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University of Limerick
Niall Gilbride
Dynamic Jet Fuel Hedging
using an Error Correction
Model with GARCH Error
Terms
MSc in Computational Finance
2014-2015
This dissertation is solely the work of the author and submitted in partial fulfilment of the
requirements of the MSc in Computational Finance, University of Limerick, Ireland
Dynamic Jet Fuel Hedging using an Error
Correction Model with GARCH Error Terms
Niall Gilbride 09008201
MSc in Computational Finance
2014-2015
Dr. Bernard Murphy
Word Count: 16,500
i
Abstract
Jet fuel hedging has been shown to stabilise airline profits and increase airline value. This
research analyses the cross hedging performance of Brent crude oil, West Texas Intermediary
crude oil, low sulphur gasoil and NYMEX heating oil in reducing airlines spot price exposure
to Gulf54 grade jet fuel. Three econometric techniques popular in the cross hedging literature
are applied to the cross-hedge portfolios with the added constraint that the portfolios must be
able to qualify for hedge accounting status. The analysis shows that hedging on a weekly
horizon is not sufficient for any of the cross-hedge portfolios to meet hedge accounting
standards and that instead hedging must be accomplished on monthly and quarterly horizons.
The results indicate that cross-hedging Gulf54 grade jet fuel using futures on Brent crude oil
provides the best hedging option for airlines across all horizons when the constraint of
meeting hedge accounting standards is taken into account.
ii
Acknowledgements
I would sincerely like to thank Dr. Bernard Murphy for his feedback and support, not only
throughout the process of carrying out this study, but throughout the year in all aspects of the
MSc. in Computational Finance for which this research represents the conclusion.
As this study is not the product of any one area of my academic work this year and rather a
culmination of the academic progress made since last September, I must also thank Dr.
Finbarr Murphy, Orla McCullagh and Dr. Weiou Wu for the progress that was made under
their guidance. I would also like to thank Prof. Zeno Adams and Mathias Gerner on whose
research this study is based, in particular Prof. Adams and for his correspondence during the
summer months.
iii
Table of Contents Abstract ....................................................................................................................................... i
Acknowledgements .................................................................................................................... ii
Table of Tables .......................................................................................................................... v
Table of Figures ........................................................................................................................ vi
Table of Equations .................................................................................................................... vi
Research Question ..................................................................................................................... 1
Introduction ................................................................................................................................ 2
Jet Fuel Hedging ........................................................................................................................ 3
SFAS 133 ................................................................................................................................... 4
Preliminary Considerations ........................................................................................................ 7
Hedge Ratio & Hedging Performance ................................................................................... 7
Hypothesis Testing & Statistical Significance ....................................................................... 9
Regression Model .................................................................................................................... 11
Unit Root Testing ................................................................................................................. 11
Regression Analysis ............................................................................................................. 13
Residual Series ......................................................................................................................... 15
Autocorrelation .................................................................................................................... 15
Heteroscedasticity ................................................................................................................ 16
Error Correction Model............................................................................................................ 17
Co-Integration ...................................................................................................................... 17
Error Correction Analysis .................................................................................................... 19
Basis Risk............................................................................................................................. 22
GARCH Extension................................................................................................................... 21
GARCH Analysis................................................................................................................. 21
Results & Discussion ............................................................................................................... 23
SFAS 133 ............................................................................................................................. 23
iv
Unit Root Testing ................................................................................................................. 27
Residual Diagnostics ............................................................................................................ 29
Cointegration Analysis......................................................................................................... 32
Optimal Hedge Ratio ........................................................................................................... 34
Hedge Performance .................................................................................................................. 37
Brent Crude Oil .................................................................................................................... 38
Low Sulphur Gasoil ............................................................................................................. 40
West Texas Intermediary Crude Oil .................................................................................... 42
Home Heating Oil ................................................................................................................ 44
Conclusions .............................................................................................................................. 45
Bibliography ............................................................................................................................ 46
Appendix A .............................................................................................................................. 50
Appendix B .............................................................................................................................. 51
MATLAB Import Data ........................................................................................................ 51
MATLAB Define Parameters .............................................................................................. 51
MATLAB Loop Arguments ................................................................................................ 51
MATLAB SFAS 133 ........................................................................................................... 51
MATLAB Unit Root Analysis ............................................................................................. 52
MATLAB OLS Regression Model ...................................................................................... 53
MATLAB Residual Diagnostics .......................................................................................... 53
MATLAB Error Correction Model ...................................................................................... 54
MATLAB ECM GARCH Extension ................................................................................... 54
MATLAB Dynamic Hedge Ratio ........................................................................................ 55
MATLAB Calculate P&L .................................................................................................... 57
v
Table of Tables
Table 1 Commodity Hedge Horizons ........................................................................................ 5
Table 2 Unit Root Hypotheses ................................................................................................. 12
Table 3 R-Square Value based on Weekly Hedging Horizon ................................................. 23
Table 4 R-Square Value based on Monthly Hedge Horizon ................................................... 23
Table 5 R-Square Value based on Quarterly Hedge Horizon .................................................. 23
Table 6 Augmented Dickey Fuller Test Levelled Series ......................................................... 27
Table 7 Kwiatkowski Phillips Schmidt Shin Test Levelled Series .......................................... 28
Table 8 Augmented Dickey Fuller Test Differenced Series .................................................... 28
Table 9 Kwiatkowski Phillips Schmidt Shin Test Differenced Series .................................... 29
Table 10 Residual Series Stationarity Tests............................................................................. 29
Table 11 Residual Series LBQ & ARCH Analysis ................................................................. 31
Table 12 Residual Series Heteroscedasticity Analysis ............................................................ 32
Table 13 Cointegration Analysis ............................................................................................. 32
Table 14 ECM Residual Speed of Adjustment ........................................................................ 34
Table 15 Minimum Variance Hedge Analysis Brent ............................................................... 38
Table 16 Minimum Variance Hedge Analysis Gasoil ............................................................. 40
Table 17 Minimum Variance Hedge Analysis WTI ................................................................ 42
Table 18 Minimum Variance Hedge Analysis Heatoil ............................................................ 44
vi
Table of Figures
Figure 1 Jet-Brent OLS Regression Plot .................................................................................. 14
Figure 2 SFAS Analysis Weekly Time Horizon...................................................................... 24
Figure 3 SFAS Analysis Monthly Time Horizon .................................................................... 25
Figure 4 SFAS Analysis Quarterly Time Horizon ................................................................... 26
Figure 5 Autocorrelation Plots Brent Gasoil ........................................................................... 30
Figure 6 Commodity Spot Prices 2009-2015 .......................................................................... 33
Figure 7 Dynamic Hedge Ratio Brent Crude Oil..................................................................... 34
Figure 8 Dynamic Hedge Ratio LS Gasoil .............................................................................. 35
Figure 9 Residual Autocorrelation Plots .................................................................................. 50
Figure 10 Residual QQ Plots ................................................................................................... 50
Figure 11 Residual Histograms ................................................................................................ 50
Table of Equations
Equation 1 Optimal Hedge Ratio ............................................................................................... 7
Equation 2 Optimal Number of Contracts ................................................................................. 8
Equation 3 P&L Series............................................................................................................... 8
Equation 4 Profit and Loss Spot ................................................................................................ 8
Equation 5 Profit and Loss Futures ............................................................................................ 9
Equation 6 Annualised Variance Gasoil .................................................................................... 9
Equation 7 Linear Model ......................................................................................................... 13
Equation 8 Error Correction Model ......................................................................................... 20
Equation 9 Conditional Variance ............................................................................................. 21
1
Research Question
The research question posed is which commodity out of Brent crude oil, WTI crude oil,
low sulphur gasoil and home heating oil is most effective when utilised in a cross-hedge
in order to reduce an airlines price exposure to Gulf54 grade jet fuel. The work intends to
build on similar research carried out by (Adams & Gerner, 2012) who evaluated the
performance of the four cross-hedge commodities listed above using (i) an OLS
regression model, (ii) an Error Correction model (ECM) and (iii) an ECM with GARCH
error terms. This study hopes to build on this and other previous work by conducting the
similar research to (Adams & Gerner, 2012) but will also incorporate analysis on whether
each potential cross-hedge can qualify for hedge accounting classification under
contemporary FAS 133 regulations. Each of the three models will be fitted to the data and
all three will be used in back-testing. The best performing commodity will be the one
which qualifies for hedge accounting under FAS 133 guidelines and produces the lowest
annualised variance for a portfolio containing 150,000 gallons of jet fuel when back-
tested over a varying number of years for each commodity leading up to 2015.
It is hoped that this study will contribute to the current literature in the field in two
existing ways and one new way. It is expected the results will support the evidence that
error correction models perform better than OLS regression models when hedging price
exposure in commodity portfolios using futures contracts. This research will also
contribute to the debate about which crude oil based commodity provides the best cross-
hedging instrument for jet fuel under current market conditions. Finally it is hoped that
the research will draw attention to the fact that hedging jet fuel on a weekly horizon, as is
the practice in a number of studies, does represent the best option for airlines as futures
hedging using the chosen products on this horizon may not be eligible for hedge
accounting status under the current regulations.
2
Introduction
The passenger airline industry is growing at a rate of 5% per annum and is by and large a
competitive industry. While jet fuel fluctuates in price regularly and accounts for the
second largest expenditure for passenger airlines, the competitive pressures associated
with the passenger airline business mean that increases in jet fuel prices cannot easily be
passed on to consumers for fear that increases in charges will result in a loss of market
share. The resulting exposure to this volatile commodity represents a significant source of
uncertainty for airline profits. Jet fuel hedging is the practice of entering into financial
contracts in order the fix the future price of jet fuel purchases. This reduces an airlines
exposure to fluctuating fuel prices, depending of the percentage of jet fuel hedged. This
would usually be done through buying jet fuel ahead of time but unfortunately due to the
lack of a liquid market for jet fuel futures, airlines are left unable to buy fuel at a fair price
ahead of time and so hedging must instead be done via futures on other commodities. The
rationale is that if the prices of jet fuel and another commodity with a liquid futures
market tend to move up and down together, then buying futures on the other commodity
will allow the hedger to profit if prices jointly increase and then use this profit to offset
the corresponding increase in jet fuel prices. The practice of hedging using futures on a
different underlying asset is known as a cross-hedge and in the aviation industry is
generally accomplished through buying futures on crude oil, gasoil and home heating oil
due to the similar price characteristics they share with jet fuel. This does not mean that
the prices of these commodities are the same or even similar, rather their prices are highly
correlated and tend to move upwards and downwards together.
The proposed dissertation is influenced by the earlier work of (Adams & Gerner, 2012)
who apply least squares regression and error correction techniques in order to evaluate the
cross hedging performance of Brent crude oil ,West Texas Intermediary crude, NYMEX
heating oil and low sulphur gasoil for hedging spot price exposure to Gulf54 grade jet
fuel. The research is concerned with which commodity of Brent, WTI, gasoil and heating
oil forms the most effective cross-hedge with jet fuel while being eligible to qualify for
hedge accounting status under current regulations. Three models are utilised for testing
hedge effectiveness are a least squares regression model, an error correction model and an
error correction model with GARCH error terms. The most effective cross-hedge is the
one that leads to the lowest annualised variance in a portfolio containing 150,000 gallons
of Gulf54 jet fuel and a model estimated quantity of the cross-hedging product.
3
Jet Fuel Hedging
Jet fuel hedging is undertaken by a large number of airlines annually including Ryanair in
Europe and Southwest Airlines in the United States. Jet fuel is a volatile commodity and
from the early 2000’s the spot price of Gulf54 grade jet fuel rose from $0.53 per gallon to
over $4 per gallon at its peak in 2008, an increase of 647%. The price has since fallen
64% to $1.42 per gallon. Jet fuel accounts for the second largest expenditure for airlines
and since its price can drift substantially over time it’s found by (Carter, et al., 2006) that
if airlines can better manage the cost of fuel then they can more accurately estimate
budgetary costs and forecast future income. Fuel hedging also has positive benefits for the
overall value of the airline. Analysing US airlines over an eleven year period (Carter, et
al., 2006) find that participation in fuel hedging strategies is associated with increases in
airline value of between 5% and 10% while (Cobbs & Wolf, 2004) also determine that
using derivatives to hedge fuel costs creates a competitive advantage for the hedged
airline.
There are a number of strategies available to airlines when proposing hedging solutions to
the cost of jet fuel and (Cobbs & Wolf, 2004) suggest a strategy which is dynamically
managed throughout the life of the hedge achieves the greatest results and note that many
airlines prefer the use of over-the-counter derivatives such as collar structures and swaps
due to their customisability. Using these types of instruments airlines attempt to lock in
prices when they feel jet fuel prices are at a low point and cap prices when they feel they
are at a high point. While the use of these instruments represent a complex hedging
strategy (Carter, et al., 2004) note that their use has been quite successful for Southwest
Airlines. Not every airline will have the expertise to manage a derivative portfolio in
order to hedge fuel costs and so for other businesses hedging through the purchase of
commodity futures may be more appropriate.
Futures contracts allow one to enter into an agreement to buy a standardised quantity of
some product (say 1,000 gallons of Brent Crude Oil) for a certain price at a certain time in
the future. Unlike a Call Option which gives the buyer the right but not the obligation to
take delivery of the product, a futures contract is legally binding. As a result if an airline
enters into a futures contract to buy jet fuel at this time next year and the subsequent price
is below that which was agreed in the contract then the airline still has to pay the higher
price. From this perspective (Morrell & Swan, 2006) note that airlines hedge in order to
4
stabilize fuel prices, rather than speculate on whether the price of jet fuel will increase or
decrease in the future. A hedge is simply taking a position which minimises the price
variance of the asset being hedged. For example a $100 asset which is perfectly hedged
would never change in value from $100 regardless of market events. It’s also worth
noting that most futures are not settled with physical delivery of the underlying assets and
rather are settled for cash at maturity.
SFAS 133
Statements of Financial Accounting Standards 133 or SFAS 133 are an accounting
standard relating to ‘Accounting for Derivative Instruments and Hedging Activities’.
SFAS 133 was originally implemented as (FASB, 1998) which was introduced in order to
deal with some of the shortcomings of its predecessor SFAS 80. During the lifespan of a
derivatives hedge the value each derivative may deviate largely from its’ initial cost and
as a result the traditional accounting methods that required instruments to be booked and
carried at historical cost were not sufficient to reflect the true value of these derivative
assets whose prices changed daily. From this the first major doctrine of SFAS 133 is that
all derivative instruments are marked-to-market, or represented at their fair value. SFAS
133 also requires businesses to match gains in a derivative instrument with losses in a
hedged asset which has the effect of preventing companies from booking gains in either
the hedged item or derivative while failing to report losses on the other side of the hedge
until quarter close.
