1. Drake DRAKE UNIVERSITY Fin 284 Futures Markets Fin 284 Fixed Income Analysis

2. Drake Drake University Fin 284 Derivatives Basic Definition Any Asset whose value is based upon (or derived from) an underlying asset. The performance of the derivative is dependent upon the performance of the underlying asset. Risk Management Since a derivatives performance is based on an underlying asset they can often be used to decrease the risk associated with changes in the spot price of an asset.

3. Drake Drake University Fin 284 Basic Types of Derivative Contracts Forward Contracts Agreement between two parties to purchase or sell something at a later date at a prie agreed upon today Futures Contract Same idea as a forward, but the contract trades on an exchange and the counter party is not set.

4. Drake Drake University Fin 284 Brief History of Derivatives Markets 1100’s Forward contracts were used by Flemish traders who gathered -- a letter de faire- forward contract specifying delivery at a later date 1600’s Japan -- Cho-ai-mai (Rice Trade on Book) Essentially futures contracts on rice designed to manage the volatility in rice prices caused by weather, warfare and other risks. Netherlands -- formal futures markets developed to trade tulip bulbs in 1636 Options also appeared in Amsterdam during the 1600’s 1863 -- Confederacy issued 20 year bonds denominated in French francs and convertible to cotton (a dual currency cotton indexed bond)

5. Drake Drake University Fin 284 Brief History Continued. Organized Exchanges in US Chicago Board of Trade Established in 1848 to bring farmers and merchants together. Futures Contracts were first traded on the CBOT1865. Developed the first standard contract Chicago Mercantile Exchange Started as the Chicago Produce Exchange in 1874 for trade in perishable agricultural products. In 1919 it became the Chicago Mercantile Exchange (CME). Introduced a contract for S&P 500 futures in 1982. NYMEX 1872 KCBOT 1876

6. Drake Drake University Fin 284 Other US Exchanges NYBOT Coffee Sugar and Cocca Exchange New York Futures Exchange Minneapolis Grain Exchange Philadelphia Board of Trade

7. Drake Drake University Fin 284 Payoff on Forward Contracts Long Position Agreeing to buy a specified amount (The Contract Size) of a given commodity or asset at a set point in time in the future (The Delivery Date) at a set price (The Delivery Price) Payoff The payoff will depend upon the spot price at the delivery date. Payoff = Spot Price – Delivery Price

8. Drake Drake University Fin 284 Example Assume you have agreed to buy L 1,000,000 in 3 months at a rate of L 1 = $1.6196 Spot RateSpot – Delivery PricePayoff $1.65$1.65-$1.6196=$0.0304$30,400 $1.6169$1.6196-$1.6196=00 $1.55$1.55-$1.6196=$0.0696-$69,600

9. Drake Drake University Fin 284 Example Graphically Spot Price Payoff 1.55 1.61961.650.0304 -.0696

10. Drake Drake University Fin 284 Payoff: Short Position Agreeing to sell a specified amount (The Contract Size) of a given commodity or asset at a point of time in the future (The Delivery Date) at a set price (The Delivery Price). Payoff on Short position Since the position is profitable when the price declines the payoff becomes: Payoff = The Delivery Price – The Spot Price

11. Drake Drake University Fin 284 Long vs. Short For a long position to exist (someone agreeing to buy) there must be an offsetting short position (someone agreeing to sell). Assume that you held the short position for the previous example: sell L 1,000,000 in 3 mos at a rate of L 1 = $1.6196 Spot RateSpot – Delivery PricePayoff $1.65$1.6196-$1.65=-$0.0304-$30,400 $1.6169$1.6196-$1.6196=00 $1.55$1.6196-$1.55= $0.0696$69,600

12. Drake Drake University Fin 284 Example Graphically Spot Price Payoff 1.55 1.61961.650.0304 -.0696

13. Drake Drake University Fin 284 Contract Goals The goal of the contract is to decrease risk, assume that you had to pay L 1,000,000 in 3 months for the shipment of an input. You are afraid that the $ price will increase and you will pay a higher price. Similarly the other party may be afraid that the $ price will decrease (maybe they are receiving a payment in 3 months)

14. Drake Drake University Fin 284 Determining the delivery price The delivery price will be determined by the participants expectations about the future price and their willingness to enter into the contract. (Today’s spot price most likely does not equal the delivery price). What else should be considered? They should both also consider the time value of money

15. Drake Drake University Fin 284 Future and Forward contracts Both Futures and Forward contracts are contracts entered into by two parties who agree to buy and sell a given commodity or asset (for example a T- Bill) at a specified point of time in the future at a set price.

