Hedging Strategies in Futures Markets

1
Hedging Strategies in Futures Markets

• Fin 288
• Fixed Income Analysis

2
Hedge Terminology

• Short Hedge
• A short hedge occurs when the hedger already owns
  an asset or will own an asset soon and expects to
  sell it at some date in the future. In this case
  the hedger will take a short position in the
  futures market, guaranteeing the price in the
  future at which the asset can be sold.

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Short Hedge Example

• You have agreed to sell 10,000 bushels of corn on
  July 1 at the spot price on that day.
• You are afraid that the price of corn may
  decrease between now and July 1.
• The current futures price for delivery of corn in
  July is 2.10.
• The current spot price of corn is approximately
  1.79 a bushel.

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Recent Corn Spot Prices
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Short Hedge

• By taking agreeing to take a short position 20
  futures contracts you decrease the impact of a
  price decline.
• Assume that on July 1 the spot price for corn is
  1.60. You will sell your corn for
• (1.6)(10,000)16,000
• Assume that the contract expires on July1 so the
  futures price equals the spot price. You can
  close out the futures contract making
• (2.10-1.6) (10,000) 5,000

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The two positions combined

• You made 16,000 in the spot and 5,000 in the
  futures market for a total of 21,000.
• Given that you still sold 10,000 bushels of corn
  You have effectively received 21,000/10,000
  2.10 per bushel (ignoring transaction costs)

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Short Hedge

• What if the spot price of corn is 2.40 on July
  1?
• You sell your corn for
• (2.4)(10,000) 24,000
• In the futures market when you close out the
  contract you loose
• (2.1-2.4)10,000 -3,000
• The total amount you receive is 21,000

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Impact of Hedge

• Regardless of the changes in the spot price the
  result of the hedge is that you have received
  21,000 for 10,000 bushels of corn.
• Note If you had not hedged you would have been
  better off when the price increased without the
  hedge.

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Hedge Terminology

• Long Hedge
• A long hedge occurs when the hedger knows that it
  will be necessary to purchase a given asset at a
  point in the future and wants to lock in the
  future price today. The alternatives to the
  hedge are buying the asset in the future at the
  market price or purchasing it today and holding
  onto it until the asset is needed in the future.

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Long Hedge

• Similar to the short hedge by simultaneously
  entering into a long position and the spot market
  you can fix the price to be paid in the future.

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Assumptions

• The hedge worked because of three assumptions
• The underlying asset in the futures market is the
  same as the asset in the spot market.
• The end of the exposure matches the delivery date
  exactly
• The contract was closed out at the futures price
  prior to delivery

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Basis Risk

• The basis is a hedging situation is defined as
  the Spot price of the asset to be hedged minus
  the futures price of the contract used. When the
  asset that is being hedged is the same as the
  asset underlying the futures contract the basis
  should be zero at the expiration of the contract.
  
• Basis Spot - Futures

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Basis Risk

• The easiest way to illustrate the basis risk is
  with an exampleLet St represent the spot
  price at time tFt represent the futures price at
  time tbt represent the basis at time t

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Basis Risk Illustration

• Assume we enter into a short hedge at time t 1
  and close out the hedge at time t 2. The
  profit on the futures position will equal F1-
  F2The total price paid received from the hedge
  is then
• S2 F1 - F2
• By definitionb1 S1-F1 and b2 S2-F2

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Basis Risk

• By rearranging the price equation S2 F1 - F2
  F1 (S2- F2) F1 b2
• When the hedge is entered into F1 is known but b2
  is unknown.
• The fact that b2 is not known represents the
  basis risk.

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Basis Risk Long Hedge

• The same expression holds for a hedger
  undertaking a long hedge.Loss on Hedge F1-F2
  price paid is S2 F1-F2
• Again the effective price paid is F1b2 where b2
  is unknown when the hedge is taken out.

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Mismatch of Maturities 1

• Assume that the maturity of the contract does not
  match the timing of the underlying commitment.
• Assume that our short hedger for Corn has agreed
  to sell corn on the spot market on October 15.
  However, the months that corn delivery are
  available are March, May, July, September and
  December.

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New Short Hedge

• To hedge the position you now need to take out a
  short position for the September futures
  contract.
• The current futures price for September is 2.28
  a bushel.
• The contract will now need to be closed out on
  October 15, prior to when the futures price and
  spot price converge.

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New Short Hedge

• What if the Spot price on October 15 is 1.90 and
  the futures price for December Delivery is 2.10?
• You sell 10,000 bushels for 1.90 each or
• 19,000
• You close out the futures position and profit
• (2.28-2.10)(10,000) 1,800
• The total price received is 20,800 or 2.08 a
  bushel.

