARCH. Journal of Applied Econometrics, 3 (4), 279-294.

Coakley, J., Dollery, J., & Kellard, N. (2007). The Role of Long Memory in Hedging
Effectiveness. Computational Statistics & Data Analysis, 52 (6), 3075 – 3082.

Cunado, J., & De Gracia, F. P. (2004). Oil Prices, Economic Activity and Inflation:
Evidence for Some Asian Countries. Working Paper, Universidad de Navarra Facultad de
Ciencias Económicas y Empresariales, Barcelona: Spain.
October 16-17, 2009
Cambridge University, UK
38
9th Global Conference on Business & Economics

ISBN : 978-0-9742114-2-7
Dark, J. (2007). Basis Convergence and Long Memory in Volatility When Dynamic
Hedging with Futures. Journal of Financial and Quantitative Analysis, 42 (4), 1021-1040.

Dickey, D. A., & Fuller, W. A. (1979). Distribution of Estimators for Autoregressive
Time Series with a Unit Root. Journal of the American Statistical Association, 74 (366),
427-431.

Energy Information Administration: Official Energy Statistics from the U.S Government.
(2009). World Proved Reserves of Oil and Natural Gas, accessed April 13, 2009,
[available at http://www.eia.doe.gov/emeu/international/reserves.html]

Engle, R. (2001). GARCH 101: The Use of ARCH/GARCH Models in Applied
Econometrics. Journal of Economic Perspectives, 15 (4), 157–168.

Engle, R. F., & Granger, C. W. J. (1987). Co-integration and Error Correction:
Representation, Estimation, and Testing. Econometrica, 55 (2), 251-276.

Engle, R. F., & Kroner, K. F. (1995). Multivariate Simultaneous Generalized Arch.
Econometric Theory, 11 (1), 122-150.

Floros, C., & Vougas, D. V. (2004). Hedge Ratios in Greek Stock Index Futures Market.
Applied Financial Economics, 14 (15), 1125 – 1136.

Gagnon, L., Lypny, G. J., & McCurdy, T. H. (1998). Hedging Foreign Currency
Portfolios. Journal of Empirical Finance, 5 (3), 197-220.

Geweke, J., & Porter-Hudak, S. (1983). The Estimation and Application of Long
Memory Time Series Models, Journal of Time Series Analysis, 4 (4), 221-238.

Granger, C., & Joyeux, R. (1980). An Introduction to Long-Memory Models and
Fractional Differencing, Journal of Time Series Analysis, 1 (1), 15–39.
October 16-17, 2009
Cambridge University, UK
39
9th Global Conference on Business & Economics

ISBN : 978-0-9742114-2-7
Gregory, A. W., & Hansen, B. E. (1996). Residual-Based Tests for Cointegration in
Models with Regime Shifts. Journal of Econometrics, 70 (1), 99-126.

Grossman, S. J., & Shiller, R. J. (1981). The Determinants of the Variability of Stock
Market Prices. American Economic Review, 71 (2), 222-227.

Johnson, L. L. (1960). The Theory of Hedging and Speculation in Commodity Futures.
Review of Economic Studies, 27 (3), 139-51.

Karanasos, M., & Kim, J. (2005). The Inflation-Output Variability Relationship in the
G3: A Bivariate GARCH (BEKK) Approach. Risk Letters, 1 (2), 17-22.

Karolyi, G. A. (1995). A Multivariate GARCH Model of International Transmissions of
Stock Returns and Volatility: The Case of the United States and Canada. Journal of
Business & Economic Statistics, 13 (1), 11-25.

Kliesen, K. L. (2006). Rising Natural Gas Prices and Real Economic Activity. Federal
Reserve Bank of St. Louis Review, 88 (6), 511-26.

Kroner, K. F., & Sultan, J. (1993). Time-Varying Distributions and Dynamic Hedging
with Foreign Currency Futures. The Journal of Financial and Quantitative Analysis, 28
(4), 535 -551.

Kwiatkowski, D., Phillips, EC. B., Schmidt, P., & Shin, Y. (1992). Testing the Null
Hypothesis of Stationarity against the Alternative of a Unit Root: How Sure Are We that
Economic Time Series Have a Unit Root?. Journal of Econometrics, 54, 159-178.

Lee, J., Huang, C. J., & Shin, Y. (1997). On Stationary Tests in the Presence of Structural
Breaks. Economics Letters, 55 (2), 165-172.
October 16-17, 2009
Cambridge University, UK
40
9th Global Conference on Business & Economics

ISBN : 978-0-9742114-2-7
Lee, J., & Strazicich, M. C. (2004). Minimum LM Unit Root Test with One Structural
Break, Working Paper, Appalachian State University, Boone: NC, accessed April 3,
2009, [available at http://econ.appstate.edu/RePEc/pdf/wp0417.pdf]

Lee, K., & Ni, S. (2002). On the dynamic effects of oil price shocks: a study using
industry level data. Journal of Monetary Economics, 49 (4), 823-852.

