1. Financial Engineering Interest Rates and interest rate derivatives Options, Futures, and Other Derivatives 6th Edition, John C. Hull 2005, Chapter 4# Neftci, Chapter 4.7

2. Chapter 4 Interest Rates

3. Types of Rates Treasury rates LIBOR rates Repo rates 3

4. Treasury Rates Rates on instruments issued by a government in its own currency 4

5. LIBOR and LIBID LIBOR is the rate of interest at which a bank is prepared to deposit money with another bank. (The second bank must typically have a AA rating) LIBOR is compiled once a day by the British Bankers Association on all major currencies for maturities up to 12 months LIBID is the rate which a AA bank is prepared to pay on deposits from anther bank 5

6. Bond Pricing To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate For example, the theoretical price of a two-year bond providing a 6% coupon semiannually is 6

7. Bond Yield The bond yield is the discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond Suppose that the market price of the bond in our example equals its theoretical price of 98.39 The bond yield (continuously compounded) is given by solving to get y=0.0676 or 6.76%. 7

8. Par Yield The par yield for a certain maturity is the coupon rate that causes the bond price to equal its face value. In our example we solve 8

9. Par Yield continued In general if m is the number of coupon payments per year, d is the present value of $1 received at maturity and A is the present value of an annuity of $1 on each coupon date (in our example, m = 2, d = 0.87284, and A = 3.70027) 9

10. Data to Determine Zero Curve (Table 4.3, page 82) 10 Bond PrincipalTime to Maturity (yrs) Coupon per year ($) * Bond price ($) 1000.25097.5 1000.50094.9 1001.00090.0 1001.50896.0 1002.0012101.6 * Half the stated coupon is paid each year

11. The Bootstrap Method An amount 2.5 can be earned on 97.5 during 3 months. Because 100 = 97.5 e 0.10127×0.25, the 3-month rate is 10.127% with continuous compounding Similarly the 6 month and 1 year rates are 10.469% and 10.536% with continuous compounding 11

12. The Bootstrap Method continued To calculate the 1.5 year rate we solve to get R = 0.10681 or 10.681% Similarly the two-year rate is 10.808% 12

13. Zero Curve Calculated from the Data (Figure 4.1, page 84) 13 Zero Rate (%) Maturity (yrs) 10.127 10.46910.536 10.681 10.808

14. Duration (page 89-90) Duration of a bond that provides cash flow c i at time t i is where B is its price and y is its yield (continuously compounded) 14

15. Key Duration Relationship Duration is important because it leads to the following key relationship between the change in the yield on the bond and the change in its price 15

16. Key Duration Relationship continued When the yield y is expressed with compounding m times per year The expression is referred to as the “modified duration” 16

17. Bond Portfolios The duration for a bond portfolio is the weighted average duration of the bonds in the portfolio with weights proportional to prices The key duration relationship for a bond portfolio describes the effect of small parallel shifts in the yield curve What exposures remain if duration of a portfolio of assets equals the duration of a portfolio of liabilities? 17

18. Convexity The convexity, C, of a bond is defined as This leads to a more accurate relationship When used for bond portfolios it allows larger shifts in the yield curve to be considered, but the shifts still have to be parallel 18

19. Forward Rates 19 The forward rate is the future zero rate implied by today’s term structure of interest rates

20. Formula for Forward Rates Suppose that the zero rates for time periods T 1 and T 2 are R 1 and R 2 with both rates continuously compounded. The forward rate for the period between times T 1 and T 2 is This formula is only approximately true when rates are not expressed with continuous compounding 20

21. Application of the Formula 21 Year ( n )Zero rate for n -year investment (% per annum) Forward rate for n th year (% per annum) 13.0 24.05.0 34.65.8 45.06.2 55.56.5

22. Forward vs Zero Rates We have So R 2 – R 1 > 0  R F > R 2 ….. 22

23. Upward vs Downward Sloping Yield Curve …For an upward sloping yield curve: Fwd Rate > Zero Rate > Par Yield …For a downward sloping yield curve Par Yield > Zero Rate > Fwd Rate See other notes for formal relation between there three rates and for how to handle bid-ask spreads 23