In order to qualify for hedge accounting under SFAS 133 it’s necessary for businesses to
demonstrate that each hedge is liable to be highly effective in reducing the risk exposure
of the underlying portfolio. It is noteworthy that (FASB, 1998) does not specify any one
measure of hedge effectiveness to be used in order to qualify for hedge accounting rules
and that the appropriateness of any assessment of hedge effectiveness can depend on the
nature of the risk being hedged. Once the measure of hedge effectiveness has been chosen
by the hedging party, (FASB, 1998) states that if the hedge fails this effectiveness test at
any time then the hedge ceases to qualify for hedge accounting rules. The (FASC, 2008)
note that the advantages of hedge accounting outweigh the disadvantages of fair value
accounting for financial instruments and there are a number of reasons why companies
want their hedge portfolios to qualify for hedge accounting rules. The principal advantage
is that it recognises the earnings effects of the hedging instrument and the hedged item in
5
the same financial period and also recognises their earnings effects in proportion to each
other which results in lower earnings volatility for companies. The lower earnings
volatility is in part due to the fact that when operating under hedge accounting rules
earnings and losses on hedged assets are deferred and recorded as offsetting gains or
losses in the value of the hedging instruments whereas outside these standards gains and
losses must be recorded individually and immediately (FASC, 2008).
In the absence of a specific recommendation much of the accounting profession has
adopted the 80-125 dollar offset ratio standard as a measure of hedge effectiveness. Under
current guidelines if one demonstrates that the R-Squared value obtained from regressing
price changes in the hedged asset on price changes in the hedging instrument is greater
than 0.8 then the hedge is deemed highly effective and will qualify for hedge accounting.
Hedge effectiveness is generally measured by the beta coefficient of a least squares
regression and must demonstrate that its value is greater than 0.8 at all times during the
life of the hedge in order to qualify for hedge accounting rules.
This method is criticised by (Charnes, et al., 2003) as during periods of low volatility the
R-Square value may fall below 0.8 and disqualify companies from hedge accounting and
its associated benefits. Its suggested by (Juhl, et al., 2012) that the time frame for back-
testing price changes should match the hedge horizon and so for each potential cross
hedge the back-testing period will be subject to the availability of futures on each
commodity. As of July 2015 the (CMEGroup, 2015) lists futures contracts on Brent and
WTI until December 2022, gasoil until December 2021 and heating oil until January
2019. It’s empirically shown by (Juhl, et al., 2012) that regressing price changes
generates an R-Squared value that converges towards one as the time horizon is extended.
The findings suggest that using monthly as opposed to weekly price changes enhances the
ability of the underlying commodities to qualify for hedge accounting as does back-
testing over a longer time period. As a result commodities with the farthest extending
futures markets (Brent and WTI) should have an advantage over gasoil and heating oil in
qualifying for hedge accounting.
Brent WTI Gas Heat
Hedge Horizon Dec-22 Dec-22 Dec-21 Jan-19
Back-test Horizon (years)
6.5 6.5 5.5 3.5
Table 1 Commodity Hedge Horizons
6
(Juhl, et al., 2012) determine that if the commodity time series are co-integrated and that
price data is available for an adequate sample period then the hedger can be confident of
meeting hedge accounting standards and that with regards to hedge accounting it doesn’t
matter whether a least squares regression model or an error correction model are used to
determine hedge effectiveness as both will yield an R-Square that approaches one as the
hedge horizon is increased. In another study on optimal hedge ratio for cross-hedging jet
fuel (Adams & Gerner, 2012) perform hedge adjustments using a weekly horizon which
they note reduces commodity price risk without having to overspend on transaction fees
and is in line with other studies1. These studies do not account for the impact of hedge
accounting and while operating on a weekly horizon offers the opportunity to adjust the
hedge on a regular basis it may not display eligibility for hedge accounting under (FASB,
1998).
Starting with a weekly time horizon if the R-Square value lies below the 0.80 threshold
then the time horizon for the hedging product will be extended to monthly and re-
evaluated to see if the hedge meets SFAS accounting standards. The time horizon may
subsequently be extended up to a maximum of three months as each quarter is the
minimum time frame for which hedge effectiveness must be reported under FAS 133
guidelines. The time horizon for which each commodity qualifies for hedge accounting
(i.e. weekly, monthly or quarterly) will be the time frame carried forward for optimal
hedge ratio analysis. Due to the varying availability of futures contracts on Brent, WTI,
Gasoil and Home Heating oil and how this applies to (FASC, 2008), it will be assumed
that the hedge horizon for each commodity will be maximised in order to run the longest
possible back-test for each commodity. Thus the commodities Brent and WTI which have
the farthest extending futures will also have the longest back-test period.
1 (Clark, et al., 2003) (Coffey, et al., 2000)
7
Preliminary Considerations
Hedge Ratio & Hedging Performance
The hedge ratio is the proportion of the size of the position taken in futures contracts
compared to the size of the exposure. (Hull, 2015) notes that because jet fuel futures are
not actively traded airlines must hedge their exposure to jet fuel using futures on other
commodities in what is known as a cross-hedge. In a regular futures hedge when the asset
underlying the futures contract is the same as the asset being hedged the hedge ratio
would naturally be 1.0. Thus to hedge the cost of 10,000 barrels of crude oil the hedger
would have to buy 10 futures contracts, which corresponds to that exact amount. The goal
of hedging is to minimise the price variance of the hedged position. When the asset
underlying the futures contracts is not the same as the asset being hedged, their prices are
unlikely to be 100% correlated and so applying a hedge ratio of 1.0 may not be the best
solution.
Finding the hedge ratio which produces the minimum variance in the cross-hedge
portfolio is clearly dependent on the relationship between the price of the asset underlying
the futures contracts and the price of the asset being hedged. Studies such as (Ederington,
1979) were among the earliest to utilise the technique of regressing cash and futures
prices and use the resulting x-coefficient to determine the optimal cross-hedge ratio. This
has become the conventional approach in determining optimum hedge ratio though it has
been improved upon a number of times. In (Nelson & Plosser, 1982) it’s demonstrated
that non-stationary data (i.e. cash and futures price levels) can lead to inaccurate results in
regression testing. To account for these findings (Benninga, et al., 1984) introduce the
usage log returns in regression testing as they are trend stationary and this technique is
still widely used today. Through the regression method optimum hedge ratio is acquired
by finding the x-coefficient of an ordinary least squares (OLS) regression. (Hull, 2015)
describes the formula for optimal hedge ratio ℎ∗ as
ℎ∗ = 𝜌𝜎𝑆
𝜎𝐹
Equation 1 Optimal Hedge Ratio
In Equation 1 above 𝜎𝑆 corresponds to the standard deviation of the change in spot price,
𝜎𝐹 represents the standard deviation of the change in futures price and 𝜌 is the coefficient
of correlation between the spot and the futures prices. It’s maintained within the SFAS
133 regulations that the back-tested for a period for testing hedge effectiveness should be
8
equal to the life of the hedge, thus if the hedge will last for five years, then the five years’
leading up to June 2015 will be used for back-testing. The optimal number of contacts to
be bought for hedging may be determined by Equation 2 below. In Equation 2 the 𝑁∗
variable represents the optimal number of futures contracts required for hedging, while
𝑄𝐴 is the size of the position being hedged and 𝑄𝐹 corresponds to the size of one futures
contract (Hull, 2015).
𝑁∗ = ℎ∗𝑄𝐴
𝑄𝐹 Equation 2 Optimal Number of Contracts
In another jet fuel hedging study (Adams & Gerner, 2012) research the hedging
performance of a number of commodities by analysing the annualised variance of a
portfolio containing jet fuel and a model calculated amount of the cross-hedging product.
The best performing hedge was determined by the lowest annualised variance produced
by the total profit-and-loss (P&L) series for each portfolio and this methodology is
followed again in this study. The total P&L series is a product of the P&L on each leg of
the cross hedge and is shown as Equation 3.
𝑃&𝐿𝑇𝑜𝑡𝑎𝑙 = 𝑃&𝐿𝑆𝑝𝑜𝑡 + 𝑃&𝐿𝐹𝑢𝑡𝑢𝑟𝑒𝑠 Equation 3 P&L Series
There will be a some deviations from the methodology of (Adams & Gerner, 2012)
whereby instead of using a $1,000,000 investment in jet fuel, the portfolio will instead
contain 150,000 gallons of Gulf 54 grade jet fuel as this was roughly the amount of jet
fuel consumed by Southwest Airline per month in 2014 (Transtats, 2015). Also the total
P&L will be calculated as a monetary amount rather than as a percentage change in
prices. The profit and loss on each leg of the hedge is shown in Equation 4 and Equation 5.
The P&L is calculated on either a weekly, monthly or quarterly horizon depending on the
results from the SFAS 133 analysis as this will determine the hedge horizon that each
cross-hedge commodity must operate on in order to be in line with hedge accounting
regulations. In the Equation 4 and Equation 5 below ℎ∗ represents the optimal number of
futures contracts needed to hedge the jet fuel exposure. The variables 𝐽𝑡 and 𝐹𝑡
correspond to the spot prices of jet fuel and the cross-hedging product at the time the
hedge is placed. The variables 𝐽𝑡+𝑇 and 𝐹𝑡+𝑇 correspond to the time horizon used for
hedging and are varied between 1, 3, 6, 9and 12 months for each hedge.
𝑃&𝐿𝑆𝑝𝑜𝑡 = 150,000 ∗ (𝐽𝑡
− 𝐽𝑡+𝑇) Equation 4 Profit and Loss Spot
9
𝑃&𝐿𝐹𝑢𝑡𝑢𝑟𝑒𝑠 = (𝐹𝑡+𝑇
− 𝐹𝑡) ∗ℎ∗ ∗ 150,000
𝑄𝐹
Equation 5 Profit and Loss Futures
The total returns series is a sum of Equation 4 and Equation 5 for each time period during
the back-testing window. The best performing cross-hedge will be the one which
produces the lowest annualised variance as measured by the standard deviation of the
total profit and loss series multiplied by the square root of number of periods per year.
For example the annualised variance equation for gasoil being hedged on a monthly
horizon is given in Equation 6, and the annualised variance will be calculated for each of
the three models across all time horizons.
𝐴𝑛𝑛 𝑉𝑎𝑟 = 𝑠𝑡𝑑(𝑇𝑜𝑡𝑎𝑙 𝑃&𝐿𝐺𝑎𝑠𝑜𝑖𝑙) ∗ √12 Equation 6 Annualised Variance Gasoil
The methodology for determining the optimal hedge ratio also differs from that of
(Adams & Gerner, 2012) in that the price levels are transformed to first differences
instead of log returns before being input into the OLS, ECM and ECM-GARCH models.
The reason for this approach is due to the way the P&L is calculated. Determining
optimal hedge ratio using log returns would yield erroneous P&L results as using log
returns does not capture the magnitude of the changes in monetary value for each
commodity, i.e. a positive 1% return of Gasoil does not perfectly hedge a 1% loss on jet
fuel because their prices are vastly different and this must be captured in the P&L
calculation. The final P&L methodology change from (Adams & Gerner, 2012) in how
the dynamic cross hedge ratio is calculated. In the analysis instead of calculating a new
optimal cross hedge ratio for each year based on all the previously available data, the 𝛽
value is calculated for each hedge placed during back-testing based on the most recent 5,
10 and 15 years of data to contrast the differences (if any) resulting from including more
years of data in determining optimal hedge ratio.
Hypothesis Testing & Statistical Significance
The methodology utilises a number of hypotheses tests throughout. Hypotheses are
investigated by specifying a null hypothesis and investigating whether this condition can
be accepted or rejected based on the evidence presented. In hypothesis testing the null
hypothesis is presumed to be true, until the data presented provides sufficient evidence to
show that it is not. There are two outcomes to hypothesis testing, rejecting the null and
failing to reject the null. The terminology here is important, as the null can never truly be
10
accepted, only rejected or not rejected based on the sample provided. Think in terms of a
murder trial where the null hypothesis is that the accused is innocent. There are two
outcomes, the first is that the court may be provided with sufficient evidence to reject this
hypothesis, i.e. the accused is guilty. On the other hand the trial may be provided be
insufficient evidence to reject this hypothesis. This doesn’t mean that the null hypothesis
is true and that the accused is innocent, just that there is not sufficient evidence to
disprove the null and the trial has found them not guilty i.e. it has failed to reject the null.
When carrying out a hypothesis test one must specify a significance level which also
represents the ‘size’ of the test. The test ‘size’ is the probability of falsely rejecting the
null i.e. the probability of a type I error (Alexander, 2008). A type I error is the false
rejection of a true null hypothesis, i.e. detecting an affect that is not present. There is also
a type II error which is failing to reject a false null hypothesis, i.e. failing to detect and
affect that is present. Thought of another way a type I error is finding an innocent person
guilty while a type II error is finding a guilty person innocent, and when testing a type I
error is more serious. Increasing the significance level reduces the size of the test but also
increases the probability of a type II error, thus decreasing its’ power which is a tests
ability to correctly reject the null hypothesis (Alexander, 2008).
The critical values for each hypothesis test represent the upper and lower percentiles of
the sample. Thus if the significance level is 5%, the critical values represent the points
between which 95% of the data lies. As a result choosing a lower significance level (of
say 90%) increases the tests chance of rejecting the null hypothesis but suffers from
losing statistical power making the results less convincing (Banerjee, et al., 1993). For the
null hypothesis to be rejected the test statistic must fall within the critical region (be more
extreme than the critical values) and produce a pValue which is lower than the
significance level. For a one sided hypothesis test such as the (Dickey & Fuller, 1979) test
for a unit root the test statistic must fall below the critical value.
11
Regression Model
A theoretical framework for determining the optimal futures position in a cross hedge is
described in (Anderson & Danthine, 1981) who specify that the proportion of output that
should be hedged in each contract, i.e. the cross-hedge ratio, is determined by the slope
coefficient output from regressing cash prices on futures prices. For most practical
purposes (Conroy, 2003) notes that one can simply use the spot prices for the cross-hedge
commodities in determining optimal hedge ratio and so that is the method pursued in this
paper. For practical purposes however the cross-hedge product spot prices will still be
referred to as futures prices for distinction from the jet fuel spot prices. The price levels
for each commodity represent a non-stationary array of data. The research of (Nelson &
Plosser, 1982) demonstrated that non-stationary data can lead to inaccurate results in
regression testing, and so to account for these findings (Benninga, et al., 1983) introduce
the usage log returns in place of cash and futures prices and this method is found to lead
to more accurate results when using regression methods for commodity hedging (Myers
& Thompson, 1989).