16. Drake Drake University Fin 284 Futures vs. Forwards Future contracts are traded on an exchange, Forward contracts are privately negotiated over- the-counter arrangements between two parties. Both set a price to be paid in the future for a specified contract. Forward Contracts are subject to counter party default risk, The futures exchange attempts to limit or eliminate the amount of counter party default risk.

17. Drake Drake University Fin 284 Other Forward Contract Risks One goal of the negotiation is to specify exactly the type, quantity, and means of delivery of the underlying asset. The chance that an asset different than anticipated might be delivered should be eliminated by the contract. Futures contracts attempt to account for this problem via standardization of the contract.

18. Drake Drake University Fin 284 Futures Contracts Long Position: Agreeing to purchase a specified amount of a given commodity or asset at a point in time in the future at a set price (the futures price) Short Position: Agreeing to sell a specified amount of a given commodity or asset at a point of time in the future for a set price (the futures price).

19. Drake Drake University Fin 284 Standardization of Futures Contracts To promote confidence in the system and eliminate counter party default risk, future contracts are highly standardized.

20. Drake Drake University Fin 284 Specifications of Futures Contract The Asset The Contract Size Delivery Arrangements Delivery Months Price Quotes Price Limits Position Limits

21. Drake Drake University Fin 284 Contract Specifications Asset Quality and type of asset are specified to guarantee specific product is delivered. Contract Size The amount of asset that is to be delivered for one contract Delivery Arrangements More important for commodities than financial assets. Specify how delivery occurs and location.

22. Drake Drake University Fin 284 Contract Specifications Delivery Months When delivery will occur (and during what part of the month delivery can occur) Price Quotes Contract must specify the units for the price quote (1/32 of a dollar etc) Also implicitly establishes the minimum fluctuation for the price of the contract.

23. Drake Drake University Fin 284 Contract Specifications Price Limits Designed to add stability to the market, limits on the maximum fluctuation in price that can occur during a trading day. Position Limits Limits the number of contracts that can be entered into by a speculator. Speculator –attempting to profit from a movement in the market Hedger – attempting to offset an underlying spot position.

24. Drake Drake University Fin 284 Does Delivery need to take place? No – most contracts will be closed out. Closing out a contract is simply taking the opposite (short if you are long or vice versa) position. The change in the futures price will be your gain or loss. With a futures contract your counter party does not remain the same. It does not matter who takes the opposite position. This is not the case for a forward contract.

25. Drake Drake University Fin 284 Forward Contracts Futures Contracts Private contract between Traded on two parties an exchange Not StandardizedStandardized Usually a single delivery date Range of delivery dates Settled at the end of contractSettled daily Delivery or final cash Contract is usually closed settlement usually takes place out prior to maturity

26. Drake Drake University Fin 284 Important Terminology Open Interest The number of contracts that are currently open (both a short and long position exist). What happens to open interest if a new long position is taken out? It could Increase It could decrease It might not change. The answer depends on whether both the long and short positions are new, or closing out or one of each.

27. Drake Drake University Fin 284 Margin Requirements To limit counter party default risk, the futures exchange requires participants to place funds in a margin account when the contract is taken out. Some Terminology: Initial Margin: The original amount deposited in the margin account Maintenance margin: The amount that must remain in the margin account Margin call – Notice that the margin account has dropped below the maintenance margin, more money must be added to the account

28. Drake Drake University Fin 284 Margin Example Example: An investor has taken a long position in gold (agreed to buy gold at some date in the future). Assume that the agreement is for 2 gold contracts each contract consists of 100 ounces of gold. The futures price is $400 per ounce. This implies that the participant would need 200*400 = $80,000 to purchase gold at the expiration of the contract.

29. Drake Drake University Fin 284 Margin Example If the futures price for gold decreases to $398, the investor would suffer a loss if the contract is closed out. The loss would total (400 - 398)200 = $400. The fear is that if at the expiration of the contract the price is 398, the participant will not honor the contract since it would result in a loss of $400.

30. Drake Drake University Fin 284 Margin Example To counteract this the investor is ask to put a sum of money into a margin account lets assume $2,000 per contract or $4000 total. When the futures price declines the loss of $400 is taken from the margin account of the investor and given to a participant that took a short position.

31. Drake Drake University Fin 284 Margin Example The value of the contract is marked to market each day, and the margin account is adjusted. The margin is effectively guaranteeing that the position is covered. If the level of the account falls below the maintenance margin the investor is required to put more funds into the account this is known as a margin call. The extra funds provided are the variation margin, if they are not provided the broker will close out the account.

