Transcript Lecture 5 Hedging

Lecture 7
Hedging with Futures
Primary Texts
Edwards and Ma: Chapters 5 & 6
CME: Chapter 5
Hedging Fundamentals

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Hedging: The activity of trading futures with the objective of
reducing or controlling price risk (due to uncertainty about
future price levels) is called hedging.
Output Price Risks: A farmer who is growing corn and
planning to sell it in six months (after harvest) cannot be
certain about what the price of corn will be in six months.
Input Price Risks: An airline career that wish to set passenger
fares that remain fixed for the next six months cannot be
certain about what the price of jet fuel will be in six months.
Both output and input price risks can be reduced or controlled
by hedging.
However, quantity risk cannot be controlled by hedging.
Hedging Fundamentals
The Basic Long and Short Hedges

Hedging typically involves taking a position in futures that is
opposite either to

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There are two basic types of hedges: Short hedge and long hedge
Short Hedge: A short hedge occurs when a firm which owns or
plans to purchase or produce a cash commodity sells futures to
hedge the cash position.

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A position that one already has in the cash market, or
A futures cash obligation that one has or will incur
Cash price risk is declining cash prices
Long Hedge: A long hedge occurs when a firm which plans to
purchase a cash commodity in either cash or forward market
purchase futures to hedge the future cash position.

Cash price risk is increasing cash prices
Hedging Fundamentals
Short Hedge (with Zero Basis Risk)
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Suppose it is June 01. A cotton farmer in Lubbock planted cotton in
April, and expects to harvest 100,000 lbs of cotton in October.
On June 01, the cash price for cotton is 55 cents/lb in the local
market and
October NYBOT Cotton futures settled at 57 cents/lb.
The farmer is worried that cash price of cotton at harvest (in
October) may decline significantly.
The farmer may hedge against the declining price risk by short
hedging.
To fully cover her expected cash position at harvest, the cotton
farmer needs to short 2 NYBOT Cotton futures (because the size of
NYBOT cotton futures is 50,000 lbs.)
Perfect Hedging
Short Hedge (with Zero Basis Risk)
Date
Cash Transaction
Futures Transaction
Basis
June 01
Local cash price =
¢/lb
Exp. cash position =
100,000 lbs
Fut. Price =
¢/lb
− 2 ¢/lb
Oct. 19
Harvest = 100,000 lbs
Cash price =
¢/lb
Sell cash cotton at
¢/lb
Fut. Price =
¢/lb
− 2 ¢/lb
Gain/Loss
Rev./Profit
Net Return (Cash Revenue + profit/loss from futures transaction)
Net realized price = Net Return /Cash position
Perfect Hedging
Short Hedge (with Zero Basis Risk)
Date
Cash Transaction
Futures Transaction
Basis
June 01
Local cash price = 55 ¢/lb
Exp. cash position =
100,000 lbs
Fut. Price = 57 ¢/lb
Short 2 NYBOT cotton
futures at 57 ¢/lb
− 2 ¢/lb
Oct. 19
Harvest = 100,000 lbs
Cash price = 50 ¢/lb
Sell cash cotton at 50 ¢/lb
Fut. Price = 52 ¢/lb
Long 2 NYBOT cotton
futures at 52 ¢/lb
− 2 ¢/lb
Gain/Loss
Loss = (50−55=) −5 ¢/lb
Gain = (57−52=) 5 ¢/lb
0
Revenue/Profit Rev. = $0.50×100,000
= $50,000
Profit = $0.05× 2×50,000
= $5,000
Net Return (Cash Revenue + profit/loss from futures transaction)
$55,000
Net realized price = Net Return /Cash position (= $55,000/100,000)
55 ¢/lb
Perfect Hedging
Long Hedge (with Zero Basis Risk)
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Suppose that a beef packer in Amarillo has a plant-capacity of
slaughtering 1,000 fed cattle per month. Each fed cattle weight
approximately 1,200 lbs.
It is Oct 19. The cash price for live cattle is 75 cents/lb in the local
market, and Feb CME Live Cattle futures settled at 85 cents/lb.
The beef packer plans to purchase live cattle in February from the
local market, but is worried that cash price for live cattle may
increase significantly in February.
The beef packer may hedge against the increasing price risk by long
hedging.
To fully cover her expected cash position in February, the beef
packer needs to long 30 CME Live Cattle futures (because the size
of CME Live Cattle futures is 40,000 lbs.)
Perfect Hedging
Long Hedge (with Zero Basis Risk)
Date
Cash Transaction
Oct 22
Local cash price =
¢/lb Futures Price =
Exp. cash position =
1,200,000 lbs
¢/lb
Feb 01
Local cash price = 80 ¢/lb
Purchase 1,000 cattle
@ 50 ¢/lb
Futures Price =
¢/lb
Gain/Loss
Loss =
Gain =
¢/lb
Payment/Profit Payment =
=
Futures Transaction
¢/lb
Basis
Profit =
=
Net Payment (Cash payment - profit/loss from futures transaction)
Net price paid = Net Payment /Cash position (=
)
¢/lb
Perfect Hedging
Long Hedge (with Zero Basis Risk)
Date
Cash Transaction
Futures Transaction
Basis
Oct 22
Local cash price = 75 ¢/lb
Exp. cash position =
1,200,000 lbs
Fut. Price = 85 ¢/lb
Long 30 CME LC
futures cont. @ 85 ¢/lb
− 10 ¢/lb
Feb 01
Local cash price = 80 ¢/lb
Purchase 1,000 cattle
@ 50 ¢/lb
Fut. Price = 90 ¢
Short 30 CME LC fut.
contracts @ 90 ¢/lb
− 10 ¢/lb
Gain/Loss
Loss = (75−80=) −5 ¢/lb
Gain = (90−85=) 5 ¢/lb
0
Payment/Profit Payment = $0.80×1,200,000 Profit=$0.05×30×40,000
= $960,000
= $60,000
Net Payment (Cash payment - profit/loss from futures transaction)
$900,000
Net price paid = Net Payment /Cash position (= 900,000/1,200,000)
75 ¢/lb
The Basis of 1998 Live Cattle Cash and Futures Prices
2.00
0.00
-2.00
-4.00
-6.00
-8.00
-10.00
-12.00
Basis
Hedging with Basis Risk
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When the basis remains unchanged, it is simple to construct
predictable, no-risk hedge.
Unfortunately, perfect hedging is not usual in reality
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However, a change in the basis can affect the results of hedging
Short Hedge
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Like cash and futures prices, basis may change as well
Basis may expand (absolute basis becomes larger) or shrink (absolute
basis becomes smaller)
Basis expands – net realized price is lower than the initial cash price
Basis shrinks – net realized price is higher than the initial cash price
Long Hedge