The use of regression testing also provides a measure of explanatory power between the
dependent and independent variables. The R-Squared value indicates the percentage of
variation in jet fuel returns which can be explained by the variation in the returns of the
cross-hedging products. While regression analysis has the advantage of being straight
forward to implement (Cecchetti, et al., 1988) note that this process for estimating
optimal hedge ratio suffers from major drawbacks. This research cites that no adjustment
can be made for the fact that the joint distribution of the time series varies over time and
that this important characteristic cannot be captured via the OLS approach. Other issues
are noted by (Kroner & Sultan, 1993) who show that if the time series are co-integrated
that the regression is misspecified as it involves over differencing the data and losing
information regarding the long run relationship between the spot and futures prices. This
can lead to a downward bias in hedge ratio estimation and as a result under-hedging.
Unit Root Testing
In time series analysis a trend-stationary series is one whose error term follows a
stationary process while an integrated time series is one whose error term follows a
random walk (DeJong, et al., 1992). Tests for stationarity are carried out on each of the
12
series as (Granger & Newbold, 1974) show that two integrated time series which are
completely unrelated can demonstrate an apparently significant relationship when one is
regressed on the other. The findings of (Granger & Newbold, 1974) led econometric
analysis to become interested in transformations to induce stationarity. Stationary series
have a number of favourable characteristics such as a finite variance and finite time
between crossings of the series mean, as well as this stationary series have the important
property that certain functions of the sample values converge to constants as the number
of sample values increases (Banerjee, et al., 1993). This means that for stationary time
series the mean and variance of the sample being tested converge towards the true mean
and variance of the process as the sample size is increased.
Shown below in Table 2 are the hypotheses for a unit root test where the variables display
either a stationary I(0) or a non-stationary I(1) trend. Its’ noted by (Alexander, 2008) that
if the test statistic falls outside the critical region and the null hypothesis is not rejected
that another unit root test should be performed of the first differenced time series to
analyse its order of integration. Differencing involves calculating the differences between
consecutive observations in a time series. The first difference of a time series 𝑥 at period 𝑡
is 𝑥𝑡 − 𝑥𝑡−1. If the price levels for any of the commodities analysed are found to be non-
stationary and 𝑦𝑡, the first differenced time series of that commodity, is subsequently
found to be stationary then the time series being analysed is termed to be integrated of the
order I(1). If the sample does not achieve stationarity until its second differences (the first
difference of 𝑦𝑡) then the sample is said to be integrated of the order I(2). Thus unit root
testing is not only useful in identifying stationary trends in data but also helps to identify
the order of integration of each time series.
Levels 𝐻0: 𝑋𝑡 ~ 𝐼(1) 𝐻1: 𝑋𝑡 ~ 𝐼(0)
First Differences 𝐻0: ∆𝑋𝑡 ~ 𝐼(1) 𝐻1: ∆𝑋𝑡 ~ 𝐼(0)
Table 2 Unit Root Hypotheses
As part of the methodology the Augmented Dickey-Fuller (ADF) test of (Dickey &
Fuller, 1979) will be carried out on the data to test each time series for stationarity. An
ADF test is a unit root test which returns a rejection decision based on the detection of a
unit root in the time series. The null hypothesis is that a unit root exists and the time
series is integrated, thus a rejection of the null hypothesis indicates that the time series is
trend-stationary. It is noted by (Alexander, 2008) that the ADF test is also the least
13
powerful unit root test. This is likely because the impact of type I errors have on the
results, i.e. at a 5% significance level the ADF test will incorrectly return a decision of
stationarity (where none is present) approximately once every twenty times. To account
for this the KPSS test of (Kwiatkowski, et al., 1992) will also be carried out on each time
series. The KPSS test is a stationarity test in which the null hypothesis is that the time
series is trend-stationary and it is expected the KPSS results should be consistent with the
ADF tests.
Regression Analysis
Regression testing is a process which involves evaluating the linear relationship between
variables. Regression analysis typically indicates how the value of a dependent variable
changes as a result of changes in value of some independent variable. Using this
statistical method we wish to find out how a change in the value of 𝑋 affects the value
of 𝑌, or looked at in a slightly different way, can we use the observed change in the value
of 𝑋 to predict the resulting change in the value of 𝑌. The simplest example of a linear
relationship is given by the equation of a straight line in Equation 7.
𝑌𝑡 = 𝛼 + 𝛽𝑋𝑡 + 휀𝑡 Equation 7 Linear Model
In Equation 7 above 𝑌 represents the dependent variable (jet fuel spot) and 𝑋 represents
the independent variable (commodity futures) while the intercept and slope of the
regressions line of best fit between the two are denoted by 𝛼 and 𝛽 respectively. The x-
coefficient or 𝛽 value also corresponds to the optimal hedge ratio estimated by the linear
regression. Figure 1 illustrates a regression between the first differenced series for Brent
crude oil and Gulf54 grade jet fuel. From Figure 1 it’s clear that not all of the data points
will form a perfectly straight line and as a result an error term is included in Equation 7
Linear Model which defines the distance between each data point and the regression line of
best fit. If the correlation between 𝑋 and 𝑌 is low then one would observe a high variance
in the errors which are represented in Equation 7 by 휀𝑡 while a high correlation between
regression variables would yield a low variance in 휀𝑡 (Alexander, 2008).
14
Figure 1 Jet-Brent OLS Regression Plot
Ordinary least squares (OLS) is a technique for estimating the unknown parameters of a
linear regression by minimising the regression residuals. The residuals represent the
difference between the observed value and the estimated value of the quantity of interest.
In the context of cross hedging where one is attempting to understand how changes in the
price of futures contracts can predict changes in the spot price of jet fuel, the OLS
regression will attempt to fit a linear model to the data such that the errors between the
predicted change in jet fuel spot prices and actual observed changes are minimised. The
research of (Engle & Granger, 1987) among others notes that the residual series made up of the
error terms may retain important information about the relationship between the variables.
Characteristics such as its speed of adjustment towards the line of best fit and whether the error
terms are autocorrelated remain unaccounted for in the linear regression model (Kroner & Sultan,
1993).
While regression analysis is unable to capture several time series characteristics important
in determining optimum hedge ratio the technique has been shown in some scenarios to
outperform other models such as the error correction model that captures long a short run
hedge dynamics. Using futures on the Nordic power exchange (Byström, 2003)
determines that an OLS hedge outperforms more elaborate moving average and
generalized autoregressive conditional heteroscedasticity (GARCH) hedging models. In
further support of OLS hedging methods (Lien, et al., 2002) find the OLS technique
outperforms a (vector) VGARCH model when hedging a sample of currency, commodity
15
and stock index futures. This may be related to the findings of (Stock, 1987) that if time
series are co-integrated and the 𝛼′𝑦𝑡 combination of its co-integrating vector and
logarithm vector is I(0) then a simple regression is likely to yield very consistent results
of the co-integrating vector (Baille & Bollerslev, 1994). In contrast with these findings
(Ghosh, 1993) shows that an error correction model (ECM) presents a significant
improvement over the OLS regression approach when determining optimum hedge ratios
using stock index futures as this method incorporates non-stationarity, the long run
equilibrium relationship and short-run dynamics of each time series. Similarly (Chou, et
al., 1997) find an ECM outperforms the traditional regression approach when Japans
Nikkei Stock Average index futures.
Residual Series
There are some characteristics of the residual time series resulting from the OLS
regression approach which can provide important information surrounding the suitability
of each model to fit the data. Autocorrelation refers to the cross-correlation of a signal
with itself at different points in time while heteroscedasticity refers to the mean variance
in the residual series changing over time (Engle, 1982). The influence of
heteroscedasticity and autocorrelation in the error structure (residuals) of a regression
may lead to spurious results if not accounted for in the analysis (Granger & Newbold,
1974).
In large sample sizes (Alexander, 2008) notes that the presence of autocorrelation and
heteroscedasticity will not have a damaging impact on the outcome of an OLS regression
but that in a small sample size where autocorrelation and heteroscedasticity are detected a
generalised least squares approach should be used instead of the ordinary least squares
method. This has consequences surrounding the hedge horizon used for testing as if hedge
horizon is increased past a monthly basis in order to meet hedge accounting standards, it
will greatly reduce the amount of data available for testing which could potentially
increase the impact that autocorrelation and heteroscedasticity can have on the results.
Autocorrelation
Autocorrelation is the linear dependence of a variable with itself at two different points in
time, or looked at in another way an autocorrelated time series is one which exhibits
similarity with a lagged version of itself. A basic assumption underlying the OLS
16
regression is that the error terms are independent and once this assumption is satisfied the
OLS procedure is valid whether or not the time series themselves are serially correlated
(Durbin & Watson, 1950). From this we can deduce that it is not important to test the
individual commodity time series for autocorrelation but instead need to analyse the
residual series produced by each OLS regression. The presence of autocorrelation is a
problem in the least squares approach because it indicates that the error terms are not
independent and this may lead to the variance of the least square estimators in the
regressions to be too large (Anderson, 1954). The presence of autocorrelation may require
modification to the usual methods of estimation and prediction and (Cochrane & Orcutt,
1949) note that when estimates of autoregressive properties are based on autocorrelated
residuals there is a large bias towards randomness. This implies that the presence of
autocorrelation in the residual series may substantially impact the results derived from the
ECM and ECM-GARCH models which incorporate residual information, or rather it can
diminish the advantage the ECM and ECM GARCH models have over the OLS
regression because the presence of autocorrelation muddles the useful information
contained in the residuals.
Heteroscedasticity
Heteroscedasticity refers to sub-populations in a collection of random variables having
different variability’s from others, which translates to the time series demonstrating
different variances over time. In a residual series this translates to serially uncorrelated
residual errors displaying a constant unconditional variance but a non-constant
conditional variance (Engle, 1982). Before the autoregressive conditional
heteroscedasticity (ARCH) model proposed by (Engle, 1982) most econometric models
assumed constant variance. The defining feature of an autoregressive conditional
heteroscedastic (ARCH) process is the specification of a non-constant variance which is
conditional on past variance levels. Under ARCH the conditional variance of the current
error term is dependent on its’ realised past values and (Engle, 1982) notes this model has
the advantage of reflecting clustering between large and small error terms (i.e. volatility
clustering).Thus in an ARCH process periods of low volatility tend to be followed by
periods of low volatility while periods of high volatility tend to be followed by further
periods of high volatility.
17
Error Correction Model
The error correction model (ECM) represents an extension of the OLS regression and
operates on the premise that important information that predicts the future evolution of
the time series under analysis is retained in the residual time series. As a result the ECM
incorporates a lagged version of the residual errors in order to capture elements of the
short term relationship between the variables. The ECM also accounts for the long run
equilibrium relationship between the time series through considering the presence of a co-
integration relationship between the variables. The research of (Engle & Granger, 1987)
shows that if the linear combination of two non-stationary variables forms a stationary
time series, then the equilibrium condition and adjustment process to equilibrium can be
represented by an error correction model (Ghosh, 1993). The incorporation of elements
that capture some of the short and long run dynamics of the time series accounts for a
number of the disadvantages of the OLS regression noted by (Kroner & Sultan, 1993).
Co-Integration
Integrated time series are non-stationary and tend to drift randomly over time, yet some
pairs of integrated series may be defined by some long-run relationship to which the
system converges to over time (Banerjee, et al., 1993). The research of (Engle & Granger,
1987) makes note that economic time series can wander substantially and yet some series
are expected to move so that they do not drift too far apart. If two time series are co-
integrated it will be observed that over time they will tend not to drift too far apart and
that the distance between them (i.e. their residual series) will follow a stationary trend that
is normally distributed around some mean value. The presence of co-integration suggests
a long run memory between the co-integrated time series. While the series may diverge
substantially in the short run and the effects of shocks may only vanish over long time
horizons, pairs of co-integrated series will always display a tendency over time to drift
back towards their equilibrium relationship (Baille & Bollerslev, 1994). In this respect the
drift between co-integrated series must be stochastically bounded and, at some point,
diminishing over time (Banerjee, et al., 1993). In terms of the time series which represent
jet fuel and each of the cross-hedge commodity prices, carrying out tests for co-
integration will indicate whether or not their price series are linked in the long run and
forms the basis of fitting an error correction model to the data (Stock & Watson, 1988).
18
“The components of the vector 𝑥𝑡 are said to co-integrated of order 𝑑, 𝑏, denoted
𝑥𝑡 ~ 𝐶𝐼(𝑑, 𝑏), if (i) all components of 𝑥𝑡 are I(d); (ii) there exists a vector α (≠0) so that
𝑧𝑡 = 𝛼′𝑥𝑡 ~ 𝐼(𝑑 − 𝑏), b > 0. The vector α is called the co-integrating vector”
To make sense of the co-integration definition above offered by (Engle & Granger, 1987)
consider the components of the vector 𝑥𝑡 are two time series 𝑗𝑡 and 𝑓𝑡 which represent the
spot price of jet fuel and the futures price of the cross-hedging commodity. Assume the
unit root tests carried out determine that 𝑗𝑡 and 𝑓𝑡 are both I(1) integrated variables which
fulfils the first condition in the above definition. It is noted by (Engle & Granger, 1987)
that for most observations 𝛼′𝑥𝑡 will not be in equilibrium and that there exists a univariate
quantity outlined above as 𝑧𝑡 called the equilibrium error. If there exists some vector α
such that the residual error series 𝑧𝑡 formed by a linear combination of α with 𝑥𝑡 is I(0),
or trend-stationary, then the second condition is fulfilled and time series 𝑗𝑡 and 𝑓𝑡 are said
to be co-integrated.
In other words co-integration implies that deviations from equilibrium are stationary with
finite variance even though the series themselves may be non-stationary with infinite
variance. Thus if 𝑧𝑡 represents the error or distance from 𝛼′𝑥𝑡 to the long run equilibrium,
it can be used to independently verify the existence of co-integration between time series
𝑗𝑡 and 𝑓𝑡 by testing whether the residuals derived from a regressing 𝑗𝑡 on 𝑓𝑡 display
stationarity.