32. Drake Drake University Fin 284 Margin Account DayFutures Price Daily Changes Cumulative Change Margin Balance Margin Call 0 4004000 1 398-2(200) = -400-4003600 2 395.5-2.5(200)=500-9003100 3 394-1(200)=200-110029,0001100 4 395(1)200=200-9004100

33. Drake Drake University Fin 284 Note: You can withdraw any amount above the initial margin Most accounts pay a money market rate of interest Some accounts allow deposit of securities, but valued at less than face value. (treasures valued at 90% other at 50%)

34. Drake Drake University Fin 284 Role of Clearinghouse The clearinghouse serves as an intermediary that guarantees the contract. The clearinghouse is an independent corporation whose shareholders are comprised of its member firms. Each member firm maintains a margin account (similar to the traders) with the clearinghouse. In essence the clearinghouse guarantees the long and the short trader that the other side will honor the contract

35. Drake Drake University Fin 284 Patterns of Futures Prices Basis = Spot Price – Futures Price The Basis moves toward zero as the spot price matures. This eliminates arbitrage possibilities. If futures is greater than spot, you could enter short in the futures market and make a profit by buying in the spot and then delivering in futures Since everyone will attempt this demand for short positions increases and futures price decreases, also spot price would increase….

36. Drake Drake University Fin 284 Other patterns Normal Market: The futures price increase as the time to maturity increases Inverted Market: the futures price is a decreasing function of the time to maturity Comparing the futures price to the expected future spot price. Normal Backwardation: The futures price is below the expected future spot price. Contango: The futures price is above the expected futures price.

37. Drake Drake University Fin 284 Other Patterns The Futures Price over time Normal Market: The futures price increase as the time to maturity increases Inverted Market: the futures price is a decreasing function of the time to maturity Comparing the futures price to the expected future spot price. Normal Backwardation: The futures price is below the expected future spot price. Contango: The futures price is above the expected futures price.

38. Drake Drake University Fin 284 Theoretical Explanations of Backwardation Keynes and Hicks-- Speculators will only enter the market if they expect to have a positive profit. If more speculators are holding a long position, it implies that the futures price is less than the expected spot price A second explanation can be found by looking at the relationship between risk and return in the market. If thee is systematic risk involved with holding the security then the investor should be compensated for accepting the risk (nonsystematic risk can be diversified away).

39. Drake Drake University Fin 284 Theoretical Pricing of Futures Contracts The theoretical price Is based upon the elimination of arbitrage opportunities. Start with a simple example: Assume transaction costs are zero Assume that storage costs are zero You have a choice today of purchasing or selling a given asset or entering into a contract to buy or sell it in the future.

40. Drake Drake University Fin 284 Theoretical Price Assume you want to own the asset at a given point in time in the future, You can enter into a long futures position or buy the asset today and hold on to it. If you enter into the futures contract you can invest your cash today and earn interest ( r)

41. Drake Drake University Fin 284 Basic Relationship The Forward Price (F) should equal the spot price (S) plus any interest that could be received on an amount of cash equal to the spot price or:

42. Drake Drake University Fin 284 Eliminating Arbitrage If the forward price is greater than the spot plus interest an arbitrage opportunity exists. Borrow to buy the underlying asset in the spot market and take a short position in the futures contract.

43. Drake Drake University Fin 284 Numerical Example Consider an asset that is currently selling at $30 The asset has a two year futures price of $35. The risk free rate is 5% At Time 0 Borrow $30 (will need to repay 30(1.05) 2 =$33.075 Buy asset for $30 Take Short Futures Position At Time 2 Deliver Asset in Futures Receive $35 Payoff loan with 33.075 Profit = 35-33.075 =$1.925

44. Drake Drake University Fin 284 Example con’t Increased demand for short contracts, the # of participants willing to sell in two years will be greater than the number willing to buy. Those willing to sell will compete by lowering their price therefore the futures price declines...

45. Drake Drake University Fin 284 Eliminating Arbitrage Part 2 What if the futures price is less than the spot price plus interest? Short Sell the underlying asset and take a long position in the futures market

46. Drake Drake University Fin 284 Numerical example What if the futures price is $31 instead of $35? Leave the spot price at $30 and r at 5% At time 0 Short sell the asset and receive $30 Place the $30 in the bank receive $30(1.05)=$33.075 Take out a long position in the Futures Market At time 1 Receive 33.075 Buy the asset in futures market for 31 Profit = 33.075-31 =2.075

47. Drake Drake University Fin 284 Eliminating Arbitrage Now there is an excess of participants willing to take a long position but few willing to take a short position. To facilitate trading the futures price will increase. As the price increases it is more attractive to participants willing to take a short position.

48. Drake Drake University Fin 284 Eliminating Arbitrage In both cases the futures price moves toward a point where arbitrage does not exist When the futures price is 33.075 neither strategy is possible and arbitrage is eliminated

49. Drake Drake University Fin 284 Paying a known cash income The above analysis can be extended to the case where the underlying asset pays a known cash income (a treasury bond for example) We are going to assume that the in cash payment is due at the same time as the expiration of the forward contract.