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Result 2

• What if the futures price for delivery in
  December is 2.35 and the spot price is 2.20
• You sell corn in the spot market and receive
• 2.20(10,000) 22,000
• You close out the futures position and loose
• (2.28-2.35)(10,000)-700
• The total you receive is 21,300 (less than you
  would have received in the spot market alone)

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Additional Risk

• In our examples we assumed that the timing of the
  spot position was fixed. It may be the case that
  the timing of the spot position is not known with
  certainty.
• This is especially the case of a long hedger who
  knows that s/he will need to acquire an asset in
  the future, but not know the exact date.

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Minimizing Basis Risk

• Given that the actual timing of the spot asset
  may also be uncertain the standard practice is to
  use a futures contract slightly longer than the
  anticipated spot position.
• The futures price is often more volatile during
  the delivery month also increasing the
  uncertainty of the hedge
• Also the long hedger could be forced to accept
  delivery instead of closing out.

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Mismatch in Maturities 2

• Assume that instead of our original problem there
  are a string of future dates over which corn will
  be needed.
• Anticipated corn demand
• Date Amount
• May 1 15,000 Bushels
• July 1 10,000 Bushels
• September 1 20,000 Bushels

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Strip Hedge

• To hedge this risk, it is possible to hedge each
  position individually.
• On Feb 1 the firm could
• enter into short May contracts for 15,000 bushels
  
• enter into short July contracts for 10,000
  bushels
• enter into short Sept contracts for 20,000 bushels

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Strip Hedge continued

• On each date the respective hedge should be
  closed out.
• The effectiveness of the hedge will depend upon
  the basis at the time each contract is closed
  out.
• (Note in this example each hedge again
  coordinated with the maturity of a contract)

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Rolling Hedge

• Another possibility is to Roll the Hedge
• Feb 1 enter into short May contracts for 45,000
  Bushels
• May 1 enter into long March contracts for 45,000
  Bushels
• enter into short July contracts for
  30,000 Bushels
• July 1 enter into long July contracts for 30,000
  Bushels
• enter into short Sept contracts for 20,000
  Bushels
• Sept 1 enter into long Sept contracts for 20,000
  Bushels

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Rolling the Hedge

• Again the effectiveness of the hedge will depend
  upon the basis at each point in time that the
  contracts are rolled over.
• This opens the from to risk from the resulting
  rollover basis.
• When the contract is closed out there is a cost
  if there has been a loss on the position.
  Therefore there may be a dollar cost to rolling
  over the hedge (basically a margin call).

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Hedging

• So far we have assumed that the underlying asset
  is an exact match for the spot position to be
  hedged. Often this is not the case. Even if the
  asset underlying the futures contract is
  identical to the spot asset, the prices of the
  two will not always move together.
• Two questions
• What futures contract should be used?
• How many contracts should be taken out?

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Hedge Ratio

• The hedge ratio is the ratio of the size of the
  position in the futures market to the size of the
  spot exposure being hedged.
• In our examples so far we have utilized a hedge
  ratio equal to one. In other words the size of
  the futures position was the same as the size of
  the position in the underlying asset.

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Minimum Variance Hedge Ratio

• The ideal hedge ratio should be the one that
  minimizes the variance of the value of the hedged
  position.

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Minimum Variance Hedge Ratio

• DS be the change in the spot price S during a
  period of time equal to the life of the project
• DF be the change in the futures price F during a
  period of time equal to the life of the project
• sS be the standard deviation of DS
• sF be the standard deviation of DF
• r be the coefficient of correlation between DS
  and DF
• h be the hedge ratio

32
The hedge ratio

• The hedge ratio is the ratio of the amount of
  futures positions undertaken in the futures
  market to the number of positions held in the
  spot market.
• Let NA the units of asset A needed at time 2
• Let NF the number of futures contracts held to
  offset the price variation in the spot asset.
• The hedge ration is then

33
Determining the Hedge Ratio

• Assume that you are holding an NA units of an
  asset which can be stored for free and you plan
  on selling it in the future.
• To hedge the risk of a price decline you want to
  undertake a short hedge using NF futures
  contracts.

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Total value of portfolio

• When you sell the asset and close the futures
  position the total change in the value of your
  two positions will equal

35

• Given that
• You can substitute

36

• Given our earlier definitions this can be written
  as

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Hedgers Objective

• The objective of the hedger is to minimize the
  change in the value of the two positions
• NA is known at the beginning of the period and
  will not change. Therefore if the hedger can
  minimize the changes to

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Hedge positions

• We just showed the change in the short hedgers
  position is
• Likewise, the change in the long hedgers position
  is

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Minimum Variance Hedge Ratio

• We want to find the hedge ratio that minimizes
  the variance of the change in the position held
  by the hedger.
• This will depend upon the covariance between the
  spot price and futures price and the variance of
  each variable.