It’s shown by (Ghosh, 1993) that smaller than optimal futures positions are taken when
the effect of co-integration is omitted from hedging models. A number of fuel hedging
strategies are described by (Cobbs & Wolf, 2004) however they fail to acknowledge a co-
integration relationship between the jet fuel and any of the cross-hedging products as
being significant though its importance is recognised in the research by (Adams &
Gerner, 2012) on which this study is based. It’s inclusion in the analysis is deemed
essential as “co-integration is the only true indispensable component when comparing ex-
post performance of various hedge strategies” (Da-Hsiang, 1996). There are a number of
tests for co-integration which may be pursued in this study, however the simple two-step
estimation of (Engle & Granger, 1987) is followed for a number of reasons. The two-step
estimation has a very practical advantage of being the most straight-forward of co-
integration tests and is easily implemented in MATLAB. Alternative cointegration testing
methods are available such as the vector auto-regression (VAR) method of (Johansen,
19
1991) which allows for testing of co-integration rank and seasonal dummies among the
variables. However due to each potential co-integration relationship containing only a
pair of variables, the more complex approach associated with the VAR method of
(Johansen, 1991) is rejected in favour of the more straight forward two-step approach of
(Engle & Granger, 1987). The effect of co-integration when hedging jet fuel futures is
studied by (Adams & Gerner, 2012) who’s research found that an error correction model
(ECM) outperforms an OLS regression when cross-hedging jet fuel with a number of
crude oil based commodities.
Error Correction Analysis
Error correction terms may be used as a way of capturing adjustments in a dependent
variable which depended not on the corresponding level of some explanatory variable but
on the extent to which the explanatory variable deviated from its equilibrium relationship
with the dependent variable (Banerjee, et al., 1993). In other words, assuming that some
long run equilibrium relationship is present between the time series (which is why co-
integration is a prerequisite), an error correction model (ECM) may be used to estimate
adjustments in jet fuel prices based on how its time series has deviated from its’
equilibrium relationship with the cross-hedging product. It’s noted by (Engle & Granger,
1987) that for a two variable system a classic error correction model would relate the
changes in one variable to past equilibrium errors as well as past changes in both
variables. (Kroner & Sultan, 1993) note that regression models can obscure the long run
relationship between spot and futures prices while (Adams & Gerner, 2012) note that the
ECM framework accounts for the autoregressive structure of spot and forward price
changes and thus incorporates the long run relationship between both markets. It can be
viewed such that if an OLS regression estimates the linear relationship between the first
differences for jet fuel and the hedging commodity, then the error correction model estimates the
dynamic structure between the first differenced series by incorporating short and long run
information about their co-integrated relationship (Alexander, 2008). The model operates on the
idea that the short term deviations from the long run equilibrium relationship will be subsequently
corrected by the time series on its own.
The error correction model employed in the methodology is the same one employed by
(Adams & Gerner, 2012) with the substitution of first differences for log returns and takes
the form
20
∆𝑓𝑑𝑆𝑡 = 𝑐 + 𝛽∆𝑓𝑑𝐹𝑡,𝑇 + ∑ 𝛾𝑘∆𝑓𝑑𝐹𝑡−𝑘,𝐾
𝐾
𝑘=1
+ ∑ 𝛿𝑙∆𝑓𝑑𝑆𝑡−𝑙
𝐿
𝑙=1
+ 𝜆𝑒𝑡−1 + 휀𝑡
Equation 8 Error Correction
Model
Where ∆𝑓𝑑𝑆𝑡 and ∆𝑓𝑑𝐹𝑡,𝑇 represent the changes in the first differenced price series, 𝑐 represents
the intercept and 𝛾𝑘 and 𝛿𝑙 signify the short run dynamics coming from the lagged changes in the
spot price (hence 𝐹𝑡−𝑘,𝐾). The 𝜆 term represents the adjustment parameter which captures the
speed at which each time series reacts to deviations from the long run equilibrium relationship
and 𝑒𝑡 is the error term, which comes in lagged form, as each time series must first deviate from
the long run relationship before beginning its correction process. The 𝑒𝑡 term is formed by the
residual series from an OLS regression of the differenced series for jet fuel on the differenced
series for the chosen cross-hedge commodity. Finally 휀𝑡 represents the standard error while 𝛽
represents the optimal hedge ratio.
The ECM analysis will be carried out using the same rolling window as the OLS regression and
will convert the residual series produced by each loop of the OLS regression model to lagged
form for use in the ECM, as shown in the MATLAB Dynamic Hedge Ratio section of
Appendix B. Like the OLS model the ECM will be run on either a weekly, monthly or quarterly
hedging horizon based on the SFAS 133 analysis for each commodity. The model itself is based
on data provided by (Adams, 2015). Since it’s noted by (Kroner & Sultan, 1993) that an OLS can
result in downward bias which leads to under hedging we may observe a slightly higher optimal
hedge ratio output from the error correction model when compared to the ordinary least squares
regression model.
The representation theorem of (Granger, 1986) specifies that that when integrated
variables share a co-integration relationship that a vector autoregressive (VAR) model of
first differences will be misspecified as the disequilibrium term is missing from the vector
autoregressive representation. This term is also missing from the more simplified OLS
regression Equation 7 outlined above. It’s noted by (Alexander, 2008) that when lagged
disequilibrium terms are included as explanatory variables that the model becomes well
specified. As shown above in Equation 8 the error correction framework accounts for the
misspecification of OLS regression models and VAR models by including a lagged error
correction term in order to incorporate the previous periods’ equilibrium error (Ghosh,
1993). The model also includes additional information about the lagged series through a
linear combination of the lagged values of the first differenced time series for each
commodity with a term that estimates the short run dynamics.
21
GARCH Extension
The OLS regression has been described by (Engle, 2001) as the “great workhorse of
applied econometrics” however he notes that when attempting to estimate and examine
the size of the errors within a model that the questions are about volatility, for which the
standard tools are the ARCH and GARCH models. Generalised autoregressive
conditional heteroscedasticity (GARCH) is a generalisation of the ARCH process
discussed earlier whereby the variance of the error terms is assumed to follow an
autoregressive moving average (ARMA) process. GARCH models are thus an extension
of the non-constant conditional variance approach of ARCH models whereby the
conditional variance at any one time is based on a weighted sum of its passed values. The
GARCH model of (Bollerslev, 1986) is generally considered superior because it can
account for the time varying variance of the spot and futures prices during the life of the
hedge and can also account for price shocks. While the classic error correction model
incorporates the cointegrated relationship and lagged residuals in order to capture the
short and long run dynamics between the time series, the GARCH extension substitutes
the lagged residual series for GARCH residuals in order to capture the non-constant
conditional variance of the residual time series as well as the short run dynamics.
GARCH Analysis
The error correction model employed by (Adams & Gerner, 2012) also specifies a
GARCH extension to obtain greater accuracy and they state that a univariate GARCH
should be sufficient to model the optimal hedge ratio. In contrast studies such as (Kroner
& Sultan, 1993) and (Baillie & Myers, 1991) use a bivariate GARCH model as they state
that in the case of two I(1) variables being co-integrated so that their linear combination is
I(0) then they should be represented by a bivariate model that includes an error correction
term. For the purpose of simplicity a univariate GARCH approach is taken where the
non-constant conditional variance is estimated using Equation 9. In order to incorporate
GARCH residuals into the model the lagged residual series used in the regular ECM will
be transformed to its GARCH representation using Equation 9 and then analysed in the
same way as the regular ECM.
ℎ𝑡2 = 𝜇 + 𝜃1휀𝑡−1
2 + 𝜃2ℎ𝑡−12 Equation 9 Conditional Variance
22
In a study based on exchange rates (Hansen & Lunde, 2005) find that there is no evidence
to suggest that a GARCH(1,1) model is outperformed by over 330 other ARCH-type
models in its ability to model conditional variance. Dynamic hedging using an error
correction model with a GARCH error structure was carried out by (Kroner & Sultan,
1993) using a GARCH(1,1) model while (Adams & Gerner, 2012) also employ the this
type of model for capturing non-constant condition variance. For these reasons a
GARCH(1,1) will be fitted to the data in order to compute the GARCH residuals. In a
GARCH(1,1) model the conditional variance is based on the most recent observation of
the residuals and the most recent estimate of the variance rate as shown by the 𝑡 − 1
subscript for the variables in Equation 9. The GARCH model will be fitted using the
MATLAB ‘estimate’ function which estimates the GARCH(1,1) model via maximum
likelihood methods. As with the lagged residuals of the ECM the ECM-GARCH will be
based off a new set of OLS residuals generated with every loop of the model. Once the
GARCH model has been fitted using the MATLAB function the residuals will be
transformed using Equation 9. Each residual may be represented as √ℎ𝑡2 and will be
substituted for the lagged error terms in the ECM model.
Basis Risk
Basis risk is described by (Figlewski, 1984) as the varying nature of the difference
between the spot and futures prices and that hedging using futures exposes the position to
this type of risk since the evolution of the futures price over time may not match the
change in value of the cash position. Since this basis may be represented by the residual
series between the spot and futures prices, then the error correction models incorporation
of the lagged error term as well as the adjustment parameter should enable an ECM to
better account for the evolution of basis risk when compared to an OLS regression. It’s
observed by (Figlewski, 1984) that this type of risk increases as the length of the hedging
horizon decreases so it’s likely that if the hedge horizon must be increased in order to
meet hedge accounting standards as per (Juhl, et al., 2012) that this will have the effect of
reducing basis risk for all of the models analysed.
23
Results & Discussion
SFAS 133
Time (y) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
Brent 0.886 0.849 0.807 0.804 0.775 0.792 0.785 0.794 0.810 0.821 0.828 0.843 0.792 0.653 0.658 0.658
WTI 0.768 0.722 0.683 0.675 0.638 0.659 0.644 0.623 0.648 0.669 0.685 0.702 0.487 0.438 0.451 0.457
Gas 0.715 0.730 0.731 0.733 0.742 0.754 0.753 0.775 0.789 0.799 0.814 0.825 0.758 0.672 0.681 0.681
Heat 0.485 0.523 0.523 0.560 0.569 0.617 0.641 0.685 0.720 0.746 0.772 0.793 0.806 0.720 0.729 0.732 Table 3 R-Square Value based on Weekly Hedging Horizon
Time (y) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
Brent 0.976 0.945 0.930 0.918 0.912 0.907 0.887 0.894 0.897 0.904 0.902 0.915 0.883 0.908 0.902 0.901
WTI 0.745 0.717 0.698 0.683 0.657 0.711 0.673 0.669 0.666 0.689 0.700 0.740 0.710 0.776 0.777 0.776
Gas 0.886 0.883 0.879 0.872 0.873 0.877 0.870 0.879 0.889 0.890 0.893 0.898 0.878 0.904 0.906 0.907
Heat 0.759 0.775 0.741 0.737 0.702 0.733 0.737 0.755 0.769 0.787 0.798 0.817 0.834 0.871 0.873 0.874 Table 4 R-Square Value based on Monthly Hedge Horizon
Time (y) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
Brent 1.000 0.962 0.953 0.951 0.945 0.942 0.947 0.943 0.942 0.942 0.942 0.952 0.945 0.951 0.950 0.951
WTI 0.949 0.957 0.881 0.867 0.862 0.853 0.790 0.761 0.757 0.757 0.765 0.802 0.858 0.877 0.876 0.877
Gas 0.947 0.948 0.948 0.942 0.943 0.949 0.953 0.953 0.962 0.960 0.961 0.965 0.968 0.972 0.972 0.972
Heat 0.982 0.971 0.975 0.947 0.938 0.930 0.933 0.934 0.942 0.944 0.942 0.947 0.966 0.969 0.970 0.969 Table 5 R-Square Value based on Quarterly Hedge Horizon
24
Table 3, Table 4 and Table 5 on the previous page present the results of simple regression
analysis on the log returns series for jet fuel on the log returns series for each of cross-hedge
commodities. The R-Square values of each regression are presented as these values may be
presented as a means to qualify for hedge accounting status. It is recommended by (FASC,
2008) that the back-testing period for each commodity matches the hedge horizon and so the
times in years across the top of each of the tables correspond to horizons extending
backwards from June 1st 2015. Thus the results presented as 0.5 years correspond to an R-
Square derived from using data from January 1st 2015 to June 1
st 2015, 3 year results
correspond to testing from June 1st 2012 to June 1
st 2015 etc. The threshold for qualifying for
hedge accounting is an R-Square value of 0.8 and in Table 3, Table 4 and Table 5 the R-Square
results which meet this criteria are highlighted in green. Each commodity was analysed on a
hedge horizon ranging from 0.5 to 8 years and on a weekly, monthly and quarterly time
horizon. It’s important to bear in mind that the available hedge horizon for each commodity
varies under SFAS regulations varies due to differences in the availability of futures on each
cross hedge product. The time horizons are shown in Table 1 and as a reminder are 6.5 years
for Brent and WTI, 5.5 years for gasoil and 3.5 years for heating oil. Figure 2 illustrates the
results from the weekly analysis and it’s clear from the results that none of the commodities
can consistently meet (FASB, 1998) hedge accounting standards on a weekly time horizon.
Figure 2 SFAS Analysis Weekly Time Horizon
Figure 2 illustrates the data presented in Error! Reference source not found. and offers a
visual breakdown on the ability of each of the cross-hedge portfolios to achieve hedge
accounting status when used to hedge jet fuel. The R-Square values on the x-axis begin at 0.4
0.400
0.500
0.600
0.700
0.800
0.900
1.000
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
R-S
qu
are
Hedge Horizon (years)
SFAS Weekly Horizon
Brent WTI Gas Heat SFAS
25
while the commodity data is analysed on a weekly time horizon and for a hedge horizon
stretching from 0.5 to 8 years. The weekly price correlation between the commodities jumps
significantly between year 7 and year 6 (2008 and 2009) and while Brent and West Texas
Intermediate see their weekly price correlations rise overall by 2015, gasoil and heating oil
see their price correlations drop off slightly in recent years. The increase in the returns
correlations across all commodities during the time of the financial crisis is consistent with
the findings of (Silvennoinen & Thorp, 2013) who found that returns correlations between
commodities increased significantly during the financial crisis and is consistent with the
theory that asset returns are more highly correlated in a market downturn. Of the four cross-
hedging products analysed on a weekly basis only Brent and gasoil meet hedge accounting
criteria at any stage during the test period and neither do it on a consistent basis. As a results
none of the cross hedging products will be brought forward for profit and loss testing on a
weekly basis as hedging jet fuel with either Brent, WTI, gasoil or heating oil on this time
horizon can qualify for hedge accounting status.
