Lecture 6 Hedging with Futures Primary Texts Edwards and Ma: Chapters 5 & 6 CME: Chapter 5.

Lecture 6Hedging with Futures

Primary Texts

Edwards and Ma: Chapters 5 & 6

CME: Chapter 5

Hedging Fundamentals

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 FundamentalsThe Basic Long and Short Hedges

Hedging typically involves taking a position in futures that is opposite either to

A position that one already has in the cash market, or A futures cash obligation that one has or will incur

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.

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 FundamentalsShort Hedge (with Zero Basis Risk)

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.)

Date Cash Transaction Futures Transaction Basis

June 01 Local cash price = ¢/lbExp. cash position = 100,000 lbs

Fut. Price = ¢/lb − 2 ¢/lb

Oct. 19 Harvest = 100,000 lbsCash price = ¢/lbSell 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)


June 01 Local cash price = 55 ¢/lbExp. cash position = 100,000 lbs

Fut. Price = 57 ¢/lbShort 2 NYBOT cotton futures at 57 ¢/lb

− 2 ¢/lb

Oct. 19 Harvest = 100,000 lbsCash price = 50 ¢/lbSell cash cotton at 50 ¢/lb

Fut. Price = 52 ¢/lbLong 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 Short Hedge (with Zero Basis Risk)

Perfect Hedging Long Hedge (with Zero Basis Risk)

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.)


Oct 22 Local cash price = ¢/lbExp. cash position = 1,200,000 lbs

Futures Price = ¢/lb

Feb 01 Local cash price = 80 ¢/lbPurchase 1,000 cattle@ 50 ¢/lb

Futures Price = ¢/lb

Gain/Loss Loss = ¢/lb Gain = ¢/lb

Payment/Profit Payment = =

Profit = =

Net Payment (Cash payment - profit/loss from futures transaction)

Net price paid = Net Payment /Cash position (= ) ¢/lb




Oct 22 Local cash price = 75 ¢/lbExp. cash position = 1,200,000 lbs

Fut. Price = 85 ¢/lbLong 30 CME LC futures cont. @ 85 ¢/lb

− 10 ¢/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= $960,000

Profit=$0.05×30×40,000 = $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

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The Basis of 1998 Live Cattle Cash and Futures Prices

Basis

Hedging with Basis Risk

When the basis remains unchanged, it is simple to construct predictable, no-risk hedge.

Unfortunately, perfect hedging is not usual in reality Like cash and futures prices, basis may change as well Basis may expand (absolute basis becomes larger) or shrink (absolute

basis becomes smaller) However, a change in the basis can affect the results of hedging Short Hedge

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



Fut. Price = 57 ¢/lb − 2 ¢/lb


Fut. Price = ¢/lb ¢/lb

Gain/Loss

Revenue/Profit



Short Hedge (with Basis Risk)Basis Expands

Change in Basis = Long Basis – Short Basis




− 2 ¢/lb



− 4 ¢/lb

Gain/Loss Loss = (50−55=) −5 ¢/lb Gain = (57−54=) 3 ¢/lb − 2 ¢/lb


Profit = $0.03× 2×50,000 = $3,000



Short Hedge (with Basis Risk)Basis Expands




Fut. Price = 57 ¢/lb − 2 ¢/lb


Fut. Price = ¢/lb ¢/lb

Gain/Loss

Revenue/Profit



Short Hedge (with Basis Risk)Basis Shrinks





− 2 ¢/lb



− 1 ¢/lb

Gain/Loss Loss = (50−55=) −5 ¢/lb Gain = (57−51=) 6 ¢/lb + 1 ¢/lb


Profit = $0.06× 2×50,000 = $6,000



Short Hedge (with Basis Risk)Basis Shrinks




Fut. Price = 85 ¢/lb


Fut. Price = ¢/lb


Payment/Profit Payment =0.80×1,200,000 = $960,000

Profit = $ × 30×40,000 = $



Long Hedge (with Basis Risk)Basis Expands

Change in Basis = Short Basis – Long Basis




− 10 ¢/lb



− 12 ¢/lb

Gain/Loss Loss = (75−80=) −5 ¢/lb Gain = (92−85=) 7 ¢/lb − 2 ¢/lb


Profit=$0.07×30×40,000 = $84,000



Long Hedge (with Basis Risk)Basis Expands




Fut. Price = 85 ¢/lb


Fut. Price = ¢/lb


Payment/Profit Payment =0.80×1,200,000 = $960,000

Profit = $ × 30×40,000 = $



Long Hedge (with Basis Risk)Basis Shrinks





− 10 ¢/lb



− 7 ¢/lb

Gain/Loss Loss = (75−80=) −5 ¢/lb Gain = (87−85=) 2 ¢/lb + 3 ¢/lb


Profit=$0.02×30×40,000 = $24,000



Long Hedge (with Basis Risk)Basis Shrinks


Hedging FundamentalsPrice Risk versus Basis Risk

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) Typically measured by the variance (or by standard deviation)