24. Forward Rate Agreement 24 A forward rate agreement (FRA) is an OTC agreement that a certain rate will apply to a certain principal during a certain future time period

25. Forward Rate Agreement: Key Results An FRA is equivalent to an agreement where interest at a predetermined rate, R K is exchanged for interest at the market rate An FRA can be valued by assuming that the forward LIBOR interest rate, R F, is certain to be realized This means that the value of an FRA is the present value of the difference between the interest that would be paid at interest at rate R F and the interest that would be paid at rate R K 25

26. Valuation Formulas If the period to which an FRA applies lasts from T 1 to T 2, we assume that R F and R K are expressed with a compounding frequency corresponding to the length of the period between T 1 and T 2 With an interest rate of R K, the interest cash flow is R K ( T 2 – T 1 ) at time T 2 With an interest rate of R F, the interest cash flow is R F ( T 2 – T 1 ) at time T 2 26

27. Valuation Formulas continued When the rate R K will be received on a principal of L the value of the FRA is the present value of received at time T 2 When the rate R K will be received on a principal of L the value of the FRA is the present value of received at time T 2 27

28. Example An FRA entered into some time ago ensures that a company will receive 4% (s.a.) on $100 million for six months starting in 1 year Forward LIBOR for the period is 5% (s.a.) The 1.5 year rate is 4.5% with continuous compounding The value of the FRA (in $ millions) is 28

29. Example continued If the six-month interest rate in one year turns out to be 5.5% (s.a.) there will be a payoff (in $ millions) of in 1.5 years The transaction might be settled at the one-year point for an equivalent payoff of 29

30. Theories of the Term Structure Page 91-92 Expectations Theory: forward rates equal expected future zero rates Market Segmentation: short, medium and long rates determined independently of each other Liquidity Preference Theory: forward rates higher than expected future zero rates 30

31. Chapter 7 Swaps 31

32. Nature of Swaps A swap is an agreement to exchange cash flows at specified future times according to certain specified rules 32

33. An Example of a “Plain Vanilla” Interest Rate Swap An agreement by Microsoft to receive 6-month LIBOR & pay a fixed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 million Next slide illustrates cash flows that could occur (Day count conventions are not considered) 33

34. One Possible Outcome for Cash Flows to Microsoft (Table 7.1, page 150) DateLIBORFloating Cash Flow Fixed Cash Flow Net Cash Flow Mar 5, 20124.20% Sep 5, 20124.80%+2.10−2.50−0.40 Mar 5, 20135.30%+2.40−2.50−0.10 Sep 5, 20135.50%+2.65−2.50+ 0.15 Mar 5, 20145.60%+2.75−2.50+0.25 Sep 5, 20145.90%+2.80−2.50+0.30 Mar 5, 2015+2.95−2.50+0.45 34

35. Typical Uses of an Interest Rate Swap Converting a liability from fixed rate to floating rate floating rate to fixed rate Converting an investment from fixed rate to floating rate floating rate to fixed rate 35

36. Intel and Microsoft (MS) Transform a Liability (Figure 7.2, page 151) 36 IntelMS LIBOR 5% LIBOR+0.1% 5.2%

37. Financial Institution is Involved (Figure 7.4, page 152) 37 F.I. LIBOR LIBOR+0.1 % 4.985% 5.015% 5.2% IntelMS Financial Institution has two offsetting swaps

38. Intel and Microsoft (MS) Transform an Asset ( Figure 7.3, page 152) 38 Intel MS LIBOR 5% LIBOR-0.2% 4.7%

39. Financial Institution is Involved (See Figure 7.5, page 153) 39 Intel F.I.MS LIBOR 4.7% 5.015%4.985% LIBOR-0.2%

40. Quotes By a Swap Market Maker (Table 7.3, page 154) MaturityBid (%)Offer (%)Swap Rate (%) 2 years6.036.066.045 3 years6.216.246.225 4 years6.356.396.370 5 years6.476.516.490 7 years6.656.686.665 10 years6.836.876.850 40

41. Day Count A day count convention is specified for fixed and floating payment For example, LIBOR is likely to be actual/360 in the US because LIBOR is a money market rate 41