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Min Variance Hedge

• The variance of either hedge position is
• Taking the first derivative of the variance and
  setting it to zero produces the hedge ratio

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Estimating the Hedge Ratio

• The hedge ratio can be rewritten to allow easy
  estimation via regression analysis

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Regression Review

• Equation of a line Y a bX
• Graphing combinations of X and Y form a line.
• X is the independent variable and placed on the
  horizontal axis. Y the dependent variable and
  placed on the vertical axis (The value of Y
  depends upon X)
• a is the Y intercept and b the slope of the line.

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We can observe observations of X,Y and plot them

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Regression Estimates the line that best explains
the relationship between the variables

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The goal is to minimize the sum of the squared
residuals

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Estimating the Regression

• Y a bX
• The slope of the line is then equal to
• The Intercept is

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Confidence in the ResultsR-Squared (R2)

• R2 will range up to one. It is the portion of
  the relationship explained by the regression
• R-Squared (R2) correlationYX2b2sx2/sY2
• Examples
• An R2 of one implies all the points are on the
  line
• An R2 of 0.5 would mean that half of the
  relationship is explained by the line.

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Confidence in the ResultsT-statistic

• The t-statistic tells us whether or not we can
  reject the hypothesis that the variable is equal
  to zero.
• The higher the t-statistic the higher the
  confidence that we can reject the hypothesis that
  the slope is zero.
• If you cannot reject the hypothesis -- It implies
  that the dependent variable has no impact on the
  independent variable.

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T-Statistic

• A Rule of Thumb
• The confidence levels are based upon the number
  of observations, but in general
• If you have a t-statistic above 2.0 you can
  reject the null hypothesis at the 95 level.
• (With 120 observations a t-statistic of 2.36
  allows rejection at the 99 level)

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Standard Error

• Provides a measure of spread around each
  variable.
• Provides a confidence band similar to standard
  deviation)
• We can use standard error to estimate the T-
  Statistic (Assuming a normal distribution)
• T-StatinterceptA/SEA T-Statslope B/SEB

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Quick Review

• Linear Regression - Provides line the best
  describes the relationship between two variables
• R2 - Portion of relationship explained by the
  estimated line
• T-Statistic - Confidence in the estimate of the
  variable (Is is statistically significant?)
• Standard Error - Confidence Interval

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Applying the Regression to the Hedge Ratio

• The minimum variance hedge ratio could be
  estimated by b in the regression.
• (St) ? ? (Ft) ?t

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Hedging using the hedge ratio

• Assume that the airline you are working for wants
  to hedge against a possible increase in the price
  of jet fuel.
• There are not futures contracts available for jet
  fuel so a contract on a different asset must be
  used.
• What contract should be used?
• What is the associated hedge ratio?

http//corporate.bmo.com/rm/commodity/images/Hedgi
ng_JetKero_Prices.pdf
54
Contract options

• NYMEX futures contracts trade on Unleaded
  Gasoline, Light Sweet Crude Oil, Brent Crude Oil,
  Heating Oil, Natural Gas, and Propane
• High correlation of spot prices for Heating Oil
  and Jet Fuel indicate it might be a good
  candidate for the contract (Correlation .994
  from Jan 1995 to October 2004).

http//corporate.bmo.com/rm/commodity/images/Hedgi
ng_JetKero_Prices.pdf
55
A Hypothetical Hedge

• Assume you know that the airline has average
  consumption of 100 Million gallons each month and
  you want to hedge the price of Jet Fuel for June.
• The Heating Oil contract calls for trading to
  stop on the last business day prior to the
  beginning of the delivery month. Assume you plan
  to close out your contract during June at the
  same time you make a spot market purchase for the
  month.

http//www.eia.doe.gov/oil_gas/petroleum/info_glan
ce/prices.html
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A Hypothetical Hedge

• You would need to use the July contract so you
  had the month of June to close out your position.
• The Price for July delivery on 2/3/05 is 1.2277
  per gallon
• There are 42,000 gallons (1,000 barrels) per
  contract.