Basis risk => the magnitude by which the basis deviates from the

mean (basis) Typically measured by the variance (or by standard deviation) Basis: Bt, T = CPt − FPt, T

Change in the basis: ∆Bt, T = ∆CPt − ∆FPt, T


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.

Where = Var(B ) = E[B2] – (E[B])2

= Var(CP − FP )

= Var(CP) + Var(FP ) – 2Cov(CP , FP)

= Var(CP ) = E[CP2] – (E[CP])2

= Var(FP) = E[FP2] – (E[FP])2

ρ = Correlation coefficient between CP and FP =

Cov(CP, FP ) = E[CP×FP] – E[CP]× E[FP]

FPCPFPCPB 2222

2B

2CP

2FP

FPCP

FPCPCov

),(


Basis risk is defined as the variance of the basis.

= 0, if and ρ = 1 > 0, if

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

For a hedge to be attractive, the basis risk should be significantly less than the hedger’s cash price risk.

),(2)()(

2222

FPCPCovFPVarCPVarFPCPFPCPB

2B

22FPCP

2B

22FPCP

Hedging FundamentalsAnticipated Hedging Effectiveness

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

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.

2

2

1)(

)(1

CP

B

CPVar

BVarHE

Hedging FundamentalsAnticipated Hedging Effectiveness

=> HE = 0 => no potential benefit from hedging => HE > 0 => hedging may be beneficial => HE < 0 => 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 ↑

2

2

1)(

)(1

CP

B

CPVar

BVarHE

22CPB 22CPB 22CPB

Devising a Hedging Strategy

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 A hedger wants to maximize hedging effectiveness ρ↑ => σ2

B ↓ => 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


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.


How Many Futures Contracts 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.

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 The hedge ratio that minimizes risk is defined as

Where, Q*f = the quantity of the commodity futures that minimize risk

HRQ

QNFC

Q

QNFC

Q

QHR

fc

c

c

fc

c

f

;

c

f

Q

QHR

**


∆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

But, Where, NFC* = the number of contracts that minimize risk

Qfc = the size of the futures contract

cfcfc

f QHRQQFP

CPQ

Q

Q

FP

CP

***

*

fcf QNFCQ **


The number of futures contract with which to hedge in order to achieve the minimum variance hedge is given by

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.

cfc QHRQNFC **

** HRQ

QNFC

fc

c


Estimating the Hedge Ratio The naïve method: The regression analysis:

Another way of obtaining the estimate of the (regression analysis) hedge ratio is to use the following formula

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

FP

CPMeanHR*

tt FPCPwhereHR ,*

FP

CPHR

*

Calculating Hedge Ratio from Cash and Futures Prices

CP FP Basis ∆CP ∆FP ∆CP/∆FPMarch 61.40 68.98 -7.59

April 64.17 69.32 -5.15 2.77 0.34 8.25

May 64.99 67.88 -2.90 0.82 -1.44 -0.57

June 63.49 65.11 -1.63 -1.50 -2.77 0.54

July 60.32 62.80 -2.48 -3.17 -2.31 1.37

Aug 59.03 60.12 -1.09 -1.29 -2.68 0.48

Sep 58.05 60.31 -2.26 -0.98 0.19 -5.27

Mean 61.63 64.93 -0.56 -1.45 0.80

Variance 7.08 15.58 5.23 4.28 1.97

Std. Dev. 2.66 3.95 2.07 1.40

Covariance 7.47 1.61

Corr. Coeff. 0.71 0.56

HE 0.26HR (Naïve) 0.80HR (Regression) 0.82

Lecture 6 Hedging with Futures Primary Texts Edwards and Ma: Chapters 5 & 6 CME: Chapter 5.

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Transcript of Lecture 6 Hedging with Futures Primary Texts Edwards and Ma: Chapters 5 & 6 CME: Chapter 5.