50. Drake Drake University Fin 284 Cash Income Example Suppose that you can purchase a treasury bond that makes its coupon payments yearly. If you purchase the bond it will pay a coupon payment of $35 in one year. The bond has a forward price of $950. The risk free rate is 5%.

51. Drake Drake University Fin 284 Know cash income Want to consider the coupon as a cash flow just like the forward price. Let the spot price be $930 (F + Coupon Payment) > S(1+r) T 985 = 950+35 > 930(1.05) = 976.50 What arbitrage opportunity exists?

52. Drake Drake University Fin 284 Similar to before Borrow to buy the underlying asset in the spot market and take a short position in the futures contract. At time 0 Borrow $930 Buy bond for $930 Enter into short position At time 1 Receive coupon payment = $35 Sell bond in Fut Market =$950 Receive total =985 Repay loan = 976.50 Profit = 3.50

53. Drake Drake University Fin 284 Opposite Case What if current price is 940? At time 0 Short sell bond receive $940 Invest $940 at 5% Enter into Long Position in Fut At Time 1 Receive $940(1.05) = 987 Buy bond in Fut Market =$950 Close short sale pay coupon =$35 Profit = $2

54. Drake Drake University Fin 284 No Arbitrage Again the futures price is moving toward a point where there will not be an arbitrage opportunity. (F + Coupon Payment) = S(1+r) T Rearranging F = S(1+r) T - Coupon Payment F = S (1+r) T - CP(1+r) T /(1+r) T F=(S – CP/(1+r) T )(1+r) T where CP/(1+r)T is the PV of the coupon payment

55. Drake Drake University Fin 284 Extension If cash payments come at other points in time, all you need is a generalization of the relationship above. Let I represent the PV of all coupon payments to be received during the forward contract. F = (S+I)(1+r) T

56. Drake Drake University Fin 284 Accounting for payments Consider the 1 year forward contract on a bond that matures in 5 years. Assume that the bond makes semiannual coupon payments of $40 and has a spot price of $900. The 6 month rate is 9% and the 1 year rate is 10% PV of coupon 1 = 40/(1.09) 0.5 = $38.31 PV of coupon 2 = 40/1.10 = $36.36

57. Drake Drake University Fin 284 Assume futures price is $930 F=$930 > (900-39.31-36.36)(1.1)=907.86 At time 0 Borrow $900 today Borrow 38.31 @9% for 6 mos Borrow $861.69 @ 10% for 1yr Enter into short Futures position At time 6 mos Receive the $40 coupon payment Repay 6 mo loan At time 1 year Sell Bond for $930 Receive coup pay = $40 Total = $970 Repay loan 861.69(1.1) = 947.859 Profit = $22.14

58. Drake Drake University Fin 284 Extensions If the futures price was less than the spot minus the PV of the coupons carried forward an argument similar to the earlier ones could have also been made A final case is if the income stream pays a known dividend income.

59. Drake Drake University Fin 284 Dividend income Assume that the asset pays a return of q in the future based on the current price of the asset. The equilibrium is then F = S(1+r) T /(1+q) T

60. Drake Drake University Fin 284 Storage Costs? If the asset has a storage cost (more important for commodities than financial assets), it can be viewed as a negative cash income, the no arbitrage condition would be: F = (S+U)(1+r) T Where U represents the present value of all costs.

61. Drake Drake University Fin 284 Generalization Thank of the net amount of any of the possible costs, income received, and interest as the cost of carrying the spot position to the future. It is the cost of holding the spot position instead of the future position. The equilibrium condition is then simply F = (S+C)(1+r c ) T C is any cash income / costs and r c is net interest expense

62. Drake Drake University Fin 284 Treasury Bond Future Contracts Traded on the CBOT 10 year Treasury note future Delivers 6.5 to 10 year maturity treasury notes (maturity form the first day of the delivery month).

63. Drake Drake University Fin 284 Price Quotations Quotations The quoted and cash price are not the same due to interest that accrues on the bond. In general:

64. Drake Drake University Fin 284 Example Assume that today is March 5, 2002 and that the bond matures on July 10, 2004 Assume we have an 11% coupon bond with a face value of $100. The quoted price is 90-05 (or 90 5/32 or 90.15625) Bonds with a total face value of $100,000 would sell for $90,156.25.