Figure 3 SFAS Analysis Monthly Time Horizon
Figure 3 presents the R-Square analysis of the hedging commodities based on a monthly time
horizon and it’s clear from the figure that there is a significant improvement in meeting the
SFAS criteria when compared to the weekly analysis. On a monthly time horizon only Brent
crude oil and low sulphur gasoil achieve a consistently high R-Square when regressed against
jet fuel for hedge horizons extending from 0.5 to 8 years. According to the data Brent has had
a higher correlation than Gasoil with Gulf54 grade jet fuel in recent years. The results also
show that the correlation of returns between jet fuel and Brent has been increasing in recent
0.400
0.500
0.600
0.700
0.800
0.900
1.000
1.100
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
R-S
qu
are
Hedge Horizon (years)
SFAS Monthly Horizon
Brent WTI Gasoil Heat Oil SFAS
26
years while the R-Square for gasoil and jet fuel has remained relatively constant. In contrast
to the weekly findings three of the four commodities demonstrate a dip in correlation around
2008 which indicates that the drivers of monthly returns on jet fuel and the cross-hedging
products became inconsistent during the financial crisis due to the high levels of volatility in
the market. There is a significant fall in correlation between jet fuel and heating oil during the
past seven years and this trend is consistent with the weekly heating oil analysis in Figure 2,
while WTI does not meet the 0.8 R-Square criteria at any stage during testing when using a
monthly time period. The results from the monthly hedge analysis are significant enough to
carry Brent and gasoil forward for profit and loss analysis on a monthly time horizon while
WTI and Heatoil will be retained for quarterly analysis.
Figure 4 SFAS Analysis Quarterly Time Horizon
Although only WTI and heating oil were required to be analysed the other commodities are
included for comparison. Figure 4 shows the results for the quarterly horizon R-Square
analysis for each of the commodities and illustrates how three of the four cross hedging
products achieve a consistently high correlation over the 8 year hedge horizon. For Brent,
gasoil and heating oil there is a distinct trend of higher correlations across the board as well
as reduced volatility in R-Square estimates as we move from a weekly to a monthly and
finally to a quarterly time horizon. This is consistent with the findings of (Juhl, et al., 2012)
who found that extending the time horizon for a hedge increases its estimated R-Squared
value. On a quarterly basis Heatoil falls well within the acceptable range to qualify for hedge
accounting while WTI fails to meets the criteria about 30% of the time. While West Texas
Intermediary crude oil fails to consistently meet the accepted measure of hedge effectiveness
0.400
0.500
0.600
0.700
0.800
0.900
1.000
1.100
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
R-S
qu
are
Hedge Horizon (years)
SFAS Quarterly Horizon
Brent WTI Gasoil Heat Oil SFAS
27
it has demonstrated a high returns correlation with jet fuel for very recent observations. As a
result rather than exclude it completely from profit and loss analysis WTI will instead have
its max hedge horizon reduced from 6.5 to 3.5 years so that quarterly profit and loss back-
testing will only occur for the number years in which WTI could meet hedge account
standards.
Unit Root Testing
The results of the Augmented Dickey Fuller (ADF) tests for a unit root and Kwiatkowski
Phillips Schmidt Shin (KPSS) tests for stationarity are presented below. The tests are carried
out twice for each commodity, first on the levelled time series (regular prices) and then on the
first differenced time series (price changes). Note that the differenced time series does not
correspond to the log price return and represents a different transformation of the data which
is consistent with (Dickey & Fuller, 1979). The ‘h’ variable in the following tables represents
a logical value of 1 (TRUE) or 0 (FALSE) based on a rejection decision on the null
hypothesis for each test. A result of TRUE indicates the null is rejected while a result of
FALSE indicates the tests failure to reject the null. The default significance level for both
tests was left unchanged at 5%. Unlike the SFAS analysis which lists results for each hedge
commodity (as each had its correlation measured with jet fuel), the unit root tests are carried
out on each of the individual commodities time series including jet fuel. Daily data was used
for the period June 1st 2000 to June 1
st 2015.
Jet Brent WTI Gasoil Heat Oil
h FALSE FALSE FALSE FALSE FALSE
pValue 0.4814 0.5384 0.4921 0.5836 0.5481
stat -0.4657 -0.3099 -0.4363 -0.1866 -0.2836
cValue -1.9416 -1.9416 -1.9416 -1.9416 -1.9416
Table 6 Augmented Dickey Fuller Test Levelled Series
The results of the ADF tests carried out on the levelled time series for each commodity are
presented in Table 6. The results show the tests failure to reject the null hypothesis of a unit
root against the autoregressive alternative which indicates that there is not sufficient evidence
to dismiss the null hypothesis and thus each series must be considered non-stationary. The
failure to reject the null is indicated by the test statistic (stat) falling within the critical values
(cValue) in each test while the results are also not significant as indicated by the pValues for
each test lying above the default confidence interval. The high pValues indicate a strong
28
decision of failure to reject the null hypothesis for each of the commodities with gasoil price
levels demonstrating the strongest presence of a unit root and jet fuel levels the weakest.
Jet Brent WTI Gasoil Heat Oil
h TRUE TRUE TRUE TRUE TRUE
pValue 0.0100 0.0100 0.0100 0.0100 0.0100
stat 20.1403 17.7791 22.5057 19.3080 16.8177
cValue 0.1460 0.1460 0.1460 0.1460 0.1460
Table 7 Kwiatkowski Phillips Schmidt Shin Test Levelled Series
The null hypothesis tested in the KPSS analysis is that the time series data is trend stationary,
i.e. rejection of the null indicates a non-stationary trend in the data. Table 7 shows the results
of KPSS tests performed on the same sets of price level data as the initial ADF tests. The
results indicate a rejection of the null hypothesis of trend stationarity which is consistent with
the findings of the ADF tests in Table 6. The results are significant as indicated by the pValue.
From these results we can determine that the levelled time series are non-stationary and
integrated of order 𝐼(𝑑), where 𝑑 ≠ 0.
Jet Brent WTI Gasoil Heat Oil
h TRUE TRUE TRUE TRUE TRUE
pValue 0.0010 0.0010 0.0010 0.0010 0.0010
stat -61.8583 -64.7162 -64.5304 -59.8894 -63.4317
cValue -1.9416 -1.9416 -1.9416 -1.9416 -1.9416
Table 8 Augmented Dickey Fuller Test Differenced Series
Once the levelled time series analysis was complete the price data for each commodity was
transformed to first differences and retested in order to determine the order of integration of
each time series. The results of ADF tests on the first differences of each time series are
presented in Table 8 and the analysis indicates that the first differenced time series for each
commodity are trend stationary. There is a strong rejection of the null hypothesis of a unit
root based on the test statistic and the results are highly significant as indicated by the very
low pValue. The test statistics show that the first differenced series of Brent and WTI
demonstrate the strongest stationary trend while the first differenced time series for gasoil
demonstrates the weakest stationary trend which is consistent with the previous findings that
gasoil prices displayed the greatest evidence of a unit root before the data transformation.
29
Jet Brent WTI Gasoil Heat Oil
h FALSE FALSE FALSE FALSE FALSE
pValue 0.1000 0.1000 0.1000 0.1000 0.1000
stat 0.0516 0.0814 0.0537 0.0850 0.0643
cValue 0.1460 0.1460 0.1460 0.1460 0.1460
Table 9 Kwiatkowski Phillips Schmidt Shin Test Differenced Series
Finally KPSS tests were carried out again using the first differenced time series data and the
results shown in Table 9 indicate a failure to reject the null hypothesis for all commodities.
The failure to reject the null shows that there is not enough evidence to suggest each time
series is not trend stationary, however the pValues for the test results shown in Table 9
indicate that they are not significant and carry a higher than acceptable probability of a type I
error, i.e. the pValues suggest there is a 10% chance that the analysis has falsely rejected the
null and so reduces our confidence in the findings. Although the KPSS tests on the first
differenced time series carry a higher than usual probability of a Type I error the highly
significant results from the Augmented Dickey Fuller tests are deemed strong enough to
suggest that the first differenced time series for each commodity are integrated of the order
I(0). From this we can deduce that the levelled time series is integrated of the order 𝐼(𝑑),
where 𝑑 = 1 as it took just one difference of the data for it to to become trend stationary.
Residual Diagnostics
The residual time series derived from the OLS regression models have a number of
characteristics which we are interested in analysing and the results of tests for stationarity,
autocorrelation and heteroscedasticity are presented in Table 10, Table 11 and Table 12. Due to
the lowering of confidence in some of the KPSS test results received previously, the KPSS
tests for stationarity are replaced by the Phillips-Perron test for a single unit root in an attempt
to achieve more statistically significant results.
ADF PP
Brent WTI Gas Heat Brent WTI Gas Heat
h 1 1 1 1 1 1 1 1
pValue 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001
stat -22.403 -12.107 -26.309 -16.686 -22.403 -12.107 -26.309 -16.686
cValue -1.942 -1.944 -1.942 -1.944 -1.942 -1.944 -1.942 -1.944
Table 10 Residual Series Stationarity Tests
Table 10 presents the results of stationarity tests performed on the residual time series formed
from regressing the first differenced time series of Brent, WTI, gasoil and heating oil on the
first differenced time series for jet fuel, and when discussing to each of the commodities in
30
the results, we are referring to its residual series with jet fuel. The time horizons for the
residual series are consistent with the SFAS analysis – Brent and gasoil residuals are obtained
from the monthly data while WTI and heating oil residuals are derived from the quarterly
data. Both tests reject the null hypothesis of a unit root in residual series for each cross-hedge
and have highly significant results. Gasoil exhibits the most extreme test statistic which
suggests that its residuals offer the greatest evidence of being trend stationary when
compared to the other commodities. WTI and heating oil offer the least evidence of
stationarity though this is to be expected since there is less data being analysed as testing is
performed on quarterly data. As mentioned earlier the trend stationary results for each cross-
hedge provide a strong indication that each of the commodities series are cointegrated with
jet fuel.
Ljung-Box Q (LBQ) tests detect autocorrelation within the residual time series and their
results are illustrated in Figure 5 for Brent and gasoil and in Appendix A for WTI and heating
oil. Under the LBQ test the null hypothesis is that the series exhibits no autocorrelation for a
fixed number of lags. The LBQ tests results for all commodities are also presented in the left
panel of Table 11 and reject the null hypothesis for all commodities which indicates that there
is significant evidence of autocorrelation within each of the residual series. This is consistent
with the plots in Figure 5 and Figure 9 which illustrate the autocorrelation functions falling
outside of the confidence bounds for all commodities. The low pValues for each test present a
strong rejection decision of the null which indicates that we can be confident that there is
significant evidence of autocorrelation in each of the residual series analysed.
Figure 5 Autocorrelation Plots Brent Gasoil
31
Gasoil exhibits a much higher degree of autocorrelation than Brent based on the pValues in
Table 11. This is not obvious in the plots in Figure 5 which illustrates the presence of
autocorrelation in Brent in up to 6 lags, though only the first 2 lags for gasoil. The results
suggest that the information retained in the residuals which will be utilised by the ECM and
ECM-GARCH models may be less useful than first thought as the residual series are subject
to being biased towards randomness since the data was not transformed to account for the
effect of autocorrelated errors. The gasoil hedge is most at risk followed by the Heatoil hedge
based on the pValues from the LBQ analysis.
LBQ ARCH
Brent WTI Gas Heat Brent WTI Gas Heat
h 1 1 1 1 1 1 1 0
pValue 4.5E-03 1.6E-03 5.2E-10 1.8E-05 5.5E-04 1.8E-09 0.0E+00 1.9E-01
stat 40.384 43.822 85.116 57.410 47.173 81.983 140.819 25.215
cValue 31.410 31.410 31.410 31.410 31.410 31.410 31.410 31.410
Table 11 Residual Series LBQ & ARCH Analysis
The right hand panel of Table 11 also lists the results of tests for autoregressive conditional
heteroscedasticity (ARCH). The ARCH testing was carried by performing LBQ tests on the
squares of the residual series. The rejection decisions for Brent, WTI and gasoil suggest that
there are significant ARCH effects in their respective residual series and the extremely low
pValues suggest the rejection decision is supported by strong evidence. The analysis fails to
detect sufficient evidence of ARCH effects in the heating oil residuals which suggests that the
ECM-GARCH model may not hold an advantage over the other models when applied to
hedging the jet-heating oil portfolio.
The results detecting the existence of autocorrelation and ARCH effects in the residuals for
the cross hedges involving Brent, WTI and gasoil indicates the presence of a short run
relationship between successive terms in each of their residual time series. The short run
relationship is that the magnitude of the variances in one period effect the variance levels in
the periods immediately after. The strong detection of ARCH effects in these commodities
suggest that the ECM-GARCH model which can capture these effects may be better able to
predict the optimal cross-hedge ratios for the Brent, WTI and gasoil portfolios.
32
HETEROSCEDASTICITY
Brent WTI Gas Heat
h 1 1 1 1
pValue 5.05E-08 0.000206 1.11E-16 1.42E-05
stat 29.69743 13.7722 69.19904 18.84536
cValue 3.841459 3.841459 3.841459 3.841459
Table 12 Residual Series Heteroscedasticity Analysis
Finally tests for residual heteroscedasticity indicate that the null of no residual
heteroscedasticity at the default 5% significance level should be rejected in favour of an
ARCH(1) model, which is consistent with the findings presented in Table 11. All
commodities reject the null hypothesis of no residual heteroscedasticity. The extremely low
pValues for Brent and gasoil suggests that these residual series display substantial evidence
of ARCH effects while WTI and heating oil also strongly reject the null hypothesis. Each
commodity had a GARCH model fitted for 1 to 4 lags and the best fit determined. In each
case 1 lag provided the best fit for the data. This is consistent with the LBQ and ARCH
analysis and indicates that a GARCH(1,1) model would provide the best fit for the data which
supports the assertions made in the literature review and methodology section.
Cointegration Analysis
The Engle-Granger cointegration test investigates the null hypothesis that no cointegration
relationship exists between Gulf54 grade jet fuel and each of the cross-hedging products
analysed. The analysis is undertaken as a two-step procedure and the pair of results for each
cross-hedge presents the findings from (i) a static regression on the levels of the variables and
(ii) a regression in an error corrected form. The analysis tests for cointegration between the
levelled time series as we are concerned with establishing a cointegration relationship
between two time series integrated of order I(1).
