DUE ON Tuesday 9 June 2020 at 17h00
Learning Objectives
By completing this case you will:
– Learn to work with futures and spot data
– Estimate hedge ratios
– Compute hedged and unhedged valuation changes with historical data
– Back-test and discuss possible hedging strategies
– Consider how hedge ratio estimates can be employed in dynamic settings
– Communicate complex analysis to a non-technical audience
Learning Activities
By completing this case you will:
– Estimate and/or compute hedge ratios
– Compute cash flows to hedging strategies using energy futures contracts
– Comment on the consequences of changing market conditions on possible
hedging strategies
– Write a brief report for a non-technical audience
You are an associate at a commercial bank. One of your colleagues has sent the
following email:
“Thanks again for sending Hull’s chapter on hedging with futures. As you know
we do a lot of work with energy-intensive companies, and one of his examples
seemed especially relevant. His cross-hedging strategy (jet fuel and heating oil)
is interesting, but we were wondering what would happen in turbulent
markets? For example, in October 2018 crude oil reached $76.50 to drop down
to $44 by December 2018 and more recently it crashed and fall to a negative
territory (-$37.63) on 20 April 2020. What would happen to someone using
Hull’s proposed strategy over this time? Would it make sense to consider other
energy futures as part of a hedging strategy (i.e. we have been wondering
about including crude oil futures in addition to heating oil)?
Any thoughts you have would be greatly appreciated.”
/ 2
Your colleague has limited quantitative skills and you feel that you can best
demonstrate these concepts through a clear example, based on data taken from the
period she noted.
You consider what might happen over this time to someone who wanted to hedge the
price risk of 5,000,000 gallons of jet fuel. You note that the futures data is taken from
NYMEX where heating oil contracts are quoted in USD per gallon (contract size of
42,000 gallons) and crude oil is quoted in USD per barrel (contract size of 1,000
Your assistant has downloaded spot jet fuel, crude oil futures and heating oil futures
daily prices from June 01, 2016 through April 21, 2020 (see spreadsheet “Combined”
in the file “Daily Energy Data – 25762 DRM – AUT 2020.xlsx”). You double check that
their use of the “lookup” function in Excel was done properly so that the combined
data is correctly lined up through time.
Your aim is to assess two factors of hedging jet fuel price changes with futures
contracts; a) the stability of the estimated hedge ratios over time and b) the role of
different hedging instruments in the performance of the hedge. To draw your
conclusions, you perform the following investigations:
i) You compute and compare the hedge ratios and the optimal number of futures
contracts by using two data samples; June 2016-June 2018 and June 2018-April 2020.
For the hedging strategy, you consider three type of hedging applications in terms of
hedging instruments: hedge with heating oil futures only, hedge with crude oil
futures only, and, hedge with both heating oil and crude oil futures. How stable the
hedge ratios have been over the time?
ii) You assess the out-of-sample hedge performance of these hedge ratios. By using
the hedge ratio estimated from the June 2016-June 2018 dataset, you hedge jet fuel
price changes in two periods; June 2018-June 2019 and June 2019-April 2020 and
then compare the hedging performance over these two periods (by comparing
standard deviations of unhedged and hedged positions).
iii) What are the main issues you have identified with this hedging application and
what are your suggestions to deal with those?
iv) First time in the history, oil markets experienced negative prices. Explain what
happened to the crude oil spot and futures markets on 20 April 2020. Explain why
not all oil prices (e.g. Brent, WTI) went to negative territories. News announcements
and online articles should be used to support your discussion.
/ 3
Submission Details
Please submit both your team’s report AND the spreadsheet you used for
your quantitative analysis. Both documents will be reviewed in assessing
your work. However, please write your report so that the reader does not
have to refer to the spreadsheet to understand your key points.
Please submit your team’s cover page for the submission, the written
report and the spreadsheet you used for your quantitative analysis by an
online submission under Case Study on Canvas by Tuesday 9 June 2020
17h00. The file names of these documents should include your team
surnames / number.
Late submissions attract a penalty of 5 marks per day.
Your submission cannot be more than 6 pages in total (including cover
sheet) with at least one-inch margins and double-spaced 12-point type.
You are welcome to discuss your basic approach with current DRM
students but all the analysis and the writing up must be from your team.
Note that reports will be checked for plagiarism via Turnitin.
Canvas will not accept submissions after the due date and time.
Please be sure to cite the work of others where appropriate.
The quality of your writing is as important as the technical accuracy of
your report.
/ 4
Case Study

Compute price changes across
relevant time periods
Compute hedge ratios and optimal
number of contracts (with each
futures singly across periods)
Compute hedge ratios and optimal
number of contracts (with both
futures across periods)
Compute hedging outcome /
st. dev. of cash flows between scenarios
Compute hedging outcome /
st. dev. of cash flows with both futures
Executive summary (to start the report)/1
Describing the basic approach to your
Discuss choice of futures
contracts for the hedge
Discuss and explain hedging
performance under different scenarios
Discuss issues (see factors a) and b) in
case study) and how to deal with
Explain negative oil prices in Apr 2020/3
BONUS: Report lodged as requested/1

/ 5
Technical Note: Hedge Ratios and Why Hull’s Formulae are Used
This section is designed to give you a bit more detail on where the formulae in Hull are
used for constructing cross hedges. It is not part of the assignment, rather designed
to give you a bit more insight into how these procedures are meant to work.
Using the jet fuel example an airline would face the uncertainty of the spot jet fuel
price at some future date; we denote this value as ST. They use an associated futures
contract to hedge (there is no actively traded jet fuel futures available). Let h be the
number of gallons of heating oil gone long through a futures contract per gallon of jet
fuel to be purchased. In this case the net cost of a gallon of jet fuel would be (F0 and
FT are the per gallon heating oil futures prices at the start and end of the hedging
period, respectively):
C S h F F T T T = – – ( ) 0
The variance of this net cost is:
( ) ( )
( ) ( ) ( )
( )
2 ,
Var C Var S h F F
Var S h Var F hCov S F
= – –
= + –
Taking the derivative (with respect to h) to minimize the variance of the cost of
acquiring the jet fuel, while longing futures, we have:
( )
Var CT 2 2 , 0 hVar F Cov S F ( T T T ) ( )
= – =
Which yields:
( )
( )
* , T
Cov S F
Var F

= =
This matches Hull equation (3.1). When we adjust for the size of the position to be
hedged (in gallons) and the size of a futures contract (in gallons) we have:
* * A
N h
This matches Hull’s expression (3.2).
/ 6
Hull also notes the following (page 82):
“The parameters , F, and S in equation (3.1) are usually estimated
from historical data on S and F. (The implicit assumption is that the
future will in some sense be like the past.) A number of equal
nonoverlapping time intervals are chosen, and the values for each of S
and F are observed. Ideally, the length of the time intervals is the same
as the length of the time interval for which the hedge is in effect. In
practice, this sometime severely limits the number of observations that
are available, and a shorter time interval is used.”
When estimating the variances and covariances (or regression coefficients) we often
use price changes taken with frequencies much shorter than the hedging horizon (e.g.
daily data used for estimates when the hedge might be planned for a year). Consider
computing a variance over T days. From the equations above we would be interested
in Var S ( T ). We can write this as follows:

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