25762 DERIVATIVES AND RISK MANAGEMENT CASE STUDY

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25762
DERIVATIVES AND RISK MANAGEMENT
CASE STUDY
Due on Thursday 14 May 2026 at 11:59pm
Learning Objectives
By completing this case you will:
– Learn to work with futures and spot price data
– Identify markets in backwardation and in contango
– 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
– Compute and interpret convenience yields
– Communicate complex analysis to a non-technical audience
Learning Activities
By completing this case you will:
– Discuss implications of markets in backwardation and in contango
– 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
Task
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, crude
oil prices crashed and fall to a negative territory (-$37.63) on 20 April 2020 and
skyrocketed to $122.11 by June 2022, while the recent conflict in Middle East brings
uncertainty. We also observed oil markets switching between backwardation and
contango. What would happen to someone using Hull’s proposed strategy over these
volatile times 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.”
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 30,000,000 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
barrels).
Your assistant has downloaded spot jet fuel, crude oil futures and heating oil futures
daily prices from January 2019 through March 2024 (see spreadsheets in the file
“Energy Data – 25762 DRM – Aut 2026.xlsx”). You double check that the combined data
is correctly lined up through time.
Your aim is to assess two factors in 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) Compute and compare the hedge ratios and the optimal number of futures
contracts by using three sample periods: January 2019-December 2021, January
2020-June 2024, and January 2023-June 2024. For the hedging strategy, you consider
three types 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. Report
the hedge ratios over these three data sample periods for the three types of hedging
instruments. Discuss the in-sample hedging performance of these hedging strategies
(compare standard deviations of unhedged and hedged positions).
ii) Assess the out-of-sample hedge performance of these hedge ratios as follows:
a. By using the hedge ratio estimatesfrom January 2019-December 2021 dataset,
you hedge jet fuel price changes from January 2022 to February 2023.
b. By using the hedge ratio estimates from January 2020-June 2024 dataset, you
hedge jet fuel price changes from July 2024 to March 2026.
c. By using the hedge ratio estimates from January 2023-June 2024 dataset, you
hedge jet fuel price changes from July 2024 to February 2026.
Compare the out-of-sample hedging performance over these three sample periods
for the three types of hedging instruments. (compare standard deviations of
unhedged and hedged positions). Which period and which hedging instruments
provide the most effective hedge Why
iii) What are the main issues you have identified with this hedging exercise and what
are your suggestions to deal with those
iv) First time in the history, oil futures markets experienced negative prices. Explain
what happened to the crude oil spot and futures markets on 21 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 response.
/ 2
v) In April 2020, oil futures markets undergone a super contango. Describe what
happened on 21 April 2020 in oil futures markets and what was the response of the
market based on the futures curve on this day (see Table 1 below). News
announcements and online articles should be used to support your response.
vi) Using the futures price data on 26 March 2026 in Table 1, calculate the convenience
yield of crude oil implied by each available futures contract (i.e., 10 contracts in
total). The spot crude oil price on 26 March 2026 was $94.43 per barrel. Assume that
the yield curve was flat, and the risk-free rate was 3.2% per annum with continuous
compounding. Typically, the cost of storing oil for one month corresponds to 0.55%
of the spot oil price payable in advance. Please round off the time-to-delivery to the
nearest month.
a. Plot these implied convenience yields with the time-to-delivery on the x-axis.
b. Comment on any patterns you observe in these convenience yields and explain
to your client what the convenience yield is and what it tells us about the state
of the crude oil market in March 2026. You might want to reflect on the market
conditions and/or the forward curve in your discussion. News announcements
and online articles can be used to support your discussion.
vii) Compare the market conditions on 21 April 2020 and 26 March 2026 and discuss
the implications of these markets’ conditions on risk management practices.
/ 3
Table 1: NYMEX Light Crude Oil futures prices on 21 April 2020 (top panel)
and 26 March 2026 (bottom panel)
(Sources: http://ifs.marketcenter.com/quick_reference.jsp,
https://futures.tradingcharts.com/futures/quotes/CL.html and CME Group
https://www.cmegroup.com/markets/energy/crude-oil/light-sweet-crude.html)
/ 4
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 Thursday 14 May
2026, 11:59pm. 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 10 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.
/ 5
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25762
DERIVATIVES AND RISK MANAGEMENT
Case Study
GRADE BREAKDOWN
Item Points
Computations/Spreadsheet
Compute price changes across relevant time periods /1
Compute hedge ratios and optimal number of contracts (with
each futures singly across periods)
/2
Compute hedge ratios and optimal number of contracts (with
both futures contracts across periods)
/2
Compute in-sample hedging outcome/ st. dev. of cash flows
between scenarios
/ 1
Compute out-of-sample hedging outcome / st. dev. of cash flows
with single futures contracts & sample periods
/1
Compute out-of-sample hedging outcome / st. dev. of cash
flows with both futures contracts & sample periods
/2
Plot forward curves on two days /1
Compute convenience yields and plot them /2
Write-up
Executive summary (to start the report) /1
Describing the basic approach to your analysis /1
Discuss choice of futures contracts for the hedge /1
Discuss and explain in-sample hedging performance under
different scenarios
/1
Discuss and explain out-of-sample hedging performance under
different scenarios
/2
Discuss issues (see factors a) and b) in case study) and how to
deal with them
/2
Explain negative oil prices and super contango in Apr 2020 /2
Explain backwardation in March 2026 /2
Discuss oil super contango/backwardation and implications /1
BONUS: Report lodged as requested /1
Total /25
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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, but rather
designed to give you a bit more insight into how these procedures is 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):
The variance of this net cost is:
Taking the derivative (with respect to h) to minimize the variance of the cost of
acquiring the jet fuel, while longing futures, we have:
( )
( ) ( )
Which yields:
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:
This matches Hull’s expression (3.2).
C S h F F T T T
= ( )0
( ) ( )
( ) ( ) ( )
0
2
( )
2 ,
T T T
T T T T
Var C Var S h F F
Var S h Var F hCov S F
=
= +
Var CT
2 2 , 0 hVar F Cov S F T T T dh
= =
( )
( )
*
,
T
T
S T T
T F
Cov S F
h
Var F

= =
* * A
F
N h Q
Q
=
/ 8
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 . We can write this as follows:
If we assume that the price changes are independent and identically distributed then
the covariance between any two price changes is 0, which means we can simplify the
computation as follows:
( ) ( )
Similarly we have:
( ) ( )
( ) ( )
( )
( )
( )
( )
*
, ,
, ,
T t
T T t t
t t t t
t t
Var F TVar F
Cov S F TCov S F
TCov S F Cov S F
h
TVar F Var F
=
=

= =
This matches the computations given in Hull’s Example 3.3.
( ) T Var S ( ) ( ) 0 1
1 1
T T
T T t t t
t t
Var S Var S S Var S S Var S
= =
= = =
1
T
T t t
t
Var S Var S TVar S
=
= =

25762
DERIVATIVES AND RISK MANAGEMENT
CASE STUDY最先出现在KJESSAY历史案例。