Hedging Strategies Using
Futures
Chapter 3
Fundamentals of Futures and Options Markets, 7th Ed, Global Edition
Ch3, Copyright © John C. Hull 2010
1
Long & Short Hedges


A long futures hedge is appropriate when you
know you will purchase an asset in the future
and want to lock in the price
A short futures hedge is appropriate when you
know you will sell an asset in the future & want
to lock in the price
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
2
Arguments in Favor of Hedging
Companies should focus on the main business
they are in and take steps to minimize risks
arising from interest rates, exchange rates, and
other market variables e.g.
A company which exports iron ore should focus
on producing and selling iron ore and not worry
about selling the $US it earns from its exports
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
3
Arguments against Hedging



Shareholders are usually well diversified and
can make their own hedging decisions
It may increase risk to hedge when competitors
do not
Explaining a situation where there is a loss on
the hedge and a gain on the underlying can be
difficult
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
4
Convergence of Futures to Spot
(Hedge initiated at time t1 and closed out at time t2)
Futures
Price
Spot
Price
Time
t1
t2
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
5
Choice of Contract
When choosing a futures contact you choose
1. The delivery month that is as close as possible
to, but later than, the end of the life of the
hedge
2. The underlying asset if possible. When there is
no futures contract on the asset being hedged,
choose the contract whose futures price is
most highly correlated with the asset price.
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
6
Basis Risk


Basis is the difference between spot &
futures (St - Ft)
Basis risk arises because of the
uncertainty about the basis when the
hedge is closed out
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
7
Long Hedge



Suppose that
F1 : Initial Futures Price
F2 : Final Futures Price
S2 : Final Asset Price
You hedge the future purchase of an asset by
entering into a long futures contract
Cost of Asset
S2 + (F1 – F2) = F1 + Basis or (S2 – F2)
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
8
Short Hedge