Basis expands – net price paid is lower than the initial cash price
Basis shrinks – net price paid is higher than the initial cash price
Short Hedge (with Basis Risk)
Basis Expands
Change in Basis = Long Basis – Short Basis
Date
Cash Transaction
Futures Transaction
Basis
June 01
Local cash price = 55 ¢/lb
Exp. cash position =
100,000 lbs
Fut. Price = 57 ¢/lb
− 2 ¢/lb
Oct. 22
Harvest = 100,000 lbs
Cash price = 50 ¢/lb
Sell cash cotton at 50 ¢/lb
Fut. Price =
¢/lb
Gain/Loss
Revenue/Profit
Net Return (Cash Revenue + profit/loss from futures transaction)
Net realized price = Net Return /Cash position
¢/lb
Short Hedge (with Basis Risk)
Basis Expands
Change in Basis = Long Basis – Short Basis
Date
Cash Transaction
Futures Transaction
Basis
June 01
Local cash price = 55 ¢/lb
Exp. cash position =
100,000 lbs
Fut. Price = 57 ¢/lb
Short 2 NYBOT cotton
futures at 57 ¢/lb
− 2 ¢/lb
Oct. 22
Harvest = 100,000 lbs
Cash price = 50 ¢/lb
Sell cash cotton at 50 ¢/lb
Fut. Price = 54 ¢/lb
Long 2 NYBOT cotton
futures at 54 ¢/lb
− 4 ¢/lb
Gain/Loss
Loss = (50−55=) −5 ¢/lb
Gain = (57−54=) 3 ¢/lb
− 2 ¢/lb
Revenue/Profit Rev. = $0.50×100,000
= $50,000
Profit = $0.03× 2×50,000
= $3,000
Net Return (Cash Revenue + profit/loss from futures transaction)
$53,000
Net realized price = Net Return /Cash position (= $53,000/100,000)
53 ¢/lb
Short Hedge (with Basis Risk)
Basis Shrinks
Change in Basis = Long Basis – Short Basis
Date
Cash Transaction
Futures Transaction
Basis
June 01
Local cash price = 55 ¢/lb
Exp. cash position =
100,000 lbs
Fut. Price = 57 ¢/lb
− 2 ¢/lb
Oct. 22
Harvest = 100,000 lbs
Cash price = 50 ¢/lb
Sell cash cotton at 50 ¢/lb
Fut. Price =
¢/lb
Gain/Loss
Revenue/Profit
Net Return (Cash Revenue + profit/loss from futures transaction)
Net realized price = Net Return /Cash position
¢/lb
Short Hedge (with Basis Risk)
Basis Shrinks
Change in Basis = Long Basis – Short Basis
Date
Cash Transaction
Futures Transaction
Basis
June 01
Local cash price = 55 ¢/lb
Exp. cash position =
100,000 lbs
Fut. Price = 57 ¢/lb
Short 2 NYBOT cotton
futures at 57 ¢/lb
− 2 ¢/lb
Oct. 22
Harvest = 100,000 lbs
Cash price = 50 ¢/lb
Sell cash cotton at 50 ¢/lb
Fut. Price = 51 ¢/lb
Long 2 NYBOT cotton
futures at 51 ¢/lb
− 1 ¢/lb
Gain/Loss
Loss = (50−55=) −5 ¢/lb
Gain = (57−51=) 6 ¢/lb
+ 1 ¢/lb
Revenue/Profit Rev. = $0.50×100,000
= $50,000
Profit = $0.06× 2×50,000
= $6,000
Net Return (Cash Revenue + profit/loss from futures transaction)
$56,000
Net realized price = Net Return /Cash position (= $56,000/100,000)
56 ¢/lb
Long Hedge (with Basis Risk)
Basis Expands
Change in Basis = Short Basis – Long Basis
Date
Cash Transaction
Futures Transaction
Oct 22
Local cash price = 75 ¢/lb
Exp. cash position =
1,200,000 lbs
Fut. Price = 85 ¢/lb
Feb 01
Local cash price = 80 ¢/lb
Purchase 1,000 cattle
@ 50 ¢/lb
Fut. Price =
Gain/Loss
Loss =
Gain =
¢/lb
Payment/Profit Payment =0.80×1,200,000
= $960,000
Profit = $
=$
Basis
¢/lb
¢/lb
× 30×40,000
Net Payment (Cash payment - profit/loss from futures transaction)
Net price paid = Net Payment /Cash position (=
)
¢/lb
Long Hedge (with Basis Risk)
Basis Expands