Lien, D. (1996). The Effect of the Cointegration Relationship on Futures Hedging: A
Note. The Journal of Futures Markets, 16 (7), 773-780.

Lien, D., Tse, Y. K., & Tsui, A. K. C. (2002). Evaluating the hedging performance of the
constant-correlation GARCH model. Applied Financial Economics, 12 (11), 791-798.

MacKinnon, J. G. (1991). Critical Values for Cointegration Tests. In R.F. Engle and
C.W.J. Granger (Ed.), Long-Run Economic Relationships. London: England: Oxford
University Press, 267-276.

Mandal, A., & Payne, J. E. (2007). Long-Run Relationship between Private and Public
Savings: An Empirical Note. Journal of Economics and Finance, 31 (1), 99-103.

Moon, G. H., Yu, W. C., & Hong, C. H. (2009). Dynamic hedging performance with the
evaluation of multivariate GARCH models: Evidence from KOSTAR index futures.
Applied Financial Economics Letters, 16 (9), 913 – 919.

Nelson, D. B. (1991). Conditional Heteroskedasticity in Asset Returns: A New Approach.
Econometrica, 59 (2), 347-370.

Phillips, P. C. B., & Perron, P. 1988. Testing for a Unit Root in Time Series Regression.
Biometrika, Vol. 74, 535-547.

Said, S. E., & Dickey, D. A. (1984). Testing for Unit Roots in Autoregressive-Moving
Average Models of Unknown Order. Biometrika, 71(3), 599-608.
October 16-17, 2009
Cambridge University, UK
41
9th Global Conference on Business & Economics

ISBN : 978-0-9742114-2-7
Sephton, P. S. (1993). Optimal Hedge Ratios at the Winnipeg Commodity Exchange. The
Canadian Journal of Economics, 26 (1), 175-193.

Settlage, D. M., & Preckel, P. V. (2002). Robustness of Non-Parametric Measurement of
Efficiency and Risk Aversion. Presented paper at the Annual Meetings of the American
Agricultural Economics Association, Long Beach, CA: the American Agricultural
Economics
Association.
accessed
April
17,
2009,
[available
at
http://ageconsearch.umn.edu/bitstream/19765/1/sp02se04.pdf]

Shafiee, S., & Topal, E. (2009). When Will Fossil Fuel Reserves Be Diminished?. Energy
Policy, 37 (1), 181-189.

Silvapulle, P., & Moosa, I. A. (1999). The Relationship between Spot and Futures Prices:
Evidence from the Crude Oil Market. Journal of Futures Markets, 19 (2), 175-193.

Stein, J. L. (1961). The Simultaneous Determination of Spot and Futures Prices.
American Economic Review, 51 (5), 1012-1025.

Switzer, L. N, & El-Khoury, M. (2007). Extreme Volatility, Speculative Efficiency, and
the Hedging Effectiveness of the Oil Futures Markets. Journal of Futures Markets, 27 (1),
61 – 84.

Terry, E. (2005). Minimum-Variance Futures Hedging Under Alternative Return
Specifications. The Journal of Futures Markets 25 (6), 537–552.

Tse, Y. K. (2000). A Test for Constant Correlations in a Multivariate GARCH Model.
Journal of Econometrics, 98 (1), 107-127.

