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.

3. 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.

4. Recent Corn Spot Prices*

5. 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 $ 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
( ) (10,000) = $5,000

6. 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)

7. 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
( )10,000 = -$3,000
The total amount you receive is $21,000

8. 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.

9. 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.

10. 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.

11. 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

12. 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

13. Basis Risk
The easiest way to illustrate the basis risk is with an example: Let: St represent the spot price at time t Ft represent the futures price at time t bt represent the basis at time t

14. Basis Risk Illustration
Assume we enter into a short hedge at time t = 1 and close out the hedge at time t = The profit on the futures position will equal F1- F2 The total price paid received from the hedge is then
S2 + F1 - F2
By definition: b1 = S1-F1 and b2 = S2-F2

15. 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.

16. 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 F1+b2 where b2 is unknown when the hedge is taken out.

17. 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.

18. 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.

19. 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:
( )(10,000) = $1,800
The total price received is $20,800 or $2.08 a bushel.

20. 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
( )(10,000)=-$700
The total you receive is $21,300 (less than you would have received in the spot market alone)

21. 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.

22. 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.

23. 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 ,000 Bushels
July ,000 Bushels
September 1 20,000 Bushels

24. 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

25. 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)

26. 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

27. 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).

28. 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?

29. 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.

30. Minimum Variance Hedge Ratio
The ideal hedge ratio should be the one that minimizes the variance of the value of the hedged position.

31. 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.

34. 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

37. Hedger’s 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

38. Hedge positions
We just showed the change in the short hedgers position is
Likewise, the change in the long hedgers position is

39. 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.

40. 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

41. Estimating the Hedge Ratio
The hedge ratio can be rewritten to allow easy estimation via regression analysis

42. 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.

43. We can observe observations of X,Y and plot them

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

45. The goal is to minimize the sum of the squared residuals

46. Estimating the Regression
Y = a + bX
The slope of the line is then equal to
The Intercept is:

47. Confidence in the Results R-Squared (R2)
R2 will range up to one. It is the portion of the relationship explained by the regression
R-Squared (R2) = correlationYX2=b2sx2/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.

48. Confidence in the Results T-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.

49. 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)

50. 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-Statintercept=A/SEA T-Statslope = B/SEB

51. 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

52. Applying the Regression to the Hedge Ratio
The minimum variance hedge ratio could be estimated by b in the regression.
(St) =  +  (Ft) + t

53. 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?

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).

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.

56. 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 $ per gallon
There are 42,000 gallons (1,000 barrels) per contract.

57. Hedge Alternatives
Without using the hedge ratio you would need to enter into 100 Million / 42,0000 = 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 = or apporximately long contracts

58. Hedge Results
The current spot price of jet fuel is $ per gallon.
Assume that on June 15 you decide to close out the contract and the price of Jet Fuel is 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

59. Hedge Results Assuming a 1 to 1 hedge ratio
Spot Price of Fuel = per gallon
Gain on Hedge = =.09 per gallon
Effective cost of jet fuel = =$1.4445
Assuming a 1.07 hedge ratio
Gain on Hedge = (1.07) =.0963 per gallon
Effective cost of jet fuel = =$1.4382

60. 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!

61. 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 contracts (there is some seasonal variation to also worry about).

62. 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.

63. 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
Some of the discsussion and examples for these topics are from Kolb Futures Options and Swaps 3rd edition

64. 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.

65. Unhedged Firm Sale Price 300 500 Gold Revenue 300,000 500,000
Futures Result
Less Production Costs , ,000
Pretax Profit ,000
Tax Obligation ,000
Add Tax Credit ,000
Net Income ,000
Expected After tax Revenue ,000

66. Hedged Firm Sale Price 300 500 Gold Revenue 300,000 500,000
Futures Result 100, ,000
Less Production Costs , ,000
Pretax Profit 100, ,000
Tax Obligation , ,000
Add Tax Credit , ,000
Net Income 100, ,000
Expected After tax Revenue ,000

67. 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.

68. 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.

70. 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.

71. 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.

72. 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.

73. 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.