Change in Basis = Short Basis – Long Basis
Date
Cash Transaction
Futures Transaction
Basis
Oct 22
Local cash price = 75 ¢/lb
Exp. cash position =
1,200,000 lbs
Fut. Price = 85 ¢/lb
Long 30 CME LC
futures cont. @ 85 ¢/lb
− 10 ¢/lb
Feb 01
Local cash price = 80 ¢/lb
Purchase 1,000 cattle
@ 50 ¢/lb
Fut. Price = 92 ¢
Short 30 CME LC fut.
contracts @ 90 ¢/lb
− 12 ¢/lb
Gain/Loss
Loss = (75−80=) −5 ¢/lb
Gain = (92−85=) 7 ¢/lb
− 2 ¢/lb
Payment/Profit Payment = $0.80×1,200,000 Profit=$0.07×30×40,000
= $960,000
= $84,000
Net Payment (Cash payment - profit/loss from futures transaction)
$876,000
Net price paid = Net Payment /Cash position (= 876,000/1,200,000)
73 ¢/lb
Long Hedge (with Basis Risk)
Basis Shrinks
Change in Basis = Short Basis – Long Basis
Date
Cash Transaction
Futures Transaction
Oct 22
Local cash price = 75 ¢/lb
Exp. cash position =
1,200,000 lbs
Fut. Price = 85 ¢/lb
Feb 01
Local cash price = 80 ¢/lb
Purchase 1,000 cattle
@ 50 ¢/lb
Fut. Price =
Gain/Loss
Loss =
Gain =
¢/lb
Payment/Profit Payment =0.80×1,200,000
= $960,000
Profit = $
=$
Basis
¢/lb
¢/lb
× 30×40,000
Net Payment (Cash payment - profit/loss from futures transaction)
Net price paid = Net Payment /Cash position (=
)
¢/lb
Long Hedge (with Basis Risk)
Basis Shrinks
Change in Basis = Short Basis – Long Basis
Date
Cash Transaction
Futures Transaction
Basis
Oct 22
Local cash price = 75 ¢/lb
Exp. cash position =
1,200,000 lbs
Fut. Price = 85 ¢/lb
Long 30 CME LC
futures cont. @ 85 ¢/lb
− 10 ¢/lb
Feb 01
Local cash price = 80 ¢/lb
Purchase 1,000 cattle
@ 50 ¢/lb
Fut. Price = 87 ¢
Short 30 CME LC fut.
contracts @ 90 ¢/lb
− 7 ¢/lb
Gain/Loss
Loss = (75−80=) −5 ¢/lb
Gain = (87−85=) 2 ¢/lb
+ 3 ¢/lb
Payment/Profit Payment = $0.80×1,200,000 Profit=$0.02×30×40,000
= $960,000
= $24,000
Net Payment (Cash payment - profit/loss from futures transaction)
$936,000
Net price paid = Net Payment /Cash position (= 936,000/1,200,000)
78 ¢/lb
Hedging Fundamentals
Price Risk versus Basis Risk
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While the objective of hedging is to reduce exposure to price risk,
hedgers trade price risk for basis risk (assumes basis risk).
Hedging => reduces exposure to cash price risk, but increases
exposure to basis risk
Cash Price Risk => the magnitude by which the cash price may
deviate from the mean (cash price)
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Typically measured by the variance (or by standard deviation)
Basis risk => the magnitude by which the basis deviates from the
mean (basis)
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Typically measured by the variance (or by standard deviation)
Basis:
Bt, T = CPt − FPt, T
Change in the basis:
∆Bt, T = ∆CPt − ∆FPt, T
Hedging Fundamentals
Price Risk versus Basis Risk
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If the cash and futures prices always change by exactly the same amount,
there is no basis risk because the change in the basis is zero. When changes
in the cash and futures prices are not equal, there is basis risk.
Basis risk is defined as the variance of the basis.
 B   CP   FP  2 
2

