1. 1

2. 2

3. 3

4. 4

5. 5

6. 6 A PURE DISCOUNT BOND DOES NOT PAY CUPONS UNTIL ITS MATURITY; C = 0:

7. 7

8. 8

9. 9

10. 10

11. 11 DURATION IS THE WIEGHTED AVERAGE OF COUPON PAYMENTS’ TIME PERIODS, t, WEIGHTED BY THE PROPORTION THAT THE DISCOUNTED CASH FLOW, PAID AT EACH PERIOD, IS OF THE CURRENT BOND PRICE.

12. 12 DURATION INTERPRETED AS A MEASURE OF THE BOND PRICE SENSITIVITY

13. 13 The negative sign merely indicates that D changes in opposite direction to the change in the yield, r. Next we present a closed form formula to calculate duration of a bond:

14. 14

15. 15 Coupon Rate

16. 16

17. 17 DURATION OF A BOND PORTFOLIO V = The total bond portfolio value P i = The value of the i-th bond N i = The number of bonds of the i-th bond in the portfolio V i = P i N i = The total value of the i-th bond in the portfolio V = ΣP i N i The total portfolio value. We now prove that: D P = Σw i D i.

18. 18

19. 19

20. 20 is the weighted average of the durations of the bonds in the portfolio. The weights are the proportions the bond value is of the entire portfolio value.

21. 21 Example: Consider a portfolio of two T-bonds: BONDFV in $M N YTM COUPON T-BOND$10015yrs 6% 5% T-BOND$20030yrs 6% 15% BONDPRICE W D T-BOND $90.2 0.1673 10.4673 T-BOND $449.1 539.3 0.8327 1.0000 12.4674 D=(.1673)(10.4673) +(.8327)(12.4674) D = 12.1392

22. 22 IMMUNIZING BANK PORTFOLIO OF ASSETS AND LIABILITIES TIME 0 ASSETSLIABIABILITIES $100,000,000 $100,000,000 (LOANS) (DEPOSITS) D = 5D = 1r = 10% TIME 1 r => 12% BUT IF D A = D L THEY REACT TO RATES CHANGES IN EQUAL AMOUNTS. THE BANK PORTFOLIO IS IMMUNIZED, i.e., IT’S VALUE WILL NOT CHANGE FOR A “small” INTEREST RATE CHANGE, IF THE PORTFOLIO’S DURATION IS ZERO or: D P = D A - D L = 0.

23. 23 A 5-YEAR PLANNING PERIOD CASE OF IMMUNIZATION IN THE CASH MARKET BOND CFV M r D P A $100 $1,000 5 yrs 10% 4.17$1,000 B $100 $1,000 10 yrs10% 6.76$1,000 4.17W A + 6.76W B = 5 W A + W B = 1 W A =.677953668. W B =.322046332. V P = $200M implies: Hold $135,590,733.6 in bond A, And $64,409,266.4 in bond B. Next, assume that r rose to 12%. The portfolio in which bonds A and B are held in equal proportions will change to:

24. 24 [1 - 4.17 (.02/1.1)] 100M = $92,418,181.2 [1 - 6.76 (.02/1.1)] 100M = $87,709,090.91 TOTAL = $180,127,272.7 INVEST THIS AMOUNT FOR 5 YEARS AT 12% y CONTINUOUSLY COMPOUNDED YIELDS: $328,213,290. ANNUAL RETURN OF: AFTER 5 YEARS AT 12%: $331,267,162. ANNUAL RETURN OF: The weighted average portfolio changes to:

25. 25 Before turning to the futures markets we elaborate on the common practice of: Repurchase Agreements An integral part of trading T-bills and T- bill futures is the market for repurchase agreements, which are used in much of the arbitrage trading in T-bills. In a repurchase agreement -- also called an RP or repo -- one party sells a security (in this case, T-bills) to another party at one price and commits to repurchase the security at another price at a future date. The buyer of the T-bills in a repo is said to enter into a reverse repurchase agreement., or reverse repo. The buyer’s transactions are just the opposite of the seller’s. The figure below demonstrates the transactions in a repo.