Brent WTI Gasoil Heatoil
h TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
pValue 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001
stat -6.629 -76.869 -5.745 -49.944 -13.938 -237.268 -8.545 -85.315
cValue -3.357 -20.195 -3.398 -19.414 -3.357 -20.195 -3.398 -19.414
Table 13 Cointegration Analysis
All results in Table 13 reject the null of no cointegration and the results in all cases are
significant. From this we can determine that each commodity time series shares a
cointegrated relationship with jet fuel. Although the pValues for these tests cannot go below
0.001, the level to which each test rejects the null hypothesis are indicated by the test
33
statistics. From the results gasoil most strongly rejects the null hypothesis of no cointegration
while WTI offers the weakest rejection. The strongest evidence of cointegration is presented
by gasoil and jet fuel and consistent with the close price relationship they share as illustrated
in Figure 6 below. The results are also consistent with the results from SFAS 133 hedge
effectiveness testing which found that WTI shares the lowest correlation with jet fuel out of
the four cross hedging products.
Figure 6 Commodity Spot Prices 2009-2015
The results show that there is a long run equilibrium relationship between the variables and
that the error terms represented by the residual series between jet fuel and each of the cross
hedge products prices form a trend stationary processes. This reinforces the results obtained
in the residual series analysis which also determined the residual series in each of the cross
hedges is stationary. The cointegration testing also yielded the speed of adjustment
parameters for the time series shown in Equation 8. The results imply that gasoil heating oil
corrects towards their long run equilibrium relationships more quickly than the other
commodities and suggests that the effect of price shocks would persist longer in the Brent
and WTI time series. The results also suggest that the OLS regression may yield closer results
to the ECM models for gasoil and heating oil as the higher adjustment parameter would mean
the residuals would yield a lower variance from the long run equilibrium relationship over
time.
0
200
400
600
800
1000
1200
0
50
100
150
200
250
300
350
400
05
/09
08
/09
11
/09
02
/10
05
/10
08
/10
11
/10
02
/11
05
/11
08
/11
11
/11
02
/12
05
/12
08
/12
11
/12
02
/13
05
/13
08
/13
11
/13
02
/14
05
/14
08
/14
11
/14
02
/15
05
/15
Gas
oil
Spo
t
Jet
& B
ren
t Sp
ot
Time
Jet Fuel vs Gasoil Spot Price 2009-2015
JET BRENT GAS
34
Brent WTI Gasoil Heatoil
0.0141 0.0150 0.0533 0.0317
Table 14 ECM Residual Speed of Adjustment
The analysis now has results which indicate that information which contributes to the
evolution of the time series is retained in the process through both short and long run
relationships. This suggests that the ECM and ECM-GARCH models, which capture effects
of both the short and long run relationships within the data, should be better able to predict
the future evolution of the time series than an OLS regression model which cannot capture
these effects.
Optimal Hedge Ratio
Optimal hedge ratios were calculated based on the back-test periods for each commodity
listed in Table 1, with the downward adjustment of WTI from 6.5 years to 3.5 years based on
the SFAS 133 analysis. For each cross hedge 5, 10 and 15 years of previous data was used to
calculate optimal hedge ratio. The varying years previous data were updated on a rolling
window as the back-test moved from its beginning around 2009/2010 up to 2015.
Figure 7 Dynamic Hedge Ratio Brent Crude Oil
Figure 7 shows the results of optimal hedge ratio analysis using an ordinary least squares
model, an error correction model and an error correction model with GARCH error terms for
Brent crude oil. The data is looked at on a monthly time horizon as per the findings of the
3.650
3.700
3.750
3.800
3.850
3.900
3.950
4.000
4.050
4.100
4.150
12/08 05/10 09/11 01/13 06/14
Op
tim
al H
ed
ge R
atio
Time
Brent Crude Oil Dynamic Hedge Ratio
OLS ECM GARCH
35
FAS 133 analysis and includes 78 observations spanning the back-test period from January
2009 to June 2015. In terms of the assertion of (Kroner & Sultan, 1993) that the OLS hedge
ratio estimation can suffer from downward bias and lead to under hedging, the results for the
Brent cross hedge do not offer conclusive evidence for or against this argument. Each of the
series track each other reasonably closely, while the ECM and ECM-GARCH models
consistently estimates a smaller hedge ratio than OLS up until 2013 after which this trend is
reversed. This may be due to the increase in correlation between Brent and jet fuel observed
around this time frame in Figure 3. The standard deviation of hedge ratio estimates for the
ECM models are larger than those of the OLS estimates. This suggests the ECM and ECM-
GARCH models are more flexible in determining optimal hedge ratio and react more to
changes in the data than the OLS approach which is consistent with the fact that these models
are more dynamic in their estimation approaches. Overall all three models produce estimates
for optimum hedge ratio which vary between 3.7 and 4.1. Since the ratios are calculated from
first differenced and jet fuel prices are in $/gallon and Brent prices in $/barrel the results
indicate that it takes between 3.7 and 4.1 barrels of jet fuel to hedge exposure to 1 gallon of
jet fuel.
Figure 8 Dynamic Hedge Ratio LS Gasoil
Illustrated in Figure 8 above are the optimal hedge ratios calculated for Gasoil using OLS,
ECM and ECM-GARCH models for data spanning a back-test period from January 2010 to
June 2015. The standard deviations of the ratios are much smaller than those of Brent, though
this would be expected as the gasoil prices are in metric-tonnes vs jet fuels in $/gallon. The
lower standard deviations (relative to hedge ratio estimations) observed in the gasoil analysis
0.025
0.026
0.027
0.028
0.029
0.03
0.031
0.032
0.033
0.034
0.035
12
/09
03
/10
06
/10
09
/10
12
/10
03
/11
06
/11
09
/11
12
/11
03
/12
06
/12
09
/12
12
/12
03
/13
06
/13
09
/13
12
/13
03
/14
06
/14
09
/14
12
/14
03
/15
Op
tim
al H
ed
ge R
atio
Time
Gasoil Dynamic Hedge Ratio
OLS ECM GARCH
36
may be attributed to the almost constant correlation between gasoil and jet fuel returns when
viewed on a monthly horizon in Figure 3.
The ECM-GARCH models incorporation of heteroscedasticity is evident from Figure 8 as we
can observe the optimal hedge ratio estimates from this model react more strongly to changes
in the data, and this is characterised by sharper adjustments in its estimated values. During
the period of falling spot prices for gasoil around the beginning of 2014 illustrated in Figure 6
the OLS estimates begin to diverge from the other two models. For the most recent years of
the back-test period the optimum hedge ratio estimates for the ECM and ECM-GARCH
models decrease slightly while the OLS estimates increase significantly. The increase in OLS
hedge ratio estimates are possibly due to an increase in volatility which one associates with
negative returns in a market downturn (Heston, 1993). The slight decrease in ECM and ECM-
GARCH estimations may be as a result of the commodity time series reacting more quickly
than expected in correcting its path towards the long run equilibrium relationship as indicated
by the data presented earlier in Table 14.
37
Hedge Performance
Hedging performance is calculated based on the minimum variance of a portfolio containing
150,000 gallons of jet fuel and the optimal number of units of the cross hedging product as
calculated by the OLS model, the ECM model and the ECM GARCH model. The optimal
number of units refers to gallons for Heatoil, barrels for Brent and WTI and metric-tonnes for
Gasoil. The equivalent number of contracts for each commodity can be calculated via
Equation 2. The reason for calculating the optimal hedging position in units of the cross-
hedging product is to obtain the most accurate results for the P&L variance analysis – so it is
assumed for the analysis that partial amounts of contracts may be purchased. For example if
the ECM model estimates it will take 3860 barrels of Brent (3.86 contracts) to hedge jet fuel
exposure on a particular month, this value is not rounded to the closest whole number of
contracts. This is because each of the model estimates will be relatively close to each other –
as shown in the previous section - and rounding them all to the nearest whole number would
obviously nullify the analysis.
The methodology surrounding the purchase of incomplete numbers of contracts represents a
limitation of the model which is not consistent with the reality of hedging and one of the
assumptions of Black-Scholes - namely that the hedger can only buy whole numbers of
contracts. For each of the commodities tested the amount of historical data used to calculate
the optimal hedge ratio for each of the three models was varied between 5, 10 and 15 years.
The analysis was also carried across a number of hedge horizons – 1, 3, 6, 9 and 12 months
for Brent and Gasoil and 1, 2, 3 and 4 quarters for WTI and Heatoil. The three lowest
variances in the analysis for each commodity are highlighted in green in Table 15, Table 16,
Table 17 and Table 18 with the lowest annualised variance overall also underlined. The
annualised variances listed are in US Dollars.
38
Brent Crude Oil
Back-test Window
OLS ECM GARCH Unhedged Hedge
Horizon
5
32,362 33,511 33,874 80,128 1M
32,653 33,853 34,222 80,411 3M
32,945 34,119 34,525 79,831 6M
32,931 33,836 34,274 71,735 9M
31,967 32,722 33,177 70,536 12M
10
32,330 32,547 32,388 80,128 1M
32,614 32,849 32,684 80,411 3M
32,924 33,157 32,986 79,831 6M
32,941 33,085 32,923 71,735 9M
31,994 32,087 31,930 70,536 12M
15
32,178 32,132 31,982 80,128 1M
32,456 32,419 32,263 80,411 3M
32,760 32,713 32,549 79,831 6M
32,783 32,671 32,515 71,735 9M
31,836 31,679 31,533 70,536 12M
Table 15 Minimum Variance Hedge Analysis Brent
Presented in Table 15 above are the back-testing results for a portfolio containing 150,000
gallons of Gulf54 grade jet fuel and a corresponding amount of Brent crude oil determined by
each of the three fuel hedging models. The figures represent the annualised variance for each
cross-hedge calculated from Equation 6. In all cases the variances of the hedged position is far
lower than the variances of the unhedged exposure to jet fuel price fluctuations. The
unhedged exposures also have a much greater deviation moving between 70 and 80 thousand
dollars whereas the variances of the hedged portfolios ranges between just 31.5 and 34
thousand dollars. The results support the assertion that hedging not only reduces portfolio
variance conditional on its underlying assets but also reduces its unconditional variance. The
effect of varying the hedge horizon also has an effect of the portfolio P&L with the 1 month
and 12 month horizons producing the lowest portfolio variance for each of the data ranges
employed which suggests that the model is better at predicting ratios for 12 months-time than
for intermediate time periods.
The results also follow an interesting trend surrounding the amount of data used for back-
testing. When using 5 years of data to calculate the optimal hedge ratio for each model the
OLS analysis produces better results that the ECM and ECM GARCH models. However the
results suggest that as the amount of data used for calculating optimal hedge ratio is expanded
to 10 and 15 years the more complex ECM and ECM GARCH models approach and
39
eventually overtake the OLS model in terms of hedge performance with the ECM GARCH
model producing the lowest overall portfolio variance for Brent crude oil when hedging on a
12 month horizon. This suggests that the ECM and ECM-GARCH models are able to form
progressively better optimal hedge ratio predictions based on an expanding data set which the
models are fitted to. This is supported by (Alexander, 2008) who states that GARCH models
based on too few observations may lack robustness which is likely the reason the ECM
GARCH eventually proves to provide the best performing hedge as the number of
observations on which the model is based are increased. The findings of the ECM with
GARCH error terms are consistent with those of (Adams & Gerner, 2012) who determine the
ECM GARCH to be the best performing model for dynamic hedging across all four
commodities used in this study.
The lowest variances in the Jet-Brent portfolio are achieved at the 12 month horizon and are
loosely consistent with the results for Jet-Gasoil portfolio presented in Table 16 below. The
analysis on the Jet-Gasoil portfolio suggests that the greatest reduction in variance is around
the 9 and 12 month horizons. However as presented in Table 14 gasoil has a greater speed of
adjustment parameter than Brent which means that the cointegrated time series of gasoil and
jet fuel tend to correct towards their long run relationship more quickly than those of Brent
and jet fuel which may explain why the best predictions are 9 months ahead for gasoil and 12
months ahead for Brent.
40
Low Sulphur Gasoil
Back-test Window
OLS ECM GARCH Unhedged Hedge
Horizon
5
50,244 51,979 54,227 80,496 1M
49,681 51,446 53,774 80,829 3M
50,245 52,318 54,538 80,044 6M
48,389 47,922 48,854 70,698 9M
48,739 47,658 48,360 69,340 12M
10
49,797 53,041 55,606 80,496 1M
49,189 52,538 55,200 80,829 3M
49,892 53,392 55,946 80,044 6M
47,841 48,621 50,520 70,698 9M
48,145 48,249 50,015 69,340 12M
15
49,670 53,543 55,901 80,496 1M
49,056 53,059 55,505 80,829 3M
49,790 53,886 56,228 80,044 6M
47,636 48,839 50,625 70,698 9M
47,903 48,381 50,013 69,340 12M
Table 16 Minimum Variance Hedge Analysis Gasoil
The analysis of the Jet-Gasoil portfolio also shows that the P&L variance of the hedged
positions far outperform the P&L variance of an unhedged position. Unlike the previous
results the Jet-Gasoil analysis does not present any evidence that the ECM and ECM GARCH
models provide better optimal hedge ratio predictions as the amount of data input is
expanded. Although the greatest reduction in variance is observed using 15 years of data to
calculate optimal hedge ratio, there is no compelling evidence to suggest that using more data
can consistently lead to more accurate predictions of optimal hedge ratio for the Jet-Gasoil
portfolio. The results show that the OLS model forms the best predictor of optimal hedge
ratio for a Jet-Gasoil portfolio. Although based on the results there is really no difference
between the lowest variance achieved by the ECM and OLS models as the OLS performs
better by a total of $22.
The residual diagnostics carried out previously may shed some light on poorer performance
of the ECM and ECM GARCH models for the Jet-Gasoil portfolio. There is much more
evidence of a stationary trend in the gasoil residuals when compared to the other three
commodities as shown in Table 10 which suggest that the OLS regressions trend line would
provide a better fit to the Jet-Gasoil residuals than for any of the other commodities which
41
would result in the OLS model giving a closer prediction to the ECM and ECM GARCH
models for this portfolio than for the other cross-hedges analysed.
The degree of autocorrelation in the residuals would have had a much greater impact on the
gasoil analysis than on any of the other portfolios as the results of Ljung-Box Q tests for
residual autocorrelation presented in Table 11 indicate much stronger evidence of
autocorrelation in the Jet-Gasoil residual series than in the other series analysed. This likely
had the effect of making the Jet-Gasoil residuals more biased towards randomness than for
the other commodities and likely reduced the usefulness of the additional information they
would have provided to the ECM and ECM GARCH models. If this assertion is true then
transformation of the residuals as per (Cochrane & Orcutt, 1949) would likely have improved
the performance of the ECM and ECM GARCH models relative to the OLS approach.