65. Drake Drake University Fin 284 Example continued Coupons on treasuries are semiannual. Assume that the next coupon date would be July 10, 2000 or 54 days from March 5. The number of days between interest payments is 181 so using the actual/actual method we have accrued interest of (54/181)(5.50) = $1.64 The cash price is then $91.79625 = $90.15625 + $1.64

66. Drake Drake University Fin 284 Conversion Factors Since there are a range of bonds that can be delivered, the quoted futures price is adjusted by a conversion factor.

67. Drake Drake University Fin 284 Price based upon 6% YTM The conversion factor is based off an assumption of a flat yield curve of 6% (that interest rates for all maturities equals 6%). By comparing the value of the bond to the face value, the CBOT produces a table of conversion factors.

68. Drake Drake University Fin 284 Conversion Factor Continued The maturity of the bond is rounded down to the nearest three months. If the bond lasts for a period divisible by 6 months the first coupon payment is assumed to be paid in six months. (A bond with 10 years and 2 months would be assumed to have 10 years left to maturity)

69. Drake Drake University Fin 284 Conversion Factor continued If the bond does not round to an exact six months the first coupon is assumed to be paid in three months and accrued interest is subtracted. A bond with 14 years and 4 months to maturity would be treated as if it had 14 years and three months left to maturity

70. Drake Drake University Fin 284 Example 1 14% coupon bond with 20 years and two months to maturity Assuming a 100 face value the value of the bond would equal the price valued at 6%: The conversion factor is then 1.92459/100 = 1.92459

71. Drake Drake University Fin 284 Example 2 What if the bond had 18 years and four months left to maturity? The bond would be considered to have 18 years and three months left to maturity with the first payment due in three months. Finding the value of the bond three months from today

72. Drake Drake University Fin 284 Example 2 continued Assume the rate for three months is (1+r) 2 = 1.03 r =.014889 Using this rate it is easy to find the PV of the bond 187.329/1.014889 = 184.581 There is one half of a coupon in accrued interest so we need to subtract 7/2=3.50 184.581 - 3.50 = 181.081 resulting in a conversion factor of 181.081/100 = 1.81081

73. Drake Drake University Fin 284 Price Quote on T-Bills Quotes on T- Bills utilize the actual /360 day count convention. The quoted price of the treasury bill is an annualized rate of return expressed as a percentage of the face value.

74. Drake Drake University Fin 284 T- Bills continued The quote price is given by (360/n)(100-Y) where Y is the cash price of the bill with n days until maturity 90 day T- Bill Y = 98 (360/90)(100-98) =8.00

75. Drake Drake University Fin 284 Rate of Return The quote is not the same as the rate of return earned by the treasury bill. The rate of interest needs to be converted to a quarterly compounding annual rate. 2/98(365/90) =.0828

76. Drake Drake University Fin 284 Quoted Price The price quote on a Treasury bill is then given by 100 - Corresponding Treasury bill price quote (quoted price = 8 so futures quote =92) Given Z = the quoted futures price Y = the corresponding price paid for delivery of $100 of 90 day treasury bills then Z = 100-4(100-Y) or Y = 100-0.25(100-Z) Z = 100-4(100-98) = 92

77. Drake Drake University Fin 284 Cheapest to Deliver Bond There are a large number of bonds that could be delivered on the CBOT for a given futures contract. The party holding a short position gets to decide which bond to deliver and therefore has incentive to deliver the cheapest.

78. Drake Drake University Fin 284 Cheapest to Deliver Upon delivery the short position receives The cost of purchasing a bond is Quoted price + accrued interest By minimizing the difference between the cost and the amount received, the party effectively delivers the cheapest bond:

79. Drake Drake University Fin 284 Cheapest to deliver The bond for which is minimized is the one that is cheapest to deliver.

80. Drake Drake University Fin 284 Example: Cheapest to Deliver Consider 3 bonds all of which could be delivered Quoted Conversion Bond Price Factor 199.5 1.0382 99.5-(93.25(1.0382)) =2.69 2 143.5 1.5188 143.5-(93.25(1.5188))=1.87 3 119.75 1.2615 119.75-(93.25(1.2615))=2.12

81. Drake Drake University Fin 284 Impact of yield changes on CTD As yield increases bonds with a low coupons and longer maturities become relatively cheaper to deliver. As rates increase all bond prices decrease, but the price decrease for the longer maturity bonds is greater As yields decrease high coupon, short maturity bonds become relatively cheaper to deliver.

82. Drake Drake University Fin 284 Wild Card Play Trading at the CBOT closes at 2p.m. however treasury bonds continue to trade until 4:00pm and a party with a short position has until 8pm to file a notice of intention to deliver. Since the price is calculated on the closing price in the CBOT the party with a short position sometimes has the opportunity to profit from price movements after the closing of the CBOT. If the Bond Prices decrease after 2 pm it improves the short position.