26. 26 Party A Party B Date t - Close the Repo T-Bill P 1 = P 0 (1+r 0,t ) Transactions in a Repurchase Agreement Date 0 - Open the Repo: Party AParty B T- Bill POPO Example: T-bill FV = $1M. P 0 = $980,000. The repo rate = 6%. The repo time: t = 4 days. P 1 = P 0 [(repo rate)(n/360) + 1] = 980,000[(.06)(4/360) + 1] = 980,653.33

27. 27 A repurchase agreement effectively allows the seller to borrow from the buyer using the security as collateral. The seller receives funds today that must be paid back in the future and relinquishes the security for the duration of the agreement. The interest on the borrowing is the difference between the initial sale price and the subsequent price for repurchasing the security. The borrowing rate in a repurchase agreement is called the repo rate. The buyer of a reverse repurchase agreement receives a lending rate called the reverse repo rate. The repo market is a competitive dealer market with quotations available for both borrowing and lending. As with all borrowing and lending rates, there is a spread between repo and reverse repo rates.

28. 28 The amount one can borrow with a repo is less than the market value of the security by a margin called a haircut. The size of the haircut depends on the maturity and liquidity of the security. For repos on T-bills, the haircut is very small, often only one-eighth of a point. It can be as high as 5% for repurchase agreements on longer-term securities such as Treasury bonds and other government agency issues.Most repos are held only overnight, so those who wish to borrow for longer periods must roll their positions over every day. However, there are some longer-term repurchase agreements, called term repos, that come in standardized maturities of one, two, and three weeks and one, two, three, and six months.Some other customized agreements also are traded.

29. 29 INTEREST RATE FUTURES The three most traded interest rate futures are: TREASURY BILLS (CME) $1mil; pts. Of 100% EURODOLLARS (CME) $1mil; pts. Of 100% TREASURY BONDS (CBT) $100,000; pts. 32nds of 100%

30. 30 CONTRACT SPECIFICATIONS FOR: 90-DAY T-BILL; 3-Month EURODOLLAR FUTURES SPECIFICATIONS13-WEEK3-Month EURODOLLAR US T-BILLTIME DEPOSIT SIZE$1,000,000$1,000,000 CONTRACT GRADEnew or dated T-billsCASH SETTLEMENT with 13 weeks to maturity YIELDSDISCOUNTADD-ON HOURS(Chicago time)7:20 AM-2:00PM7:20 AM - 2:00PM DELIVERY MONTHSMAR-JUN-SEP-DEC MAR-JUN-SEP-DEC TICKER SYMBOLTBEB MIN. FLUCTUATION.01(1 basis pt).01(1 basis pt) IN PRICE($25/pt) ($25/pt) LAST TRADING DAYThe day before the2nd London business day first delivery daybefore 3rd Wednesday DELIVERY DATE1st day of spot month Last day of trading on which 13-week T-bill is issued and a 1-year T-bill has 13 weeks to maturity

31. 31 Transactions in a Cash-and-Carry Arbitrage. Repo Market Arbitrageur T-Bill Dealer Futures Market P O (MONEY) T-Bill Short Position F 0,t P O (MONEY) Transactions in a Cash-and-Carry Arbitrage. Repo Market Arbitrageur Futures Market P 0 (1+r 0t ) T-Bill Deliver T-Bill Receive F 0,t Date t Date 0 F 0,t > P 0 (1+r 0,t )

32. 32 Transactions in a Reverse Cash-and-Carry Arbitrage. Repo Market Arbitrageur T-Bill Dealer Futures Market P O (MONEY) T-Bill P0P0 Long Position F 0,t T-Bill Transactions in a Reverse Cash-and-Carry Arbitrage Repo Market Arbitrageur Futures Market P 0 (1+r 0,t ) T-Bill Take Delivery T-Bill Pay F 0,t Date t F 0,t < P 0 (1+r 0,t ) Date 0

33. 33

34. 34

35. 35

36. 36 Transactions in a Cash-and-Carry Arbitrage Repo Market Arbitrageur T-Bill Dealer Futures Market P O =$954,330.56 182-day T-Bill P 0 = $954,330.56 Short Position F Ot = $976,011.75 182-day T-Bill Transactions in a Cash-and-Carry Arbitrage Repo Market Arbitrageu r Futures Market P 0 (1+r 0,t ) = $975,561.14 91 day T-Bill Deliver 91-day T-Bill F 0,t = $976,011.75 Date t Date 0 Profit = $450,61