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Hedge Alternatives

• Without using the hedge ratio you would need to
  enter into 100 Million / 42,0000 2380.95 or
  approximately 2381 long contracts
• By running a regression using the spot price and
  futures price assume that you discover that your
  hedge ratio is 1.07 futures positions for each
  spot position. This implies a need to enter into
  107 million / 42,000 2547.62 or apporximately
  2548 long contracts

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Hedge Results

• The current spot price of jet fuel is 1.4345 per
  gallon.
• Assume that on June 15 you decide to close out
  the contract and the price of Jet Fuel is 1.5345
  per gallon.
• The effectiveness of the hedge depends upon the
  futures price for delivery of Heating oil in
  July. Assume that the futures price is 1.3077

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Hedge Results

• Assuming a 1 to 1 hedge ratio
• Spot Price of Fuel 1.5345 per gallon
• Gain on Hedge 1.3077-1.2277.09 per gallon
• Effective cost of jet fuel 1.5345-.09 1.4445
• Assuming a 1.07 hedge ratio
• Spot Price of Fuel 1.5345 per gallon
• Gain on Hedge 1.3077-1.2277(1.07) .0963 per
  gallon
• Effective cost of jet fuel 1.5345-.0963
  1.4382

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Effective Cost

• Total cost with 1 to 1 hedge 144,450,000
• Total cost with 1.07 Hedge ratio 143,820,000
• A difference of 630,000 for the month!

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Problems

• Current Open Interest for July 2005 is 8532
  contracts, there may not be enough liquidity in
  the market to cover the hedge (will there be
  enough short participants willing to take a short
  position?
• It might be difficult to close out the futures
  position, however current open interest for the
  March 2005 contract is 69970 contracts (there is
  some seasonal variation to also worry about).

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Tailing the Hedge

• Adjustments to the margin account will also
  impact the hedge and need to be made.
• The idea is to make the PV of the hedge equal the
  underlying exposure to adjust for any interest
  and reinvestment in the margin account.

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Should a firm Hedge?

• Tax incentives for Hedging
• Costs of Financial Distress as an Incentive
• Principal Agent Conflicts as an Incentive
• Principal-Agent Conflicts as a Disincnetive
• Lack of Owner Diversification as an Incentive
• Transaction Costs as a Disincentive
• Competitive Environment

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Tax incentive for Hedging

• Consider a mining firm that expects to mine 1,000
  ounces of gold bullion this year at a cost of
  300 per ounce.
• Assume that there are two possible prices for
  gold, 300 or 500 and both are equally possible.
• If the firm has positive income it can use a
  20,000 tax credit to offset taxes and it expects
  to pay a 20 tax rate.

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Unhedged Firm

• Sale Price 300 500
• Gold Revenue 300,000 500,000
• Futures Result 0 0
• Less Production Costs -300,000 -300,000
• Pretax Profit 0 200,000
• Tax Obligation 0 -40,000
• Add Tax Credit 0 20,000
• Net Income 0 180,000
• Expected After tax Revenue 90,000

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Hedged Firm

• Sale Price 300 500
• Gold Revenue 300,000 500,000
• Futures Result 100,000 -100,000
• Less Production Costs -300,000 -300,000
• Pretax Profit 100,000 100,000
• Tax Obligation -20,000 -20,000
• Add Tax Credit 20,000 20,000
• Net Income 100,000 100,000
• Expected After tax Revenue 100,000

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Cost of Financial Distress

• In the previous example the pretax expected
  income was the same for cases, but the after tax
  net income differed.
• In a perfect market, investors could diversify
  with out any costs and eliminate any risk
  associated with the change in expected profits.
• However, if a firm pursues a high risk strategy
  in the real world and goes bankrupt, there are
  high transaction costs.

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Cost of financial distress

• By Hedging the firm can minimize the cost
  associated with possible bad outcomes and
  therefore increase the value of the firm.

69
Principal Agent Conflicts as an Incentive

• In efficient markets manger (agents) act in the
  best interest of the shareholders (principals).
• Shareholders, in theory, can diversify by holding
  a portfolio of securities, if the firms fails the
  loss is limited.
• The Manager has a much larger stake in the firm
  succeeding and may be more risk averse than the
  shareholder therefore hedging when the
  shareholder would prefer not to hedge.

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Principal Agent Conflicts as an Disincentive

• The manger may also run into internal conflict if
  the hedge looses money. It is difficult to
  explain a loss in derivative markets to the board
  of directors and shareholders, even if the loss
  was associated with a hedging strategy.
• Therefore the manager may resist hedging for fear
  of perceived poor performance or even job loss.

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Lack of Owner Diversification

• It may be that the owners are not really
  diversified as assumed in efficient markets they
  would then have an incentive to pressure the
  manger to hedge to decrease risk.

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Transaction Costs

• In the long run gains and losses on hedging
  should offset each other ignoring transaction
  costs.
• However regardless of a loss or gain there is a
  transaction cost to hedging, therefore in the
  long run there may be a cost to hedging that
  decreases the value of the firm.

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Competitive Environment

• If the retail price fluctuates with the wholesale
  price of inputs then the profit margin for the
  firm stays relatively constant without hedging.
• In this case hedging may actually increase the
  volatility of income compared to competitors.