Where
2
2
CP
 FP
2
2
2
 B = Var(B ) = E[B ] – (E[B])
= Var(CP − FP )
= Var(CP) + Var(FP ) – 2Cov(CP , FP)

2
 CP = Var(CP ) = E[CP2] – (E[CP])2

2
 FP = Var(FP) = E[FP2] – (E[FP])2
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ρ = Correlation coefficient between CP and FP =
Cov(CP, FP ) = E[CP×FP] – E[CP]× E[FP]
Cov ( CP , FP )
 CP  FP
Hedging Fundamentals
Price Risk versus Basis Risk

Basis risk is defined as the variance of the basis.
2
2
2
 B   CP   FP  2  CP  FP
 Var ( CP )  Var ( FP )  2 Cov ( CP , FP )


 = 0, if  CP   FP
2
2
2



>
0,
if
CP
FP
B
2
B
2
2
and ρ = 1

The magnitude of the basis risk depends mainly on the degree of correlation
between cash and futures prices
 Higher ρ =>
Lower basis risk
 Lower ρ
=>
Higher basis risk
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For a hedge to be attractive, the basis risk should be significantly less than
the hedger’s cash price risk.
Hedging Fundamentals
Anticipated Hedging Effectiveness
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One measure of anticipated hedging effectiveness is to compare the
basis risk that the hedgers expect to assume with the price risk they
expect to eliminate.
The smaller the anticipated basis risk is compared to the anticipated
price risk, the more effective is the hedge. This measure of
effectiveness can be stated formally as
HE  1 
Var ( B )
Var ( CP )


B
2
 1
 CP
2
The closer HE is to 1, the more effective is the hedge.
The higher the cash price risk relative to the basis risk (i.e., higher
variance of cash price relative to lower variance of the basis), the
more effective is the hedge.
Hedging Fundamentals
Anticipated Hedging Effectiveness
Var ( B )
HE  1 
Var ( CP )

 B   CP
2
2
 B   CP

 B   CP



2
2
2
2
=> HE = 0 =>
=> HE > 0 =>
=> HE < 0 =>
B
2
 1
 CP
2
no potential benefit from hedging
hedging may be beneficial
hedging incurs a potential loss
Anticipated hedging effectiveness is higher the lower is the basis
risk.
The basis risk is lower the higher the correlation between the
cash and futures prices.
ρ↑
=>
σ2B ↓ =>
HE ↑
Devising a Hedging Strategy
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A hedger faces three initial decisions:
 What kind of futures to use,
 Which contract month of that futures to use, and
 How many contracts to hedge with