83. Drake Drake University Fin 284 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.

84. Drake Drake University Fin 284 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.

85. Drake Drake University Fin 284 Simple Hedge Example Assume you know that you will owe at rate equal to the LIBOR + 100 basis points in three months on a notional amount of $100 Million. The interest expenses will be set at the LIBOR rate in three months. Current three month LIBOR is 7%, Eurodollar futures contract is selling at 92.90.

86. Drake Drake University Fin 284 Simple Hedge Example 100 - 92.90 = 7.10 The futures contract is paying 7.10% Assume the interest rate may either increase to 8% or decrease to 6%

87. Drake Drake University Fin 284 A Short Hedge Agree to sell 10 Eurodollar future contracts (each with an underlying value of $1 Million). We want to look at two results the spot market and the futures market. Assume you close out the futures position and that the futures price will converge to the spot at the end of the three months.

88. Drake Drake University Fin 284 Rates increase to 8% Spot position: Need to pay 8% + 1% = 9% on $10 Million $10 Million(.09/4) = $225,000 Futures Position: Fut Price = $92 interest rates increased by.9% Close out futures position: profit = ($10 million)(.009/4) = $22,500

89. Drake Drake University Fin 284 Rates Increase to 8% Net interest paid $225,000 - $22,500 = $202,500 $10 million(.0810/4) = $202,500

90. Drake Drake University Fin 284 Rates decrease to 6% Spot position: Need to pay 6% + 1% = 7% on $10 Million $10 Million(.07/4) = $175,000 Futures Position: Fut Price = $94 interest rates decreased by 1.1% Close out futures position: loss = ($10 million)(.011/4) = $27,500

91. Drake Drake University Fin 284 Rates Decrease to 8% Net interest paid $175,000 + $27,500 = $202,500 $10 million(.0810/4) = $202,500

92. Drake Drake University Fin 284 Results of Hedge Either way the final interest rate expense was equal to 8.10 % or 100 basis points above the initial futures rate of 7.10% Should the position be hedged? It locks in the interest rate, but if rates had declined you were better off without the hedge.

93. Drake Drake University Fin 284 Simple Example 2 On January 2 the treasurer of Ajax Enterprises knows that the firm will need to borrow in June to cover seasonal variation in sales. She anticipates borrowing $1million. The contractual rate on the loan will be the LIBOR rate plus 1% The current 3 month LIBOR rate is 3.75% and the Eurodollar futures contract is 4.25%

94. Drake Drake University Fin 284 Simple Example 2 Continued To hedge the position assume the treasurer sells one June futures contract. Assume interest rates increase to 5.5% on June 13. Assume that the expiration of the contract is June 13, the same day that the loan will be taken out. The futures price will be 100-5.50 = 94.50

95. Drake Drake University Fin 284 Rates increase to 5.5% Spot position: Need to pay 5.5%+1%= 6.5% on $1 Million $1 Million(.065/4) = $16,250 Futures Position: Fut Price = $94.50 interest rates increased by 1.25% Close out futures position: profit = ($1million)(.0125/4) = $3,125

96. Drake Drake University Fin 284 Rates Increase to 5.5% Net interest paid $16,250 - $3,125 = $13,125 $1 million(.0525/4) = $13,125 which is the interest rate implied by the Eurodollar futures contract 4.25% +1% = 5.25%

97. Drake Drake University Fin 284 Assumptions The hedge worked because of three assumptions: The underlying exposure is to the three month LIBOR which is the same as the loan The end of the exposure matches the delivery date exactly The margin account did not change since the rte changed on the last day of trading.

98. Drake Drake University Fin 284 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

99. Drake Drake University Fin 284 Basis Risk On what types of contracts would you expect the basis to be negative? Positive? Why? (-) Low interest rates assets such as currencies or gold or silver (investment type assets with little or zero convenience yield. F = S(1+r) T (+) Commodities and investments with high interest rates (high convenience yield) F = S(1+r+u) T Implies it is more likely that F < S(1+r+u) T

100. Drake Drake University Fin 284 Basis Risk The easiest way to illustrate the basis risk is with an example: Let: S t represent the spot price at time t F t represent the futures price at time t b t represent the basis at time t

101. Drake Drake University Fin 284 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 F 1 - F 2 The total price paid from the hedge is then S 2 + F 1 - F 2 By definition: b 1 = S 1 -F 1 and b 2 = S 2 -F 2

102. Drake Drake University Fin 284 Basis Risk By rearranging the price equation: S 2 + F 1 - F 2 = F 1 + S 2 - F 2 = F 1 + b 2 When the hedge is entered into F 1 is known but b 2 is unknown. The fact that b 2 is not known represents the basis risk. The same expression holds for a hedger undertaking a long hedge. Loss on Hedge = F 1 -F 2 price paid is S+F 1 -F 2

103. Drake Drake University Fin 284 Mismatch of Maturities 1 Assume that the maturity of the contract does not match the timing of the underlying commitment. Assume that the loan is anticipated to be needed on June 1 instead of June 13.