37. 37

38. 38 Transactions in a Reverse Cash-and-Carry Arbitrage Repo Market Arbitrageur T-Bill Dealer Futures Market P O = $954,330.56 182-day T-Bill P O = $954,330.56 Long Position F O,t = $973,809.04 182-day T-Bill Transactions in a Reverse Cash-and-Carry Arbitrage Repo Market Arbitrageu r Futures Market P 1 = $975,561.13 91 day T-Bill Take Delivery 91-day T-Bill F 0,t = $973,809.04 Date t Date 0 PROFIT = $1,752.09

39. 39

40. 40 LET THE YIELD ON THE SHORT-TERM BILL BE 8%: THEORETICAL RATE = 11.74% IS LESS THAN 12.50% = ACTUAL MARKET RATE REVERSE CASH-AND-CARRY

41. 41

42. 42 To minimize dV/dr with respect to N, set: Next, we use the following substitutions for

43. 43 Recall that:

44. 44 Normally, the ratios of the yields sensitivities to the interest rate, r, are assumed to be zero. Thus: This is the price sensitivity hedge ratio.

45. 45

46. 46 EURODOLLAR FUTURES These are futures on the interest earned on Eurodollar three-month time deposits. The rate used is LIBOR - London Inter-Bank Offer Rate. These time deposits are non transferable, thus, there is no delivery! Instead, the contracts are CASH SETTLED.

47. 47

48. 48

49. 49 Arbitrage with Eurodollar Futures

50. 50 Quasi Arbitrage or, how to borrow capital using Eurodollar Futures.

51. 51

52. 52 PROFIT: 20[970,575-968,675] = $38,000

53. 53 On November 1, 2000, a firm agrees to borrow $10M for 12 months, beginning December 19, 2000 at LIBOR + 100bps. A STRIP HEDGE WITH EURODOLLARS FUTURES DATE CASH FUTURES F 11.1.00 LIBOR 8.44% Short 10 DEC91.41 Short 10 MAR91.61 Short 10 JUN91.53 Short 10 SEP91.39 12.19.00 LIBOR 9.54% Long 10 DEC90.46 3.13.01 LIBOR 9.75% Long 10 MAR90.25 6.19.01 LIBOR 9.44% Long 10 JUN90.56 9.18.01 LIBOR 8.88% Long 10 SEP91.12

54. 54 PERIOD: 1 2 3 4 RATE a : 10.54% 10.75% 10.44% 9.88% INTEREST b : $263,500 $268,750 $261,000 $247,000 FUTURES c : $23,750 $34,000 $24,250 $6,750 NET d : $239,750 $234,750 $236,750 $240,250 EFFECTIVE RATE e : 9.59% 9.39% 9.47% 9.61% MEAN RATE UNHEDGED10.40% HEDGED 9.52% a.LIBOR + 100 BPS b.($10M)(RATE)(3/12) c.(PRICE CHANGE)(25)(100)(10) d.b - c e.(NET/10M)(12/3)(100%)

55. 55 A STACK HEDGE WITH EURODOLLAR FUTURES: DATA ON NOVEMBER 11, 2000 VOLUMEOPEN INTEREST DEC 00 46,903185,609 MAR 01 29,236127,714 JUN 01 5,788 77,777 SEP 01 2,672 30,152 DECISION: STACK MAR FUTURES FOR JUN AND SEP. ROLL OVER AS SOON AS OPEN INTEREST REACHES 100,000

56. 56 THE STACK HEDGE DATE CASHFUTURESF. POSITION 11.1.00 8.44%S 10 DEC 91.41S10DEC S 30 MAR 91.61S30MAR 12.19.00 9.54%L 10 DEC 90.46S30MAR 1.12.01 9.47%L 20 MAR 90.47S10MAR S 20 JUN 90.42S20JUN 2.22.01 9.95%L 10 JUN 89.78S10MAR S 10 SEP 89.82S10JUN S10SEP 3.13.01 9.75%L 10 MAR 90.25S10JUN S10SEP 6.19.01 9.44%L 10 JUN 90.56S10SEP 9.18.01 8.88%L 10 SEP 91.12NONE