Although Table 12 shows that the Jet-Gasoil residuals also display a much high level of
heteroscedasticity than the other cross-hedges which can be accounted for in the ECM
GARCH model it’s clear that the damaging effect of autocorrelation in the residual series has
had a greater impact on the results based on the ECM GARCH approach proving to be the
worst performing model for the Jet-Brent portfolio.
The higher levels of heteroscedasticity in the Jet-Gasoil residuals may also explain why the
trend of increasing input data to produce more accurate results is not observed in the Gasoil
cross-hedge as it is in the Brent cross-hedge, as the much stronger evidence of
heteroscedasticity means that the conditional variance of the Jet-Gasoil residuals is much
more dependent on the variances immediately preceding each point than other the better
model specification associated with including more data from 10 and 15 years in the past.
Overall the cross-hedging portfolio containing gasoil does not perform as well as the Brent
portfolio with a difference of approximately $16,000 between the best performing hedges
with each commodity. The results so far indicate the Brent is the best cross-hedging product
for jet fuel at all of the time horizons analysed and provide some contrast with the results of
(Adams & Gerner, 2012) who find that gasoil proves to be the best cross-hedging product up
to three months and is overtaken by Brent for hedge horizons greater than three months.
42
West Texas Intermediary Crude Oil
Since the SFAS analysis determined that the WTI and Heatoil hedges can only meet
regulations on a three month horizon, the size of the jet fuel position being hedged by these
commodities was increased three fold to 450,000 gallons for the analysis. This is to reflect
the jet fuel required for each three month interval being hedged every quarter.
Table 17 presents the findings from the profit and loss analysis for the Jet-WTI portfolio.
While the results regarding the annualised variance of the hedged versus the unhedged
positions is consistent with the findings for Brent and gasoil, the increase in the hedge
horizon to once per quarter has had a drastically negative impact on the overall effectiveness
of WTI as a cross-hedging product. It is notable though that the WTI price movements
displayed the lowest correlation with Jet Fuel price movements on a monthly and quarterly
horizon before the profit and loss analysis took place as shown in Figure 3 and Figure 4.
Among all of the cross-hedging products analysed in the study, WTI produced the highest
annualised variance in the hedging portfolio indicating that it is the least effective cross-
hedging product analysed.
Back-test Window
OLS ECM GARCH Unhedged Hedge
Horizon
5
163,611 158,632 161,812 303,325 1Q
159,720 155,913 157,849 313,505 2Q
161,080 157,598 159,163 329,582 3Q
170,201 167,024 168,267 206,799 4Q
10
164,740 161,216 163,836 303,325 1Q
161,063 158,552 160,298 313,505 2Q
162,446 160,157 161,615 329,582 3Q
171,540 169,592 170,657 206,799 4Q
15
164,860 161,735 164,350 303,325 1Q
161,119 158,926 160,798 313,505 2Q
162,446 160,428 162,079 329,582 3Q
171,482 169,765 171,073 206,799 4Q
Table 17 Minimum Variance Hedge Analysis WTI
The results for the Jet-WTI cross indicate the error correction model produces the best results
when hedging on a quarterly horizon. The lowest variances were produced on a hedge
horizon of 6 to 9 months and using only 5 years of data. The results indicate that the inclusion
of more data in fitting each of the models does not produce more accurate predictions of
optimal hedge ratio. This is consistent with the findings of the SFAS 133 analysis presented
43
in Figure 4 which shows that the correlation between the returns series for WTI and Jet Fuel
have been much higher in the past four years when compared the eight year period analysed.
The results suggest that even though only the previous eight years were used in the SFAS
analysis that the correlation between WTI and Jet Fuel also remained at a lower level going
back to fifteen years.
The Jet Fuel and WTI time series presented the least evidence of cointegration as shown in
Table 13 and the residual diagnostics indicated that the Jet-WTI residual series also presented
the least evidence for stationarity as well as the lowest levels of heteroscedasticity out of all
of the potential cross-hedges analysed. The low levels of cointegration may explain the poor
performance overall of WTI when compared to the P&L results for heating oil shown in Table
18, which demonstrated the second highest level of cointegration after gasoil. The residual
analysis surrounding the stationarity and heteroscedasticity tests may explain the greater
performance of the ECM over the other two models as the low degree of stationarity and
heteroscedasticity between the jet fuel and WTI time series may have detracted from the
strengths of the OLS and ECM-GARCH models.
44
Home Heating Oil
Back-test Window
OLS ECM GARCH Unhedged Hedge
Horizon
5
63,702 64,314 69,348 326,588 1Q
64,785 65,166 68,715 336,264 2Q
64,995 65,004 67,461 350,555 3Q
65,109 65,171 70,625 270,233 4Q
10
65,212 67,404 75,030 326,588 1Q
66,493 68,628 75,731 336,264 2Q
66,948 68,974 75,841 350,555 3Q
66,701 69,131 78,645 270,233 4Q
15
64,455 66,488 70,958 326,588 1Q
65,767 67,709 71,762 336,264 2Q
66,203 68,020 71,770 350,555 3Q
65,565 67,964 73,714 270,233 4Q
Table 18 Minimum Variance Hedge Analysis Heatoil
Although the unhedged variances are larger for that of home heating oil than the other
quarterly hedge using WTI, the annualised variances for the Jet-Heatoil portfolio represent a
vast improvement on the WTI hedge performance. The results show that the OLS regression
produces the lowest annualised variance across all hedge horizons, though the error
correction model does produce one of the three lowest overall variances. The Jet-Heatoil
residual series displayed the second highest overall evidence of autocorrelation and the
second lowest overall evidence of heteroscedasticity. Although the jet fuel and heating oil
time series displayed the second highest level of cointegration, the effect of autocorrelation
on the information retained in the residuals as well as the weaker effect of heteroscedasticity
likely reduced the advantage that the ECM and ECM GARCH models had in being able to
capture some of the short and long run dynamics in the relationship between the Jet Fuel and
Heatoil time series.
It’s notable that in general the differences between the estimates from each model were small
and that in reality when adding the constraint of the hedger not being able to purchase partial
numbers of contracts that the differences between the models may be nullified.
45
Conclusions
This research investigated the hedging performance of Brent crude oil, West Texas
Intermediary crude oil, low sulphur gasoil and NY Harbour home heating oil when used as
cross-hedging instruments for Gulf54 grade jet fuel. The cross hedging products were
analysed under the constraints of meeting hedge accounting standards and compared using an
OLS regression model, an error correction model and an error correction model with GARCH
error terms. In all cases the use of each of the cross hedging products reduces the hedgers risk
when compared to an unhedged long position in jet fuel, though hedging exposure on a
weekly horizon is misspecified as doing so will block the portfolio from being able to qualify
for hedge accounting status under the current regulations. For Brent crude oil and low sulphur
gasoil hedging may be carried out on a monthly time horizon, though for WTI and heating oil
this horizon must be pushed out to three months in order to meet hedge regulations.
The results from the model performance show that the ECM-GARCH model produced the
lowest overall annualised variance for a portfolio containing 150,000 litres of Gulf54 grade
jet fuel and a Brent crude oil as a cross-hedging product. As well as this the model was
improved by expanding the amount of data used in specifying the GARCH parameters. It was
found the regular ECM model produced the best results for WTI while the OLS regression
produced the best results for gasoil and heating oil. The better than expected performance of
the OLS regression is significant but cannot be taken as substantial evidence of greater
overall performance as the effect of autocorrelation in the residuals was not accounted for
before performing analysis using the ECM and ECM-GARCH models and it’s likely that had
the residual data been transformed that it would have affected the overall results.
46
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50
Appendix A
Figure 9 Residual Autocorrelation Plots
Figure 10 Residual QQ Plots
Figure 11 Residual Histograms
51
Appendix B
MATLAB Import Data
addpath(genpath('C:\Users\Niall\Documents\College\Masters\Dissertation\Data')); % add
selected folders and subfolders to path
DAILY = readtable('EIADAILY.csv'); % read in data
WEEKLY = readtable('EIAWEEKLY.csv');
MONTHLY = readtable('EIAMONTHLY.csv');
QUARTERLY = readtable('EIAQUARTERLY.csv');
MATLAB Define Parameters
COMM = 2; %2 = BRENT; 3 = WTI; 4 = GAS; 5 = HEAT
HOR = 1; % futures horizon in months
GAL = 150000; % no of gallons of Jet Fuel to Hedge
WIN = 15; % rolling window in years from which to calculate optimal hedge ratio
MATLAB Loop Arguments
if COMM == 3
LOOP = 3;
elseif COMM == 5
LOOP = 5;
else LOOP = 1;
end
% WTI & Heatoil both using quarterly prices vs. Brent and Gasoil both using monthly prices
% this is to save space repeating each loop 3 times - i.e. can use 'if COMM=LOOP, else'
% and avoid having to say 'if 3,elseif 5, else'
MATLAB SFAS 133
TH = [WIN,WIN,WIN,WIN]; %time horizons matrix
BT = [6.5 3 5.5 3.5]; % max backtest horizons as per SFAS 133
% Weekly Horizon
WEEKLY_DBL = table2array(WEEKLY(:,2:end)); % convert data series to double
WEEKLY_RETS = price2ret(WEEKLY_DBL); % convert to 15 years of returns
RSQW = zeros(1,4); % matrix to store R_Square Values for weekly horizon
LW = length(WEEKLY_RETS); % W to denote weekly
% fit linear regression
for k = 2:5
BTW = BT(1,k-1)*52; %length of SFAS back test & reset to match commodity on each loop
SFASW = (WEEKLY_RETS(LW-BTW:end,:)); %create matrix for SFAS horizon & resize on each
loop
FAS133W = fitlm(SFASW(:,1),SFASW(:,k));
RSQW(1,k-1) = FAS133W.Rsquared.Adjusted;
end
% Monthly Horizon
MONTHLY_DBL = table2array(MONTHLY(:,2:end)); % convert data series to double
MONTHLY_RETS = price2ret(MONTHLY_DBL);
RSQM = zeros(1,4); %matrix to store R_Square Values for monthly horizon
LM = length(MONTHLY_RETS);
52
for k = 2:5
BTM = BT(1,k-1)*12; %length of SFAS back test & reset to match commodity on each loop
SFASM = (MONTHLY_RETS(LM-BTM:end,:)); %create matrix for SFAS horizon & resize on each
loop
FAS133M = fitlm(SFASM(:,1),SFASM(:,k));
RSQM(1,k-1) = FAS133M.Rsquared.Adjusted;
end
% Quarterly Horizon
QUARTERLY_DBL = table2array(QUARTERLY(:,2:end)); % convert data series to double
QUARTERLY_RETS = price2ret(QUARTERLY_DBL);
RSQQ = zeros(1,4); %matrix to store R_Square Values for monthly horizon
LQ = length(QUARTERLY_RETS);
for k = 2:5
BTQ = BT(1,k-1)*4; %length of SFAS back test & reset to match commodity on each loop
SFASQ = (QUARTERLY_RETS(LQ-BTQ:end,:)); %create matrix for SFAS horizon & resize on each
loop
FAS133Q = fitlm(SFASQ(:,1),SFASQ(:,k));
RSQQ(1,k-1) = FAS133Q.Rsquared.Adjusted;
end
MATLAB Unit Root Analysis
DAILY_DBL = table2array(DAILY(:,2:end)); % first perform on levelled time series
% Augmented Dickey Fuller Test
ADF_LEVELS = zeros(4,5); %unit root levels results matrix
for i = 1:5
[h,pValue,stat,cValue] = adftest(DAILY_DBL(:,i)); % null = unit root present
ADF_LEVELS(1,i) = h;
ADF_LEVELS(2,i) = pValue; % default significance level = 0.05
ADF_LEVELS(3,i) = stat;
ADF_LEVELS(4,i) = cValue;
end
Variables = {'h';'pValue';'stat';'cValue'};
JET = ADF_LEVELS(:,1);
BRENT = ADF_LEVELS(:,2);
WTI = ADF_LEVELS(:,3);
GAS = ADF_LEVELS(:,4);
HEAT = ADF_LEVELS(:,5);
ADF_LVLTable = table(JET,BRENT,WTI,GAS,HEAT,'RowNames',Variables);
% Kwiatkowski-Phillips-Schmidt-Shin test
KPSS_LEVELS = zeros(4,5);
for i = 1:5
[h,pValue,stat,cValue] = kpsstest(DAILY_DBL(:,i));% null = stationary
KPSS_LEVELS(1,i) = h;
KPSS_LEVELS(2,i) = pValue;% default significance level = 0.05
KPSS_LEVELS(3,i) = stat;
KPSS_LEVELS(4,i) = cValue;
end
JET = KPSS_LEVELS(:,1);
BRENT = KPSS_LEVELS(:,2);
WTI = KPSS_LEVELS(:,3);
GAS = KPSS_LEVELS(:,4);
HEAT = KPSS_LEVELS(:,5);
KPSS_LVLTable = table(JET,BRENT,WTI,GAS,HEAT,'RowNames',Variables);
% perform tests on first differences
DAILY_DIFF = diff(DAILY_DBL);
53
% Augmented Dickey Fuller Test
ADF_DIFF = zeros(4,5); %unit root levels results matrix
for i = 1:5
[h,pValue,stat,cValue] = adftest(DAILY_DIFF(:,i));
ADF_DIFF(1,i) = h;
ADF_DIFF(2,i) = pValue;
ADF_DIFF(3,i) = stat;
ADF_DIFF(4,i) = cValue;
end
JET = ADF_DIFF(:,1);
BRENT = ADF_DIFF(:,2);
WTI = ADF_DIFF(:,3);
GAS = ADF_DIFF(:,4);
HEAT = ADF_DIFF(:,5);
ADF_FDTable = table(JET,BRENT,WTI,GAS,HEAT,'RowNames',Variables);
% Kwiatkowski-Phillips-Schmidt-Shin test
KPSS_DIFF = zeros(4,5);
for i = 1:5
[h,pValue,stat,cValue] = kpsstest(DAILY_DIFF(:,i));% null = stationary
KPSS_DIFF(1,i) = h;
KPSS_DIFF(2,i) = pValue;
KPSS_DIFF(3,i) = stat;
KPSS_DIFF(4,i) = cValue;
end
JET = KPSS_DIFF(:,1);
BRENT = KPSS_DIFF(:,2);
WTI = KPSS_DIFF(:,3);
GAS = KPSS_DIFF(:,4);
HEAT = KPSS_DIFF(:,5);
KPSS_FDTable = table(JET,BRENT,WTI,GAS,HEAT,'RowNames',Variables);
MATLAB OLS Regression Model
MONTHLY_DIFF = diff(MONTHLY_DBL); % first differenced time series
QUARTERLY_DIFF = diff(QUARTERLY_DBL);
for COM = COMM
if COM == LOOP % determine length returns series for Jet Fuel based on hedge commodity
JET = (QUARTERLY_DIFF(:,1));
FUT = (QUARTERLY_DIFF(:,COM));
else
JET = (MONTHLY_DIFF(:,1));
FUT = (MONTHLY_DIFF(:,COM));
end
end
OLSREG = fitlm(FUT,JET);
OHR_OLS = table2array(OLSREG.Coefficients(2,1)); % output x-coef (optimal hedge ratio)
MATLAB Residual Diagnostics
RESIDS = table2array(OLSREG.Residuals); % OLS residuals
% stationarity tests
% ADF test
[h,pValue,stat,cValue] = adftest(RESIDS(:,4));
RESIDS_ADFTable = table(h,pValue,stat,cValue);
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% PP test
[h,pValue,stat,cValue] = pptest(RESIDS(:,4));
RESIDS_PPTable = table(h,pValue,stat,cValue);
% autocorrelation tests
[h,pValue,stat,cValue] = lbqtest(RESIDS(:,4)); % null = no autocorrelation
RESIDS_LBQTable = table(h,pValue,stat,cValue);
[~,~,bounds] = autocorr(RESIDS(:,4),[],2); % determine confidence bounds for autocorr
% test for significant ARCH effects
[h,pValue,stat,cValue] = lbqtest(RESIDS(:,4).^2); % 1 indicates significant ARCH effects
RESIDS_ARCHTable = table(h,pValue,stat,cValue);
% test for heteroscedasticity
[h,pValue,stat,cValue] = archtest(RESIDS(:,4)); % null = no conditional heteroscedasticity
RESIDS_HETTable = table(h,pValue,stat,cValue);
% determine suitable number of lags for each commodity
numLags = 4;
logL = zeros(numLags,1);
for k = 1:numLags
Mdl = garch(0,numLags);
[~,~,logL(k)] = estimate(Mdl,RESIDS(:,4),'Display','off');
% doing for quarterly returns same values
end
fitStats = aicbic(logL,1:numLags);
ARCH_LAGS = find(min(fitStats));
MATLAB Error Correction Model
Engle-Granger Cointegration Test null hypothesis = no cointegration
[EGC_h,EGC_pValue,EGC_stat,EGC_cValue,EGC_reg] = egcitest...