Suppose that
F1 : Initial Futures Price
F2 : Final Futures Price
S2 : Final Asset Price
You hedge the future sale of an asset by
entering into a short futures contract
Price Realized
S2 + (F1 – F2) = F1 + Basis or (S2 – F2)
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
9
Basis Risk Examples p 56 - 57
Ex 3.3 p 56: Short Hedge
A firm in the US exports goods to Japan on Mar 1
They expect to receive 50m Yen at the end of July.
They will have to sell these yen for $US
At Mar 1 the Sept futures price is F1 = 0.780.
Exchange rates are shown as cents per yen.
A value of 0.78 means we can sell 100 yen for 78
cents and 1000 yen sells for $7.80
(A higher exchange rate value means more $
income for US exporting firm)
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
10
Basis Risk Examples p 56 - 57
At the end of July the Spot price is S2 = 0.720.
The new Sept Futures price is now F2 = 0.725.
At the end of July we find the Basis is
S2 - F2 = (0.72 – 0.725) = - 0.005
At this date the Gain on futures is
F1 - F2 = (0.78 – 0.725) = 0.055
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
11
Basis Risk Examples p 56 - 57
There are 2 ways to find the net exchange rate
Method 1:
July Spot Price plus Gain on futures
S2 + (F1 – F2) = 0.72 + 0.055 = 0.775
Method 2:
March Futures Price plus Basis in July
F1 + (S2 – F2) = 0.78 - 0.005 = 0.775
Work through Ex 3.4 p 57
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
12
Cross Hedging (Ch 3.4)
When it is not possible to obtain a futures contract for
a commodity itself we might use a futures contract on
a similar commodity e.g. using Heating Oil futures for
Jet Fuel.
(underlying asset vs. hedged asset)
The Futures price will change in a similar but not
exactly the same way as the Spot price of the original
commodity.
The hedge ratio h* is the ratio of the value of the
futures contracts (heating oil) and the value of the
original commodity (jet fuel) we are trying to hedge.
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
13
Optimal Hedge Ratio
Our aim is to buy or sell futures in the related
commodity so our futures contracts come as
close as possible to hedging changes in the
spot prices of the original commodity.
The optimal hedge ratio formula
h*  
S
F
will lead us to buy that amount of futures
contracts that minimizes the differences
between changes in spot prices and changes in
the values of the futures contracts.
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
14
Optimal Hedge Ratio
In the formula
S
h*  
F
S is the standard deviation of changes in the spot
price during the hedging period DS,
F is the standard deviation of changes in the
futures price during the hedging period DF
 is the coefficient of correlation between
changes in spot prices DS and futures prices DF.
Check the Appendix p 74 - 78
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
15
Optimal Hedge Ratio
The hedge ratio
h*  
S
F
can be calculated in 2 ways
Both methods use the values during the hedging
period shown in Table 3.2 on p 61 and in the
Excel file Table 3.2 where
the spot price changes DS are in C7:C21 and
the futures price changes DF are in B7:B21
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
16
Optimal Hedge Ratio
Method 1:
Calculate S, F and  and use them in
S
h*  
F
S in cell C24 enter =STDEV(C7:C21)
F in cell B24 enter =STDEV(B7:B21)
 in cell B25 enter =CORREL(B7:B21,C7:C21)
To calculate h* in cell B27 enter
=B25*C24/B25
We find
0.928*0.0263/0.0313 = 0.778
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
17
Optimal Hedge Ratio
Method 2:
The formula for the optimal hedge ratio
h*  
S
F
is equal to the estimated slope of the linear
regression function
DS = a + b DF + e
To find h* in cell B27 enter
=LINEST(C7:C21,B7:B21)
We will obtain the same value 0.778
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
18
Optimal Hedge Ratio
INTERPRETATION 1:
The value h* = 0.778 indicates that if we have an
amount we wish to hedge of $1,000,000 we will
buy futures contracts worth
0.778 of 1,000,000 or 778,000
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
19
Optimal Hedge Ratio
INTERPRETATION 2:
When we square the correlation coefficient  we
obtain what is called
R2 or the Coefficient of Determination (= 2)
Here 0.9282 equals 0.862
It is now called the hedge effectiveness
It shows the proportion of the variation in DS that
is explained by variation in DF
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
20
Optimal Number of Contracts
In our Jet Fuel example we have the following
information
QA is the amount we wish to hedge
(2,000,000 gals of jet fuel)
QF is the amount we listed on the futures contract
(42,000 gals of heating oil)
The required number of contracts N* is
h*Q
A  0.778* 2000000  37.03 (37)
N *
42000
Q
F
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
21
Tailing the Hedge


There are two ways of determining the number
of contracts to use for hedging are
 Compare the quantity to be hedged with the
futures contract quantity as we just did
 Compare the value to be hedged with the
value of one futures contract
Values are the products of prices and quantities
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
22
Tailing the Hedge
In our example the spot price pS is (1.94) and
VA the $ amount we wish to hedge is pS* QA
VA = 1.94*2,000,000 = $3,880,000
The futures price pF is (1.99) and
VF the $ amount for a futures contract is pF* QF
VF = 1.99*42,000 = $83,580
Here the required number of contracts is
h*V
A  0.778*3880000  36.11(36)
N *
V
83550
F
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
23
Tailing the Hedge
The required number of futures contracts is
h*V
A  0.778*3880000  36.11(36)
N *
V
83550
F
Why do the two methods give slightly different
answers namely 37 and 36 ? Our value formula
h*V
h*pS Q
h*Q p
A
A 
A S
N *
pF
V
pF Q
Q
F
F
F
is equal to our quantity formula multiplied by the
price ratio pS / pF and 1.94/1.99 is 0.975
This process is called tailing the hedge
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
24
Hedging Using Index Futures
(Page 61 - 67)
To hedge the risk in any share portfolio
the number of contracts that should be
shorted is
VA
N* = b
VF
where
VA is the current value of the portfolio,
β is the Portfolio Beta
VF is the value of a futures contract
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
25
Hedging Using Index Futures
(Page 61 - 67)
In the special case where a portfolio is
well diversified and has b = 1 now to
hedge the risk in this portfolio the number
of contracts that should be shorted is
VA
N* =
VF
where
VA is the current value of the portfolio,
VF is the value of a futures contract
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
26
Hedging Using Index Futures
(Page 61 - 67)
Comparing the stock index contracts formula
N* = b V A
VF
with the formula for the number of commodity
futures contracts
VA
N* h*
VF
we see that h* = b which agrees our definition
of b for any financial asset.
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
27
Reasons for Hedging an Equity
Portfolio