57. 57 PERIOD: 1 2 3 4 RATE(%) a : 10.54 10.75 10.44 9.88 INTEREST($) b :263,500 268,750 261,000 247,000 FUTURES($) c : 23,750 34,000 28,500 28.500 (3,500) 16,000 (32,500) NET($) d :239,750 234,750 236,000 235,000 EFFECTIVE RATE (%) e : 9.59 9.39 9.44 9.40 MEAN RATE UNHEDGED10.40% HEDGED 9.46% a.LIBOR + 100 BPS b.($10M)(RATE)(3/12) c.(PRICE CHANGE)(25)(100)(10) d.b - c e.(NET/10M)(12/3)(100%). This completes the example. Next, we show an example of a STRIP hedge with T-bill futures:

58. 58

59. 59

60. 60 LONG-TERM INTEREST RATE FUTURES The U.S. T-BOND FUTURES trades on the CBOT The underlying assets are Treasury Bonds with long-term maturity. It is among the most successful futures contracts of all existing contracts. On any give day, there are 40 to 50 different T-bonds traded in the cash market and most of these are deliverable against a T-bond futures position, which makes this market extremely liquid.

61. 61 SPECIFICATIONS OF U.S. TREASURY BOND FUTURES CONTRACTS EXCHANGECBOT DATE OF INTRODUCTIONAUGUST 22, 1997 TICKET SYMBOLUS CONTRACT SIZE$100,000 FACE VALUE CONTRACT MONTHSMAR. JUN. SEP. DEC. PRICE QUOTATIONPOINTS AND 1/32 OF A POINT. PRICES ARE BASED ON 6% COUPON RATE WITH 20 YEARS TO MATURITY(8% <2000) TICK SIZE1/32 OF A POINT, = $31.25 DELIVERABLE GRADESU.S. T-BONDS THAT ARE NOT CALLABLE FOR AT LEAST 15 YEARS AND HAVE A MATURITY OF AT LEAST 15 YEARS FROM THE FIRST BUSINESS DAY OF THE DELIVERY MONTH. LAST TRADING DAY7TH BUSINESS DAY PRECEDING THE LAST BUSINESS DAY OF THE DELIVERY MONTH. DELIVERY METHODFEDERAL RESERVE BOOK-ENTRY WIRE-TRANSFER SYSTEM.

62. 62 CONVERSION FACTORS T-BOND FUTURES PRICES ARE BASED ON A 20 –YEAR BOND THAT PAYS 6% (8%, PRIOR TO 2000) COUPON RATE, SEMIANNUALLY. DELIVERABLE AGAINST A SHORT POSITION IS ANY T- BOND WITH MATURITY, OR FIRST TIME TO CALLABILITY OF 15 YEARS. THUS, THE SHORT MAY CHOSE FROM A VARIETY OF BONDS AND DELIVER THE BOND THAT WILL MINIMIZE (MAXIMIZE) THE SHORT COST (REVENUE). THIUS BONMD IS LABELED “THE CHEAPEST TO DELIVERY.” THE PROBLEM IS THAT THE ORIGINAL FUTURES PRICE CONTRACTED FOR MUST BE ADJUSTED TO CONFORM WITH THE BOND THAT IS EVENTUALLY DELIVERED. THE ADJUSMENT PUT THE BOND DELIVERED ON PAR WITH THE ORIGINAL FUTURES PRICE. THIS IS DONE WITH CONVERSION FACTORS. THE CONVERSION FACTOR OF A T-BOND REPRESENTS THE NET PRESENT VALUE OF THE COUPON PAYMENTS OF THE BOND ACTUALLY DELIVERED IN EXCESS(IN SHORT), RELATIVE TO A 20YEAR BOND WITH 6% (8%) RATE, R.