42. Confirmations Confirmations specify the terms of a transaction The International Swaps and Derivatives has developed Master Agreements that can be used to cover all agreements between two counterparties Governments now require central clearing to be used for most standardized derivatives 42

43. The Comparative Advantage Argument (Table 7.4, page 156) AAACorp wants to borrow floating BBBCorp wants to borrow fixed 43 FixedFloating AAACorp4.0%6 month LIBOR − 0.1% BBBCorp5.2%6 month LIBOR + 0.6%

44. The Swap (Figure 7.6, page 157) 44 AAACorp BBBCorp LIBOR LIBOR+0.6% 4.35% 4%

45. The Swap when a Financial Institution is Involved (Figure 7.7, page 157) 45 AAACorp F.I. BBBCorp 4% LIBOR LIBOR+0.6% 4.33% 4.37%

46. Criticism of the Comparative Advantage Argument The 4.0% and 5.2% rates available to AAACorp and BBBCorp in fixed rate markets are 5-year rates The LIBOR−0.1% and LIBOR+0.6% rates available in the floating rate market are six-month rates BBBCorp’s fixed rate depends on the spread above LIBOR it borrows at in the future 46

47. The Nature of Swap Rates Six-month LIBOR is a short-term AA borrowing rate The 5-year swap rate has a risk corresponding to the situation where 10 six-month loans are made to AA borrowers at LIBOR This is because a lender can enter into a swap where income from the LIBOR loans is exchanged for the 5- year swap rate 47

48. Using Swap Rates to Bootstrap the LIBOR/Swap Zero Curve Consider a new swap where the fixed rate is the swap rate When principals are added to both sides on the final payment date, the swap is the exchange of a fixed rate bond for a floating rate bond The floating-rate rate bond is worth par. The swap is worth zero. The fixed-rate bond must therefore also be worth par This shows that swap rates define par yield bonds that can be used to bootstrap the LIBOR (or LIBOR/swap) zero curve 48

49. Valuation of an Interest Rate Swap Initially interest rate swaps are worth close to zero At later times they can be valued as the difference between the value of a fixed-rate bond and the value of a floating-rate bond Alternatively, they can be valued (but NOT replicated!) as a portfolio of FRAs 49

50. Valuation in Terms of Bonds The fixed rate bond is valued in the usual way The floating rate bond is valued by noting that it is worth par immediately after the next payment date 50

51. Valution of Floating-Rate Bond 51 0 t*t* Valuation Date First Payment Date, Floating Payment = k * Second Payment Date Maturity Date Value = L Value = L+k * Value = PV of (L+k * ) at t *

52. Example Pay six-month LIBOR, receive 8% (s.a. compounding) on a principal of $100 million Remaining life 1.25 years LIBOR rates for 3-months, 9-months and 15- months are 10%, 10.5%, and 11% (cont comp) 6-month LIBOR on last payment date was 10.2% (s.a. compounding) 52

53. Valuation Using Bonds (page 161) 53 TimeB fix cash flow B fl cash flow Disc factor PV of B fix PV of B fl 0.254.0105.1000.97533.901102.505 0.754.00.92433.697 1.25104.00.871590.640 Total98.238102.505 Swap value = 98.238 − 102.505 = −4.267 Note: the subscripts ‘fix’ and ‘fl’ denote ‘fixed’ and ‘floating’, respectively

54. Valuation in Terms of FRAs Each exchange of payments in an interest rate swap is an FRA The FRAs can be valued on the assumption that today’s forward rates are realized 54

55. Valuation of Example Using FRAs (page 163) TimeFixed cash flow Floating cash flow Net Cash Flow Disc factorPV of B fl 0.254.0-5.100-1.1000.9753-1.073 0.754.0-5.522-1.5220.9243-1.407 1.254.0-6.051-2.0510.8715-1.787 Total-4.267 55

56. Overnight Indexed Swaps Fixed rate for a period is exchanged for the geometric average of the overnight rates Should OIS rate equal the LIBOR rate? A bank can Borrow $100 million in the overnight market, rolling forward for 3 months Enter into an OIS swap to convert this to the 3-month OIS rate Lend the funds to another bank at LIBOR for 3 months 56

