1. Drake DRAKE UNIVERSITY Fin 288 Interest Rates Futures Fin 288 Futures Options and Swaps

2. Drake Drake University Fin 288 Interest Rate Future Contracts Traded on the CBOT 30 Year Treasury Bond &30 Yr Mini 10, 5, & 2 year Treasury note futures 30 Day Fed Funds 5 & 10 year Swap German Debt Traded on CME Eurodollar Futures

3. Drake Drake University Fin 288 A quick look at contract Specifications Treasury Bonds and Notes- Range of delivery dates Fed Funds Futures Price Swaps Delivery Muni Underlying Asset

4. Drake Drake University Fin 288 Treasury Securities Since a majority of the interest rate instruments we will use are related to treasury securities, we need to discuss some basics relating to the pricing of Treasury securities.

5. Drake Drake University Fin 288 Some Pricing Issues Day Count Conventions Used to determine the interest earned between two points in time Useful in calculating accrued interest Specified as X/Y X = the number of days between the two dates Y = The total number of days in the reference period

6. Drake Drake University Fin 288 Day Count Conventions Day Count ConventionMarket Used US Treasury Bonds Corporate and Municipal Bonds US T-Bills & money Market Instruments

7. Drake Drake University Fin 288 Price Quotes for Treasury Bills Let Y d = annualized yield, D = Dollar Discount F= Face Value, t = number of days until maturity Price = F -D

8. Drake Drake University Fin 288 Price Quotes on T- Bills Note: Return was based on face value invested, not the actual amount invested. 360 day convention makes it difficult to compare to notes and bonds. CD equivalent yield makes the measure comparable to other money market instruments

9. Drake Drake University Fin 288 Accrued Interest When purchasing a bond between coupon payments the purchaser must compensate the owner for for interest earned, but not received, since the last coupon payment

10. Drake Drake University Fin 288 Price Quotations Quotations The quoted and cash price are not the same due to interest that accrues on the bond. In general:

11. Drake Drake University Fin 288 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.

12. Drake Drake University Fin 288 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

13. Drake Drake University Fin 288 Conversion Factors Since there are a range of bonds that can be delivered, the quoted futures price is adjusted by a conversion factor.

14. Drake Drake University Fin 288 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.

15. Drake Drake University Fin 288 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)

16. Drake Drake University Fin 288 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

17. Drake Drake University Fin 288 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

18. Drake Drake University Fin 288 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

19. Drake Drake University Fin 288 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

20. Drake Drake University Fin 288 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.

21. Drake Drake University Fin 288 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

22. Drake Drake University Fin 288 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

23. Drake Drake University Fin 288 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

24. Drake Drake University Fin 288 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.

25. Drake Drake University Fin 288 Cheapest to Deliver Upon delivery the short position receives The cost of purchasing a bond is Quoted bond price + accrued interest By minimizing the difference between the cost and the amount received, the party effectively delivers the cheapest bond:

26. Drake Drake University Fin 288 Cheapest to deliver The bond for which is minimized is the one that is cheapest to deliver.

27. Drake Drake University Fin 288 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

28. Drake Drake University Fin 288 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.

29. Drake Drake University Fin 288 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.

30. Drake Drake University Fin 288 Eurodollar Futures Eurodollar – dollar deposited in a foreign bank outside of the US. Eurodollar interest rate is the interest earned on Eurodollars deposited by one bank with another bank. London Interbank Offer Rate (LIBOR) – Rate at which banks loan to each other in the London Interbank Market.

31. Drake Drake University Fin 288 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.

32. Drake Drake University Fin 288 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%

33. Drake Drake University Fin 288 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.

34. Drake Drake University Fin 288 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

35. Drake Drake University Fin 288 Rates Increase to 8% Net interest paid $225,000 - $22,500 = $202,500 $10 million(.0810/4) = $202,500

36. Drake Drake University Fin 288 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

37. Drake Drake University Fin 288 Rates Decrease to 8% Net interest paid $175,000 + $27,500 = $202,500 $10 million(.0810/4) = $202,500

38. Drake Drake University Fin 288 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.

39. Drake Drake University Fin 288 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%

40. Drake Drake University Fin 288 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

41. Drake Drake University Fin 288 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

42. Drake Drake University Fin 288 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%

43. Drake Drake University Fin 288 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 rate changed on the last day of trading.