Which Futures Contract
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A hedger wants to maximize hedging effectiveness
ρ↑ => σ2B ↓ => HE ↑
The hedger chooses a futures contract the price of which is highly
correlated with the cash prices of the commodity (or asset) to be hedged
Futures and cash prices of the same commodity are generally highly
correlated.
When hedging a commodity on which no futures contract is traded, a
closely related commodity is used on which futures contract is traded –
cross-hedging
Devising a Hedging Strategy

Which Contract Month
 Typically, the prices of the nearest month futures contract are the
most highly correlated with cash prices. Thus, using the near
month futures contract reduces the basis risk the most.
 When hedging a continuous cash obligation for a long period of
time, hedgers have to decide between two alternatives:


hedging with a nearby futures contract and rolling the hedge forward
 Rolling the hedge often entails greater brokerage and execution
costs.
hedging with a more distant futures contract and rolling the hedge
less frequently.
 Using a more distant futures contract usually increases basis
risk, since its price would be less correlated with cash prices.
Devising a Hedging Strategy
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How Many Futures Contracts
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
Decide about the optimal futures position to assume - the number of
futures contract times the quantity represented by each contract
In general, the number is determined by the hedge ratio – the ratio of the
size of the futures position to the size of the cash position.
HR 
Qf
Qc


NFC  Q fc
Qc
;
NFC 
Qc
 HR
Q fc
Qc = the quantity of the cash commodity that is being hedged.
Qf = the size (quantity) of the futures position = NFC×Qfc
Qfc = the size of the futures contract
NFC = Number of futures contracts
*
Q
f
*
The hedge ratio that minimizes risk is defined as HR 
Qc

Where, Q*f = the quantity of the commodity futures that minimize risk
Devising a Hedging Strategy
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∆VH = ∆CP× Qc − ∆FP× Qf
∆VH = the change in value of the total hedged position
∆CP = The change in the cash price
∆FP = the change in the futures price
If the change in the value of the hedged position is set equal to zero
(making variability equal to zero), then
 CP
 FP


*

Qf
Qc
 Qf 
*
 CP
 FP
 Q c  Q f  HR  Q c
*
*
But, Q *f  NFC *  Q fc
Where, NFC* = the number of contracts that minimize risk
Qfc = the size of the futures contract
Devising a Hedging Strategy

The number of futures contract with which to hedge in order to
achieve the minimum variance hedge is given by
NFC
*
 Q fc  HR  Q c
NFC
*
*

Qc
 HR
*
Q fc


Estimating the Hedge Ratio
There are two general techniques to estimate the hedge ratio:



the naïve method and
the regression analysis.
Both procedures use historical price data to estimate the hedge ratio.
Devising a Hedging Strategy
Estimating the Hedge Ratio


  CP 
HR *  Mean 

  FP 
The naïve method:
The regression analysis: HR *   , where  CPt      FPt  

Another way of obtaining the estimate of the (regression analysis) hedge
ratio is to use the following formula
HR



*
  
  CP
  FP
σ∆CP = standard deviation of the changes in cash price (∆CP)
σ∆FP = standard deviation of the changes in futures price (∆FP)
ρ∆ = Correlation coefficient between ∆CP and ∆FP
Calculating Hedge Ratio from Cash and Futures Prices
CP
FP
61.40
64.17
64.99
63.49
60.32
59.03
58.05
68.98
69.32
67.88
65.11
62.80
60.12
60.31
61.63
Mean
7.08
Variance
Std. Dev. 2.66
Covariance
Corr. Coeff.
64.93
15.58
3.95
7.47
0.71
March
April
May
June
July
Aug
Sep
HE
HR (Naïve)
HR (Regression)
Basis
-7.59
-5.15
-2.90
-1.63
-2.48
-1.09
-2.26
5.23
∆CP
∆FP
∆CP/∆FP
2.77
0.82
-1.50
-3.17
-1.29
-0.98
0.34
-1.44
-2.77
-2.31
-2.68
0.19
8.25
-0.57
0.54
1.37
0.48
-5.27
-0.56
4.28
2.07
-1.45
1.97
1.40
1.61
0.56
0.80
0.26
0.80
0.82