Desire to be out of the market for a short period
of time. (Hedging may be cheaper than selling
the portfolio and buying it back.)
Desire to hedge systematic risk
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
28
Stock Index Hedge Example
(Page 64 - 67)
We need to hedge a share portfolio over the next 3
months and we have to use futures contracts
which mature in 4 months where
S&P 500 index is now at
1,000
(B13)
Futures price of S&P 500 is at 1,010
(B5)
Value of portfolio is VA $5,050,000
(B14)
Beta of portfolio is
1.5
(B19)
One contract is on $250 times the index (B16)
Risk-free interest rate Rf
4% p.a.
(B17)
Dividend yield on index DY 1% p.a.
(B18)
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
29
Example
(contd.)
What position in futures contracts on the S&P
500 is necessary to hedge the portfolio which we
have to sell shares in 3 months time?
Value of a contract is VF = 250*1010 = 252,500
We have to sell N* futures contracts where
N* = b
VA
VF
5050000
 30
= 1.5
252500
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
30
Example (contd.)
What are the possible outcomes when we buy 30
Futures contracts on the S&P 500 at 1010 ?
In Table 3.4 p 64, 65 (Excel file Table 3.4.xls) we
look at what happens in 3 months time when we
have different values for
The index or S2
900, 950, 1000, 1050, 1100 (B4:F4)
The futures price of index or F2
902, 952, 1003, 1053, 1103 (B6:F6)
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
31
Example (contd.)
Row 2 (B5:F5) Contains F1 = 1010
Row 4 (B7:F7) Contains the Total Gains on the 30
Futures contracts
N*(F1 – F2)*250
In B7 =$B$15*(B5-B6)*$B$16
Row 5 (B8:F8) Contains the Return on the Market
(both Dividends and Capital Gains)
((DY*S1)+(S2-S1))/S1
In B8 =(($B$18*$B$13)+(B4-$B$13))/$B$13
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
32
Example (contd.)
Row 6 (B9:F9) Contains the Expected Returns on
the Portfolio E(RP) based on the CAPM model
Rf + b*(RM – Rf)
In B9 =$B$17+$B$19*(B8-$B$17)
Row 7 (B10:F10) Contains the Expected Portfolio
value in 3 months
VA*(1+E(RP))
In B10 =$B$14*(1+B9)
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
33
Example (contd.)
Row 8 (B11:F11) Contains the Total value of the
Expected Portfolio value and the Gains on the
Futures position
VA*(1+E(RP)) plus N*(F1 – F2)*250
In B11 =B10+B7
Highlight B4:B11 and drag to column F
Check the final row of values !!
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
34
Changing the Portfolio Beta (p 66)
The value of a combination of shares and index
futures contracts is always about 5,100,000 while
the market index changes from 900 to 1100
As large changes in the market index produce little
or no change in the value of this combination of
shares and index futures contracts we say that this
combination has a beta of 0
How can we use futures contracts to obtain
combinations of shares and futures contracts with
beta values such as 0.75 or 2 ?
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
35
Changing the Portfolio Beta (p 66)
In our example where we have to sell in the future
and where
S&P 500 index is now at
1,000
Futures price of S&P 500 is at 1,010
Value of portfolio is VA $5,050,000
Beta of portfolio is
1.5
We get a beta of 0 or complete hedge and a beta
of 0 when we have N = 30
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
36
Changing the Portfolio Beta (p 66)
To obtain a smaller beta such as 0.75 or half of
1.5 we simply sell 15 or half of the N = 30
To obtain a larger beta such as 2.0 or one third
larger than 1.5 we now buy i.e. take a long
position in one third of 30 which is 10 contracts.
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
37
Changing the Portfolio Beta (p 66)
The general rule we follow to change b to b*
To reduce the b i.e. when b > b* we
VA
sell / short position of (b - b* )
VF
To increase the b i.e. when b > b* we
VA
buy / long position of (b - b* )
VF
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
38
Stock Picking