44. Drake Drake University Fin 288 Basis Risk revisited 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

45. Drake Drake University Fin 288 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

46. Drake Drake University Fin 288 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.

47. Drake Drake University Fin 288 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%

48. Drake Drake University Fin 288 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)

49. Drake Drake University Fin 288 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

50. Drake Drake University Fin 288 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%

51. Drake Drake University Fin 288 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.

52. Drake Drake University Fin 288 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

53. Drake Drake University Fin 288 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

54. Drake Drake University Fin 288 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.

55. Drake Drake University Fin 288 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

56. Drake Drake University Fin 288 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.

57. Drake Drake University Fin 288 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.

58. Drake Drake University Fin 288 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

59. Drake Drake University Fin 288 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

60. Drake Drake University Fin 288 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.

61. Drake Drake University Fin 288 Number of Contracts The treasurer will need to enter into: $36(.75)(.33) = $9 million Of short futures contracts

62. Drake Drake University Fin 288 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.

63. Drake Drake University Fin 288 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%

64. Drake Drake University Fin 288 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

65. Drake Drake University Fin 288 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%

66. Drake Drake University Fin 288 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

67. Drake Drake University Fin 288 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.

68. Drake Drake University Fin 288 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 rate is 5.6625%, actual is 5.75% a 8.75 BP difference

69. Drake Drake University Fin 288 Basis Risk The result of the two sources of risk: 18.75 + 8.75 = 27.5 basis points

70. Drake Drake University Fin 288 Duration: The Big Picture Calculation: Given the PV relationships, we need to weight the Cash Flows based on the time until they are received. In other words we are looking for a weighted maturity of the cash flows where the weight is a combination of timing and magnitude of the cash flows

71. Drake Drake University Fin 288 Calculating Duration One way to measure the sensitivity of the price to a change in discount rate would be finding the price elasticity of the bond (the % change in price for a % change in the discount rate)

72. Drake Drake University Fin 288 Duration Mathematics Macaulay Duration Macaulay Duration is the price elasticity of the bond (the % change in price for a percentage change in yield). Formally this would be:

73. Drake Drake University Fin 288 Duration Mathematics Taking the first derivative of the bond value equation with respect to the yield will produce the approximate price change for a small change in yield.

74. Drake Drake University Fin 288 Duration Mathematics The approximate price change for a small change in r

75. Drake Drake University Fin 288 Duration Mathematics Macaulay Duration substitute

76. Drake Drake University Fin 288 Macaulay Duration of a bond

77. Drake Drake University Fin 288 Duration Example 10% 30 year coupon bond, current rates =12%, semi annual payments

78. Drake Drake University Fin 288 Example continued Since the bond makes semi annual coupon payments, the duration of 17.3895 periods must be divided by 2 to find the number of years. 17.3895 / 2 = 8.69475 years Another interpretation of duration is shown here: Duration indicates the average time taken by the bond, on a discounted basis, to pay back the original investment.

79. Drake Drake University Fin 288 Using Duration to estimate price changes Rearrange % Change in Price Estimate the % price change for a 1 basis point increase in the yearly yield Multiply by original price for the price change -0.000820257(838.8357)=-.688061

80. Drake Drake University Fin 288 Using Duration Continued Using our 10% semiannual coupon bond, with 30 years to maturity and YTM = 12% Original Price of the bond = 838.3857 If YTM = 12.01% the price is 837.6986 This implies a price change of -0.6871 Our duration estimate was -0.6881 a difference of.0010

81. Drake Drake University Fin 288 Note: Previously yield increased from 12% a year to 12.01%. We used the Duration represented in years, 8.69475 We could have also used duration represented in semiannual periods, 17.3895. The change in yield needs to be adjusted to.0001/2 =.00005 however, the original yield (1+r) stays at 1.06. The estimated price change is then the same as before: -0.000820257(838.8357)=-.688061

82. Drake Drake University Fin 288 Modified Duration Substitute D MOD The % Change in price was given above as:

83. Drake Drake University Fin 288 Modified vs Macaulay Duration

84. Drake Drake University Fin 288 Duration - Continuous Time Using continuous compounding the bond value formula becomes And the Duration equation becomes

85. Drake Drake University Fin 288 Change in Bond Price – Continuous Time The estimated percentage change in the price of the bond is then given by letting value (V) = price (P): By rearranging the actual price change is then

86. Drake Drake University Fin 288 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

87. Drake Drake University Fin 288 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

88. Drake Drake University Fin 288 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.

89. Drake Drake University Fin 288 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