Verbeek, M. (2008). A Guide to Modern Econometrics (3rd edition). West Sussex:
England: John Wiley & Sons Ltd.
October 16-17, 2009
Cambridge University, UK
42
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
TABLES
Table 1: Unit root and stationarity tests
ADF (for 13 lags)35
KPSS (for 25 lags)36
PP (for 13 lags)37
None
C
CT
C
CT
C
CT
CCO
-0.0081
-1.4736
-1.4863
5.510*
0.419*
-1.48187
-1.66267
FCO
-0.1494
-1.8978
-1.4822
4.451*
0.343*
-1.74498
-1.21791
CNG
-0.7017
-3.1586**
-3.8011**
3.176*
0.289*
-3.16340**
-3.90671**
FNG
-1.0451
-0.2103
-2.1976
4.910*
1.095*
-0.08087
-2.03372
ΔCCO
-18.6028*
-18.6131*
-18.6354*
ΔFCO
-9.2851*
-9.2914*
-9.3704*
ΔCNG
-12.2548*
-12.2585*
-12.2720*
ΔFNG
-13.7863*
-13.8246*
-13.8973*
Critical values for ADF Test
(1%, 5%, 10%)
Critical values for KPSS
Test (1%, 5%, 10%)
Critical values for PP Test
(1%, 5%, 10%)
None: -2.58, -1.95, -1.62
Constant: -3.43, -2.86, -2.57
Constant & Trend: -3.96, -3.41, -3.12
Constant: 0.739, 0.463, 0.347
C & T: 0.216, 0.146, 0.119
Cons.: -3.437, -2.864, -2.568
C & T: -3.969, -3.415, -3.129
Note: *, **, *** represent statistically significant series for 1%, 5%, and 10%, respectively.
Table 2: GPH test
GPH
October 16-17, 2009
Cambridge University, UK
CNG
FNG
Power = 0.50000
Power = 0.50000
43
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Estimated d = 0.80867
Power = 0.40000
Estimated d = 0.87464
Estimated d = 0.92753
Power = 0.40000
Estimated d = 0.90687
Table 3: LM unit root test
Lee-Strazicich Unit Root Test, Series CCO
Lee-Strazicich Unit Root Test, Series FCO
Variable
S{1}
Constant
D(1525)
DT(1525)
Variable
S{1}
Constant
D(1525)
DT(1525)
Coefficient T-Stat
-0.0159
-3.7037
-0.0264
-0.5965
0.5315
0.3374
0.4649
2.1801
Coefficient
-0.0076
0.0610
0.8422
0.0768
T-Stat
-2.5535
1.6418
0.5995
0.5242
Lee-Strazicich Unit Root Test, Series CNG
Lee-Strazicich Unit Root Test, Series FNG
Variable
S{1}
Constant
D(855)
DT(855)
Variable
S{1}
Constant
D(1013)
DT(1013)
Coefficient T-Stat
-0.0210 -4.2486
0.0161
1.3788
0.0736
0.2211
0.0103
0.6240
Coefficient T-Stat
-0.0094
-2.8367
0.0178
1.8469
-0.5261
-2.2542
-0.0397
-2.7996
Note: The critical values for different lambda (λ) values are as follows:
λ
.1
.2
.3
.4
.5
1%
-5.11
-5.07
-5.15
-5.05
-5.11
5%
-4.50
-4.47
-4.45
-4.50
-4.51
10%
-4.21
-4.20
4.18
-4.18
-4.17
Table 4: AEG test
Series
CCO-FCO
October 16-17, 2009
Cambridge University, UK
Test
Statistics
-0.04819
Critical Values
(1%, 5%, and 10%)
-3.91
-3.34
44
-3.05
Decision
Cannot reject the null
hypothesis of no
cointegration
9th Global Conference on Business & Economics
October 16-17, 2009
Cambridge University, UK
ISBN : 978-0-9742114-2-7
45
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Table 5: Gregory-Hansen cointegration test for crude oil
Min. Tstatistics
Min. At
1%
5%
10%
FullBreak
-3.14649
1073
-5.47
-4.95
-4.68
Constant
-2.68924
1131
-5.13
-4.61
-4.34
Trend
-2.82956
1133
-5.45
-4.99
-4.72
Result
Cannot reject Null of no
cointegration
Cannot reject Null of no
cointegration
Cannot reject Null of no
cointegration
Table 6: OLS optimal hedge ratio estimations
1. CCO & FCO: (In-sample period: First 1500 observations)
Variable
Coeff
T-Stat
Signif.
Constant
0.008712
0.63458
0.52569944
DFCO
0.995664
64.39929
0.00000000
2.
CNG & FNG (In-sample period: First 1500 observations)
Variable
Coeff
T-Stat
Signif.
Constant
0.006958
0.81194
0.41682668
DFNG
0.516453
5.19907
0.00000020
Table 7: Descriptive statistics of hedge ratios
Crude Oil
Natural Gas
OLS
GARCH
OLS
GARCH
Mean
0.995665
1.033867
0.516453
0.280994
Std. Error
0.000000
0.148130
0.000000
0.105568
Minimum
0.995665
0.966644
0.516453
0.122707
October 16-17, 2009
Cambridge University, UK
46
9th Global Conference on Business & Economics
Maximum
0.995665
ISBN : 978-0-9742114-2-7
2.490628
0.516453
0.770545
Table 8: (Un)Hedged portfolio mean & variances
Number of Obs.
Unhedged
Portfolio
216
OLS Hedged
Portfolio
216
GARCH Hedged
Portfolio
216
Mean (Dollar return)
-0.163148
-0.004718
0.016524
Portfolio Variance
11.055584
1.316559
0.970718
Var. Reduction Rate
-
88.09%
91.22%
Number of Obs.
206
206
206
Mean (Dollar return)
-0.0067
-0.00051
-0.00566
Portfolio Variance
0.062585
0.061568
0.059365
Var. Reduction Rate
-
1.63%
5.15%
Crude Oil Portfolio
Natural Gas Portfolio
October 16-17, 2009
Cambridge University, UK
47
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
FIGURES
Figure 1: Crude oil price series
Figure 2: Natural gas price series
Figure 3: Crude oil (cash) return series
October 16-17, 2009
Cambridge University, UK
48
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Figure 4: Crude oil (futures) return series
Figure 5: Natural gas (cash) return series
Figure 6: Natural gas (futures) return series
October 16-17, 2009
Cambridge University, UK
49
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Figure 7: Crude oil portfolio hedge ratios
Figure 8: Natural gas portfolio hedge ratios
October 16-17, 2009
Cambridge University, UK
50
9th Global Conference on Business & Economics
October 16-17, 2009
Cambridge University, UK
ISBN : 978-0-9742114-2-7
51