(p 66 – 67)
If you think you can pick stocks that will
outperform the market, futures contract can be
used to hedge the market risk
If you are right, you will make money whether
the market goes up or down
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
39
Rolling The Hedge Forward p 67-68



When a hedge is required up until a date
after the delivery date for the futures
contract then we can use a series of futures
contracts to increase the life of a hedge
Each time we switch from 1 futures contract
to another we incur a type of basis risk
In Table 3.5 where 3 different contracts we
can call A, B and C are used there are
different F1 and F2 values for each contract
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
40
Rolling The Hedge Forward p 67-68
In slide 12 we saw that there are 2 ways to
find the net exchange rate. The first method
we used was to find the
July Spot Price plus the Gain on futures
S2 + (F1 – F2)
In the example where we buy a futures
contract and renew it 2 times we have 3
forward contracts A, B and C the formula is
S2 + (FA1 – FA2) + (FB1 – FB2) + (FC1 – FC2)
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
41
Rolling The Hedge Forward p 67-68
Our Spot Prices at the beginning and when we
sell the oil are
S1 = 69.00 and S2 = 66.00
Using the rolling hedge we can obtain
S2 + (FA1 – FA2) + (FB1 – FB2) + (FC1 – FC2)
= 66 + (68.2 – 67.4) + (67 – 66.5)
+ (66.3 – 65.9)
= 66 + 0.8 + 0.5 + 0.4 = 66 + 1.7
= 67.7
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
42
Rolling The Hedge Forward p 67-68
Futures contracts must be settled daily while
positions being hedged can be months or
years in the future.
This means that in the short term there can be
major mismatches between cash outflows for
margins in the Futures market and the final
inflows from the customers.
In the Metallgesellschaft (MG) example p 69 in
the early 1990s the firm sold futures contracts
5 to 10 years ahead (and lost over $1 billion)!
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
43
Rolling The Hedge Forward p 67-68
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
44
Rolling The Hedge Forward p 67-68
In 1992 MG offered customers Futures
contracts for heating oil and petrol with futures
prices 6 to 8 cents above current spot prices.
Their marketing campaign was very successful.
To hedge these future sales MG bought short
term futures contracts.
Shortly afterwards prices started to fall.
In Table 2.1 p 27 we see that falling spot and
then futures prices mean a firm has to pay
margin calls
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
45
Rolling The Hedge Forward p 67-68
The management at MG decided the firm could
not afford these short term cash outflows.
The firm paid customers to cancel their forward
contracts.
MG incurred losses of $1.33 billion
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
46
HOMEWORK EXERCISE

An investor has a foreign equity portfolio worth
EUR10m, which has a beta of 1.1. The investor
is concerned about short term volatility in the
next 12 months. The EUR denominated stock
index futures is priced at EUR120,000 and has a
beta of 0.95. The AUD/EUR spot and forward
rates are respectively $0.80 and $0.815. The
risk-free domestic and foreign interest rates are
respectively 6% & 4% pa.
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
47
HOMEWORK EXERCISE


Calculate the optimal hedge on the stock
index and the EUR?
During the year the EUR stock index fell
4.55%, the AUD/EUR exchange rate fell to
$0.785 and the futures price fell to
EUR110,600. Explain all possible
outcomes.
Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C. Hull 2010
48