57. Overnight Indexed Swaps continued...but it bears the credit risk of another bank in this arrangement The OIS rate is now regarded as a better proxy for the short-term risk-free rate than LIBOR The excess of LIBOR over the OIS rate is the LIBOR- OIS spread. It is usually about 10 basis points but spiked at an all time high of 364 basis points in October 2008 57

58. An Example of a Currency Swap An agreement to pay 5% on a sterling principal of £10,000,000 & receive 6% on a US$ principal of $18,000,000 every year for 5 years 58

59. Exchange of Principal In an interest rate swap the principal is not exchanged In a currency swap the principal is usually exchanged at the beginning and the end of the swap’s life 59

60. The Cash Flows (Table 7.7, page 166) 60 DateDollar Cash Flows (millions) Sterling cash flow (millions) Feb 1, 2011-18.0+10.0 Feb 1, 2012+1.08−0.50 Feb 1, 2012+1.08−0.50 Feb 1, 2014+1.08−0.50 Feb 1, 2015+1.08−0.50 Feb 1, 2016+19.08−10.50

61. Typical Uses of a Currency Swap Convert a liability in one currency to a liability in another currency Convert an investment in one currency to an investment in another currency 61

62. Valuation of Currency Swaps Like interest rate swaps, currency swaps can be valued either as the difference between 2 bonds or as a portfolio of forward contracts Only caveat is that this time cash flows need to be converted in the same currency 62

63. Swaps & Forwards A swap can be regarded as a convenient way of packaging forward contracts Although the swap contract is usually worth close to zero at the outset, each of the underlying forward contracts are not worth zero 63

64. Credit Risk A swap is worth zero to a company initially At a future time its value is liable to be either positive or negative The company has credit risk exposure only when its value is positive Some swaps are more likely to lead to credit risk exposure than others 64

65. Other Types of Swaps Floating-for-floating interest rate swaps, amortizing swaps, step up swaps, forward swaps, constant maturity swaps, compounding swaps, LIBOR-in-arrears swaps, accrual swaps, diff swaps, cross currency interest rate swaps, equity swaps, extendable swaps, puttable swaps, swaptions, commodity swaps, volatility swaps…….. 65

66. Chapter 28* Option Strategies Variations Using Interest Rate Options *Part on risk management/engineering applications

67. Interest Rate Options Interest rate option gives holder the right but not the obligation to receive one interest rate (e.g. floating\LIBOR) and pay another (e.g. the fixed strike rate L K )

68. Caps A cap is a portfolio of “caplets” Each caplet is a call option on a future LIBOR rate with the payoff occurring in arrears Payoff at time t k+1 on each caplet is N  k max(L k - L K, 0) where N is the notional amount,  k  = t k+1 - t k, L K is the cap rate, and L k is the rate at time t k for the period between t k and t k+1 It has the effect of guaranteeing that the interest rate in each of a number of future periods will not rise above a certain level

69. Caplet Payoff 69 t 0 = 0t 1 = 30t 2 = 120 days Expiry \ Valuation of option, (LIBOR 1 - L K ) Strike rate L K fixed in the contract δ = 90 days

70. Planned Borrowing + Caplet (Call on Bond)

71. Loan + Interest Rate Floorlet (Put on Bond)

72. Collar 72 Comprises a long cap and short floor. It establishes both a floor and a ceiling on a corporate or bank’s (floating rate) borrowing costs. Effective Borrowing Cost with Collar (at T  t k+1 = t k + 90) = = [L k – max[{0, L k – L K } + max {0, L K – L k }]N(90/360) = L k,CAP N(90/360)if L k > L k,CAP = L k,FL N(90/360)if L k < L k,FL = L k (90/360)if L k,FL < L k < L k,CAP Collar involves borrowing cost at each payment date of either L k,CAP = 10% or L k,FL = 8% or L k = LIBOR if the latter is between 8% and 10%.