(MONTHLY_DBL(:,[1 COMM]),'test',{'t1','t2'});
% h value of 1 rejects the nulll i.e. series are cointegrated
for COM = COMM % determine commodity being tested at apply egcitest to appropriate time
horizon
if COM == LOOP
DFt = diff(QUARTERLY_DIFF(:,COMM)); % change in differenced time series
DSt = diff(QUARTERLY_DIFF(:,1));
else
DFt = diff(MONTHLY_DIFF(:,COMM));
DSt = diff(MONTHLY_DIFF(:,1));
end
end
LAG = lagmatrix(RESIDS(:,4),1); % lag residual matrix
LAGS = LAG(1:end-1,1); %remove unused final value
EGC_ECM = fitlm([DFt LAGS],DSt); % error correction regression with lagged residuals
OHR_ECM = table2array(EGC_ECM.Coefficients(2,1)); % optimal hedge ratio (x-coef)
ECM_GAMMA = table2array(EGC_ECM.Coefficients(3,1)); % optimal hedge ratio (x-coef)
MATLAB ECM GARCH Extension
Mdl = garch(1,1);
GARCH_EstMdl = estimate(Mdl,RESIDS(:,4));
if GARCH_EstMdl.P == 0 % if no GARCH term set alpha = 0
GARCH_alpha = 0; % unelegant conversion of cell to double
else GARCH_alpha = table2array(cell2table(GARCH_EstMdl.GARCH));
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end
GARCH_beta = table2array(cell2table(GARCH_EstMdl.ARCH));
GARCH_omega = GARCH_EstMdl.Constant;
GARCH_RESIDS = zeros(length(RESIDS(:,4))-1,1);
for i = 2:length(RESIDS(:,4))
GARCH_RESIDS(i-1,1) = GARCH_omega+(GARCH_alpha*(RESIDS(i,4)-RESIDS(i-1,4)))+...
(GARCH_beta*RESIDS(i,4).^2);
end
GARCH_ECM = fitlm([DFt GARCH_RESIDS],DSt); % error correction regression with lagged
residuals
OHR_GARCH = table2array(GARCH_ECM.Coefficients(2,1));
MATLAB Dynamic Hedge Ratio
DAYS = datenum('01-June-2015')-datenum('01-June-2000');
MONTHS = round(DAYS/365)*12; % count number of months
QUARTERS = round(DAYS/365)*4; % count number of quarters
% Define Backtest Parameters
for COM = COMM % commodities have different backtest horizon
if COM == 2
START = datenum('01-January-2009'); % Brent start period
COMOD = {'Brent'}; % labels for final table
elseif COM == 3
START = datenum('01-June-2012'); % WTI reduced due to hedge effectiveness tests
COMOD = {'WTI'};
elseif COM == 4
START = datenum('01-January-2010'); % Gasoil start period
COMOD = {'Gasoil'};
else START = datenum('01-January-2012'); % Heatoil start period
COMOD = {'Heatoil'};
end
end
FINISH = datenum('01-June-2015'); % sample end period
% Define Period in Months/Quarters
SAM_START = datenum('01-June-2015')-START; % period from START to June 2015 in days
SAM_FINISH = datenum('01-June-2015')-FINISH; % period from FINISH to June 2015 in days
SAM_START_M = round((SAM_START/365)*12); % define in months
SAM_FINISH_M = round((SAM_FINISH/365)*12);
SAM_START_Q = round((SAM_START/365)*4); % define in quarters
SAM_FINISH_Q = round((SAM_FINISH/365)*4);
% Calculate Dynamic Hedge Ratios
for COM = COMM % determine whether monthly or quarterly hedge horizon
if COM == LOOP % if using WTI or Heatoil
OHR_OLS_BT = zeros(1,SAM_START_Q-(SAM_FINISH_Q-1));
OHR_ECM_BT = OHR_OLS_BT;
OHR_GARCH_BT = OHR_OLS_BT;
JET_BT = zeros(length(SAM_FINISH_Q:SAM_FINISH_Q+(TH(1,COMM-1)*4)),1); %reset matrices
FUT_BT = zeros(length(SAM_FINISH_Q:SAM_FINISH_Q+(TH(1,COMM-1)*4)),1);
for k = SAM_FINISH_Q:SAM_START_Q % loop to run backtest
JET_BT = QUARTERLY_DIFF(LQ-(k+(TH(1,COMM-1)*4)):LQ-k,1); % use previous 8 years
of
% data starting at beginning of backtest period
FUT_BT = QUARTERLY_DIFF(LQ-(k+(TH(1,COMM-1)*4)):LQ-k,COMM); %using differenced
data
% OLS
OLSREG_BT = fitlm(FUT_BT,JET_BT); % OLS regression
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OHR_OLS_BT(1,k-(SAM_FINISH_Q-1)) = table2array(OLSREG_BT.Coefficients(2,1));%
store OLS hedge ratio
% ECM
DFt = diff(FUT_BT); % delta first differences
DSt = diff(JET_BT);
RES = table2array(OLSREG_BT.Residuals(:,4));
LAG = lagmatrix(RES,1); % lagged residuals
LAGS = LAG(1:end-1,1);
ECM = fitlm([DFt LAGS],DSt);
OHR_ECM_BT(1,k-(SAM_FINISH_M-1)) = table2array(ECM.Coefficients(2,1));
% GARCH
GARCH_EstMdl = estimate(Mdl,RES);
if GARCH_EstMdl.P == 0 % if no GARCH term set alpha = 0
GARCH_alpha = 0;
else GARCH_alpha = table2array(cell2table(GARCH_EstMdl.GARCH));
end
GARCH_beta = table2array(cell2table(GARCH_EstMdl.ARCH));
GARCH_omega = GARCH_EstMdl.Constant;
GARCH_RES = zeros(length(RES)-1,1);
for i = 2:length(RES)
GARCH_RES(i-1,1) = GARCH_omega+(GARCH_alpha*(RES(i,:)-RES(i-1,:)))+...
(GARCH_beta*RES(i,:).^2);
end
GARCH_ECM = fitlm([DFt GARCH_RES],DSt);
OHR_GARCH_BT(1,k-(SAM_FINISH_M-1)) = table2array(GARCH_ECM.Coefficients(2,1));
end
else % if using Brent or Gasoil
OHR_OLS_BT = zeros(1,SAM_START_M-(SAM_FINISH_M-1));
OHR_ECM_BT = OHR_OLS_BT;
OHR_GARCH_BT = OHR_OLS_BT;
JET_BT = zeros(length(SAM_FINISH_M:SAM_FINISH_M+(TH(1,COMM-1)*12)),1); %reset
matrices
FUT_BT = zeros(length(SAM_FINISH_M:SAM_FINISH_M+(TH(1,COMM-1)*12)),1);
for k = SAM_FINISH_M:SAM_START_M % note change from quarterly to monthly horizon
% OLS
JET_BT = MONTHLY_DIFF(LM-(k+(TH(1,COM-1)*12)):LM-k,1);
FUT_BT = MONTHLY_DIFF(LM-(k+(TH(1,COM-1)*12)):LM-k,COMM);
OLSREG_BT = fitlm(FUT_BT,JET_BT);
OHR_OLS_BT(1,k-(SAM_FINISH_M-1)) = table2array(OLSREG_BT.Coefficients(2,1));
% ECM
DFt = diff(FUT_BT);
DSt = diff(JET_BT);
RES = table2array(OLSREG_BT.Residuals(:,4));
LAG = lagmatrix(RES,1);
LAGS = LAG(1:end-1,1);
ECM = fitlm([DFt LAGS],DSt);
OHR_ECM_BT(1,k-(SAM_FINISH_M-1)) = table2array(ECM.Coefficients(2,1));
% GARCH
GARCH_EstMdl = estimate(Mdl,RES);
if GARCH_EstMdl.P == 0 % if no GARCH term set alpha = 0
GARCH_alpha = 0;
else GARCH_alpha = table2array(cell2table(GARCH_EstMdl.GARCH));
end
GARCH_beta = table2array(cell2table(GARCH_EstMdl.ARCH));
GARCH_omega = GARCH_EstMdl.Constant;
GARCH_RES = zeros(length(RES)-1,1);
for i = 2:length(RES)
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GARCH_RES(i-1,1) = GARCH_omega+(GARCH_alpha*(RES(i,:)-RES(i-1,:)))+...
(GARCH_beta*RES(i,:).^2);
end
GARCH_ECM = fitlm([DFt GARCH_RES],DSt);
OHR_GARCH_BT(1,k-(SAM_FINISH_M-1)) = table2array(GARCH_ECM.Coefficients(2,1));
end
end
end
OHR_OLS_BTT = transpose(OHR_OLS_BT); % transpose optimal hedge ratio series to vertical
matrices
OHR_ECM_BTT = transpose(OHR_ECM_BT); % want this format for P&L loops
OHR_GARCH_BTT = transpose(OHR_GARCH_BT);
MATLAB Calculate P&L
for COM = COMM
if COM == LOOP
JET_PRICE = QUARTERLY_DBL(LQ-(SAM_START_Q-1):end,1);
FUT_PRICE = QUARTERLY_DBL(LQ-(SAM_START_Q-1):end,COMM);
P = 4; % time periods for annualising variances
GAL = GAL*3; % triple fuel consumption on quarterly horizon
else
JET_PRICE = MONTHLY_DBL(LM-(SAM_START_M-1):end,1);
FUT_PRICE = MONTHLY_DBL(LM-(SAM_START_M-1):end,COMM);
P = 12; % time periods for annualising variances
end
end
JET_PL = zeros(length(JET_PRICE)-HOR,1);
FUT_PL_OLS = JET_PL; % reset matrix sizes for P&L
TOTAL_PL_OLS = JET_PL;
FUT_PL_ECM = JET_PL;
TOTAL_PL_ECM = JET_PL;
FUT_PL_GARCH = JET_PL;
TOTAL_PL_GARCH = JET_PL;
%calculate P&L for chosen commodity
for i = 1:length(JET_PRICE)- HOR
JET_PL(i,1) = (JET_PRICE(i,1)-JET_PRICE(i+1,1))*GAL; % JET P&L: t1-t0*gallons
N_OLS = OHR_OLS_BTT(i,1)*GAL; % number of units of futures to buy
FUT_PL_OLS(i,1) = (FUT_PRICE(i+1,1)-FUT_PRICE(i,1))*N_OLS; % FUT P&L: t0-t1*units bought
TOTAL_PL_OLS(i,1) = JET_PL(i,1) + FUT_PL_OLS(i,1); %JET P&L + FUT P&L
N_ECM = OHR_ECM_BTT(i,1)*GAL; % repeat for ECM hedge ratios
FUT_PL_ECM(i,1) = (FUT_PRICE(i+1,1)-FUT_PRICE(i,1))*N_ECM;
TOTAL_PL_ECM(i,1) = JET_PL(i,1) + FUT_PL_ECM(i,1);
N_GARCH = OHR_GARCH_BTT(i,1)*GAL; % repeat for GARCH hedge ratios
FUT_PL_GARCH(i,1) = (FUT_PRICE(i+1,1)-FUT_PRICE(i,1))*N_GARCH;
TOTAL_PL_GARCH(i,1) = JET_PL(i,1) + FUT_PL_GARCH(i,1);
end
VAR_JET = std(JET_PL)*sqrt(P); % unhedged variance
VAR_OLS = std(TOTAL_PL_OLS)*sqrt(P); % OLS variance
VAR_ECM = std(TOTAL_PL_ECM)*sqrt(P); % ECM variance
VAR_GARCH = std(TOTAL_PL_GARCH)*sqrt(P); % GARCH variance
% store & output minimum variance hedge ratio results in table
VAR = table(VAR_OLS,VAR_ECM,VAR_GARCH,VAR_JET,HOR,WIN,'RowNames',COMOD);
Published with MATLAB® R2015a