104. Drake Drake University Fin 284 Simple Example Redone On January 2 the treasurer of Ajax Enterprises knows that the firm will need to borrow in June to cover seasonal variation in sales. She anticipates borrowing $1million. The contractual rate on the loan will be the LIBOR rate plus 1% The current 3 month LIBOR rate is 3.75% and the Eurodollar futures contract is 4.25%

105. Drake Drake University Fin 284 Simple Example 2 Continued To hedge the position assume the treasurer sells one June futures contract. Assume interest rates increase to 5.5% on June 1. Assume that the futures price has decreased to 94.75 (before it had decreased to 94.50) implying a 5.25% rate (a 25 bp basis)

106. Drake Drake University Fin 284 Rates increase to 5.5% Spot position: Need to pay 5.5%+1%= 6.5% on $1 Million $1 Million(.065/4) = $16,250 Futures Position: Fut Price = $94.75 interest rates increased by 1.00% Close out futures position: profit = ($1million)(.0100/4) = $2,500

107. Drake Drake University Fin 284 Rates Increase to 5.5% Net interest paid $16,250 - $2,500 = $13,750 $1 million(.055/4) = $13,750 which is more than the interest rate implied by the Eurodollar futures contract 4.25% +1% = 5.25%

108. Drake Drake University Fin 284 Minimizing Basis Risk Given that the actual timing of the loan 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 short hedger could be forced to accept delivery instead of closing out.

109. Drake Drake University Fin 284 Mismatch in Maturities 2 Assume that instead of our original problem the treasurer is faced with a stream of expected borrowing. Anticipated borrowing at 3 month LIBOR DateAmount Mach 1$15 Million June 1$45 Million September 1$20 million December 1$10 Million

110. Drake Drake University Fin 284 Strip Hedge To hedge this risk, it to hedge each position individually. On January 1 the firm should: enter into 15 short March contracts enter into 45 short June contracts enter into 20 short Sept contracts enter into 10 short December contracts

111. Drake Drake University Fin 284 Strip Hedge continued On each borrowing 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.

112. Drake Drake University Fin 284 Rolling Hedge Another possibility is to Roll the Hedge: January 2enter into 90 short March contracts March 1enter into 90 long March contracts enter into 75 short June contracts June 1enter into 75 long June contracts enter into 30 short Sept contracts Sept 1enter into 30 long Sept contracts enter into 10 short Dec contracts Dec 1enter into 10 long Dec contracts

113. Drake Drake University Fin 284 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.

114. Drake Drake University Fin 284 Cross 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. Two questions What futures contract should be used? How many contracts should be taken out?

115. Drake Drake University Fin 284 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.

116. Drake Drake University Fin 284 Minimum Variance Hedge Ratio The ideal hedge ratio should be the one that minimizes the variance of the value of the hedged position.

117. Drake Drake University Fin 284 Minimum Variance Hedge Ratio  S be the change in the spot price S during a period of time equal to the life of the project  F be the change in the futures price F during a period of time equal to the life of the project  S be the standard deviation of  S  F be the standard deviation of  F  be the coefficient of correlation between  S and  F h be the hedge ratio

118. Drake Drake University Fin 284 Hedge positions The change in the short hedgers position is the change in the long hedgers position is

119. Drake Drake University Fin 284 Min Variance Hedge The variance of the hedge position is Taking the first derivative of the variance and setting it to zero produces the hedge ratio

120. Drake Drake University Fin 284 Applying the Hedge Ratio Finding the optimal number of future contracts is a simple application of the minimum variance hedge ratio. The optimal number of contract should be given by: N * = h * N P /Q where N p is the size of the position being hedged (units) and Q is the size of one futures contract (units)

121. Drake Drake University Fin 284 Estimating the Hedge Ratio The hedge ratio can be rewritten to allow easy estimation via regression analysis

122. Drake Drake University Fin 284 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.

123. Drake Drake University Fin 284 We can observe observations of X,Y and plot them

124. Drake Drake University Fin 284 Regression Estimates the line that best explains the relationship between the variables

125. Drake Drake University Fin 284 The Line is the one that minimizes the sum of the squared residuals

126. Drake Drake University Fin 284 Estimating the Regression The slope of the line is then equal to The Intercept is:

127. Drake Drake University Fin 284 Applying the Regression to the Hedge Ratio The minimum variance hedge ratio could be estimated by  in the regression. (S t ) =  +  (F t ) +  t

128. Drake Drake University Fin 284 Example Now assume that the treasury has decided to borrow it the commercial paper market instead of from a financial institution. There is not a commercial paper futures contract so it must be decided what contract to use to hedge the possible interest rate change in the commercial paper market. Assume that the treasure wants to borrow $36 million in June with a one month commercial paper issue.