73. Combining options with swaps Cancelable swaps - can be cancelled by the firm entering into the swap if interest rates move a certain way Swaptions - options to enter into a swap

74. Swaptions OTC option for the buyer to enter into a swap at a future date and a predetermined swap rate  A payer swaption gives the buyer the right to enter into a swap where they pay the fixed leg and receive the floating leg (long IRS).  A receiver swaption gives the buyer the right to enter into a swap where they will receive the fixed leg, and pay the floating leg (short IRS).

75. Swaptions Example A US bank has made a commitment to lend at fixed rate $10m over 3 years beginning in 2 years time and may need to fund this loan at a floating rate. In 2 years time, the bank may wish to swap the floating rate payments for a fixed rate, Perhaps at that time, the bank may think that interest rates may rise over the 3 years and hence the cost of the fixed rate payments in the swap will be higher than at inception.

76. Example Bank might need a $10m swap, to pay fixed and receive floating beginning in 2 years time and an agreement that swap will last for further 3 years The bank can hedge by purchasing a 2-year European payer swaption, with expiry in T = 2, on a 3 year “pay fixed-receive floating” swap, at say s K = 10%. Payoff is the annuity value of Nδmax{s T – s K, 0}. So, value of swaption at T is: f = $10m[s T – s K ] [(1 + L 2,3 ) -1 + (1 + L 2,4 ) -2 + (1 + L 2,5 ) -3 ]

77. The Complications in Valuing Interest Rate Derivatives Of course, RNV can still be used (as long as NA) But we need a whole term structure to define the level of interest rates at any time The stochastic process for an interest rate is more complicated than that for a stock price Volatilities of different points on the term structure are different Interest rates are used for discounting the payoff as well as for defining the payoff 77

78. Approaches to Pricing Interest Rate Options Use a variant of Black’s model When using Black’s model we assume that the interest rate underlying each caplet/floorlet is lognormal Use a no-arbitrage model of the yield curve based Will not study in this course as it requires some dedicated formalism there are other modules devoted to this. 78

79. Appendix 79

80. Liquidity Preference Theory Suppose that the outlook for rates is flat and you have been offered the following choices Which would you choose as a depositor? Which for your mortgage? 80 MaturityDeposit rateMortgage rate 1 year3%6% 5 year3%6%

81. Liquidity Preference Theory cont To match the maturities of borrowers and lenders a bank has to increase long rates above expected future short rates In our example the bank might offer 81 MaturityDeposit rateMortgage rate 1 year3%6% 5 year4%7%

82. Instantaneous Forward Rate The instantaneous forward rate for a maturity T is the forward rate that applies for a very short time period starting at T. It is where R is the T-year rate 82

83. Example of Bootstrapping the LIBOR/Swap Curve (Example 7.1, page 160) 6-month, 12-month, and 18-month LIBOR zero rates are 4%, 4.5%, and 4.8% with continuous compounding. Two-year swap rate is 5% (semi-annual) The 2-year LIBOR/swap rate, R, is 4.953% 83

84. Comparative Advantage May Be Real Because of Taxes General Electric wants to borrow AUD Quantas wants to borrow USD Cost after adjusting for the differential impact of taxes 84 USDAUD General Electric5.0%7.6% Quantas7.0%8.0%

85. Example of Currency Swap Valuation All Japanese LIBOR/swap rates are 4% All USD LIBOR/swap rates are 9% 5% is received in yen; 8% is paid in dollars. Payments are made annually Principals are $10 million and 1,200 million yen Swap will last for 3 more years Current exchange rate is 110 yen per dollar 85

86. Valuation in Terms of Bonds (Table 7.9, page 169) TimeCash Flows ($)PV ($)Cash flows (yen)PV (yen) 10.80.73116057.65 20.80.66826055.39 30.80.61076053.22 310.07.63381,2001,064.30 Total9.64391,230.55 86 Value of Swap = 1230.55/110 − 9.6439 = 1.5430

87. Valuation in Terms of Forwards (Table 7.10, page 170) 87 Time$ cash flow Yen cash flow Forward Exch rate Yen cash flow in $ Net Cash Flow Present value 1-0.8600.0095570.5734-0.2266-0.2071 2-0.8600.0100470.6028-0.1972-0.1647 3-0.8600.0105620.6337-0.1663-0.1269 3-10.012000.01056212.6746+2.67462.0417 Total1.5430