63. 63 CBOT T-BOND CONVERSION FACTORS based on 8% futures prices YRS = YEARS TO MATURITY OR FIRST CALLABILITY M = NUMBER OF MONTHS CR = COUPON RATE CF = Conversion factor ROUND MONTHS TO: M*= 0,3,6, OR 9. Case 1: M* = 0 Case 2: M* = 3

64. 64 Case 3: M* = 6 Case 4: M* = 9

65. 65 EXAMPLE: THE CONVERSION FACTOR FOR DELIVERY OF THE 11 3/4s FOR NOVEMBER 15, 2015, ON THE DECEMBER 1999 T-BOND FUTURES CONTRACT: ON 12.1.99 YRS = 15 until 2014. M = 11 14 DAYS are ignored. THUS, YRS = 15 M is rounded off to M* = 9. FIRST COMPUTE : CF 6 = 1.3298. NOW COMPUTE:

66. 66 A bond portfolio manager decides to sell $10M FV of 11 7/8 M-19 yrs T-bonds on March 28. Currently, FEB 26, the bond sells for S=$101/$100FV. A SHORT T-BOND HEDGE TIMECASHFUTURES FEB. 2510M FV T-BONDSSELL 160 JUN CR = 11 7/8 M-19T-BOND Fs. S = 10,100,000F=70-16 Ds = 7.83Df = 7.20 Ys = 11.74%Yf = 14.92% MAR. 28S = 95.6875/$100FVLONG 16 JUN $9,568,750T-BOND Fs Opportunity loss F = 61 - 23 Futures gain: [(70-16)-(61-23)]160 =(8-25)160=($8781,25)160 =$1,405,000. Total selling price: 9,568,750 + 1,405,000 = $10,973,750

67. 67 LONG HEDGE WITH T - BOND FUTURES DATECASHFUTURES MAR. 29 LONG 110 SEP T-BOND Fs. F = 78-21 BY REGRESSION N = 110. JUL. 15 s=107 19/32 SHORT110 SEP T-BOND $10,759,375 Fs. F = 86-6 Gain from futures:110[(86-6) – (78-21)] =110[7-17] =110[$7,531.25] = $828,437.5 THUS, THE EFFECTIVE PURCHASE PRICE OF THE T- BONDS IS:$10,7593,750 - $ 828,437.5 = $9,930,937.5

68. 68 HEDGING A CORPORATE BOND ISSUE FEB. 24. DECISION: ISSUE $50M CORPORATE BONDS AT PAR VALUE ON MAR. 24. EXPECTATIONS: CR = 13.76% M = 20yrs D = 7.22 DATE CASH FUTURES 2.24 D S = 7.83 SHORT 674 FUTURES. y S = 13.6% F(JUN) = 68-11 S=$50M.D F =7.83; y F = 13.6% 5.24ISSUE BONDS LONG 674 JUN T-BOND CR=13.26%Fs. F(JUN) = 55-25 S=$907.4638/$100FV V(BOND ISSUE) = $45,373,190 Gain from futures: 674[(68-11)-(55-25)] =674[12-18]=674[$12,562.5] =$8,467,125. TOTAL VALUE=$53,840,315.

69. 69 PROJECT PLANNING: ELIMINATION OF FOREIGN INVESTMENT RISK EXPOSURE BY FORWARD TRADING OR BY THE USE OF U.S. TREASURY SECURITIES. An American firm expects an income of FC40,268,394 from a project in the foreign country. Its next project will begin only 90 days later in the same country. Thus, the above sum will be invested in the foreign country for 90 days. The interest rate in 90 days is not known and the firm would like to eliminate the risk exposure associated with the investment of the above sum for 90 days. Given current spot and forward exchange rates, as well as current and forward interest rates, the firm will choose between investing forward or using the U.S. T-bill futures.

70. 70 PROJECT PLANNING: Exchange rateSpot Interest rate Spot:.4729U.S.F.C. 90-day.4765 3-month: 14%9.15% 180-day.47826-month:16$10.22% 3-month forward rates U.S. = 18.035%.F.C. = 11.3% I.Invest 40,268,394 FC forward at the foreign country will result in a fixed and known total sum of: 40,268,394e (.113)(.25) = 41,422,197FC in 180 days. Instead, the firm could entertain the following strategy:

71. 71 DateSpot marketFutures market t=0Do nothing.1.Short 40,268,394 90-day forward at $.4765/FC. 2. Long 20 T-bill futures on the CBT for 959,394.49 3. Short $20M 180-day forward at $.4808/FC. t=90 ReceivePay $40,268,394(.4765) = 40,268,394FC.= $19,187,889.80 and take delivery of $20M face value T-bills. t = 180 T-billsTake delivery of 41,597,338 mature, Collectto close your short $20M forward position paying $20M. THE SECOND STRATEGY IS BETTER.