129. Drake Drake University Fin 284 Number of contracts part 1 You must choose what underlying contract best matches the 30 day commercial paper return. 90 Day T-Bill. 90 day LIBOR Eurodollar, 10 year treasury bond. Assume 90 day LIBOR Eurodollar has the highest correlation so it is chosen. Assume now that the treasurer for Ajax has ran the regression and that the beta is.75

130. Drake Drake University Fin 284 Number of contracts part 2 We also need to consider the asset underlying the three month LIBOR futures contract and one month commercial paper rate have different maturities. A 1 basis point movement in $1,000,000 of borrowing is $1,000,000(.0001)(30/360) = $8.33 A one basis point change in $1,000,000 of the future contract is equal to: $1,000,000(.0001)(90/360) = $25

131. Drake Drake University Fin 284 Number of contracts part 2 The change in the three month contract is three times the size of the change in the one month this would imply a hedge ratio of 1/3 IF the assets underlying both positions was the same. Both sources of basis risk need to be considered.

132. Drake Drake University Fin 284 Number of Contracts The treasurer will need to enter into: $36(.75)(.33) = $9 million Of short futures contracts

133. Drake Drake University Fin 284 The Cross Hedge On January 2 3 month LIBOR = 3.75% June Eurodollar Future price is 95.75 implying 4.24% rate Spread between spot LIBOR rate and 1 month commercial paper rate is 60 basis points This implies a 4.35% commercial paper rate.

134. Drake Drake University Fin 284 Expectations Previously Ajax hoped to lock in a 4.25% 3 month LIBOR rate or an increase of 50 basis points form the current 3.75% Keeping the 50 basis point increase constant and using our hedge ratio of.75 the goal becomes locking in a.75 (50) = 37.5 basis point increase in the commercial paper rate. This implies a one month rate of 4.35% + 37.5BP = 4.725%

135. Drake Drake University Fin 284 Results Futures Assume that on June 1 the 3 month LIBOR rate increases to 5.5% (as it did in our previous example), also assume that the futures contract price falls to 94.75. Closing out the Futures contract resulted in a profit of $2,500 per $1million. Since we have 9 $1 million contracts our profit is 9(2,500)=$22,500

136. Drake Drake University Fin 284 Results Spot LIBOR increased by 1.75 % or 175 basis points, assuming our hedge ratio is correct this implies a.75(175) = 131.25 basis point increase in the one month commercial paper rate. So the new expected one month commercial paper rate is 4.35+1.3125 = 5.6625% However assume that the relationship was not perfect ant the actual one month rate is 5.75%

137. Drake Drake University Fin 284 Results Given the 5.75% commercial paper rate the cost of borrowing has increased by $36,000,00(.0575-.0435)(30/360) = $42,000 Subtracting our profit of 22,500 in futures market the net increase in borrowing cost is: $42,000 - $22,500 = $19,200 This is equivalent to an increase of: 36,000,000(X)(30.360) = $19,500 X = 65 BP

138. Drake Drake University Fin 284 Results Using the 65 BP increase Ajax ended up paying 5% for its borrowing. The treasurer was attempting to lock in 4.725% or 27.5BP less than what she ended up paying. The 27.5 BP difference is the result of basis risk.

139. Drake Drake University Fin 284 Basis Risk Source 1 June 1 spot LIBOR was 5.5% the LIBOR rate implied by the futures contract was 5.25% a 25 BP difference Given the hedge ratio of.75 this should be a 25(.75) = 18.75 BP difference for commercial paper Source 2 Expected 1 month commercial paper rte is 5.6625%, actual is 5.75% a 8.75 BP difference

140. Drake Drake University Fin 284 Basis Risk The result of the two sources of risk: 18.75 + 8.75 = 27.5 basis points

141. Drake Drake University Fin 284 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. For N contracts this becomes Ne -rT contracts where r is the risk free rate and T is the time to maturity.

142. Drake Drake University Fin 284 Duration Hedging You can also estimate the hedge ratio using duration. We know that the change in price can be estimated using duration. Assume that we have a bond portfolio with duration equal to D P  P=-PD P  y Likewise the change in the asset underlying a futures contract should be estimated by  F=-FD F  y

143. Drake Drake University Fin 284 Duration Hedging You can combine the two to produce a position with a duration of zero. The optimal number of contracts is Must assume a bond to be delivered