CHInterestRatesandDuration(久期)(金融工建华东师范大学汤银才)

1、4.1Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityInterest Rates and Duration(久期久期)Chapter 44.2Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityTypes of Rates

2、 Treasury rates（国债利率）regarded as risk-free rates LIBOR rates (London Interbank Offer rate) (伦敦银行同业放款利率)generally higher than Treasury zero rates Repo rates (回购利率)slightly higher than the Treasury rates4.3Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Sha

3、nghai Normal UniversityZero Rates A zero rate (or spot rate), for maturity T, is the rate of interest earned on an investment that provides a payoff only at time T. In practice, it is usually called zero-coupon interest rate (零息票利率).4.4Options, Futures, and Other Derivatives, 4th edition 2000 by Joh

4、n C. Hull Tang Yincai, 2003, Shanghai Normal UniversityExample (Table 4.1, page 89) 4.5Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityBond Pricing To calculate the cash price of a bond we discount each cash flow at the appropria

5、te zero rate In our example (page 89), the theoretical price of a two-year bond with a principal of $100 providing a 6% coupon semiannually is39.98$1033330 . 2068. 05 . 1064. 00 . 1058. 05 . 005. 0eeee4.6Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Sha

6、nghai Normal UniversityBond 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 is given by solving to

7、get y=0.0676 or 6.76%.33310398390 51 0152 0eeeeyyyy .4.7Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityPar Yield The par yield (面值收益率) for a certain maturity is the coupon rate that causes the bond price to equal its face value

8、(ie. The principal). The bond is usually assumed to provide semiannual coupons. In our example we solvececececec=.22210021006 870 050 50 0581 00 0641 50 0682 0.to get 4.8Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityPar Yield (

9、continued) In general if m is the number of coupon payments per year, P is the present value of $1 received at maturity and A is the present value of an annuity(年金) of $1 on each coupon date cP mA()100100 100100PmcA4.9Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yin

10、cai, 2003, Shanghai Normal UniversitySample Data (Table 4.2, page 91) BondTime toAnnualBondPrincipalMaturityCouponPrice(dollars)(years)(dollars)(dollars)1000.25097.51000.50094.91001.00090.01001.50896.01002.0012101.64.10Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yi

11、ncai, 2003, Shanghai Normal UniversityThe Bootstrap Method(息票剥离法息票剥离法)-used to determine zero rates An amount 2.5 can be earned on 97.5 during 3 months. The 3-month rate is 4 times 2.5/97.5 or 10.256% with quarterly compounding This is 10.13% with continuous compounding Similarly the 6 month and 1 y

12、ear rates are 10.47% and 10.54% with continuous compounding 4.11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityThe Bootstrap Method （continued） To calculate the 1.5 year rate we solve to get R = 0.1068 or 10.68% Similarly the tw

13、o-year rate is 10.81%（see the equation on page 91）44104960 1047 0 50 1054 1 015eeeR.4.12Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityZero Curve Calculated from the Data (Figure 4.1, page 92) Zero Rate (%)Maturity (yrs)10.12710

14、.46910.53610.68110.8084.13Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityForward Rates(远期利率远期利率) The forward rate is the future zero rate implied by todays term structure(期限结构) of interest rates It is determined by the current z

15、ero rates The forward rate is for a specified future time period4.14Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityCalculation of Forward RatesTable 4.4, page 93 Zero Rate forForward Ratean n -year Investment for n th YearYear (

16、n )(% per annum)(% per annum)110.0210.511.0310.811.4411.011.6511.111.54.15Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityFormula for Forward Rates Suppose that the zero rates for time periods T1 and T2 are R1 and R2 with both ra

17、tes continuously compounded. The forward rate RF for the period between times T1 and T2 is121122)(221211TTTRTRRAeeAeFTRTTRTRF4.16Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityInstantaneous Forward Rate The instantaneous forward

18、 rate for a maturity T is the forward rate that applies for a very short time period starting at T. Letting gives rise to where R is the T-year rateTRTRRFTTT124.17Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityUpward vs Downward

19、 SlopingYield Curve (Figure 4.2 and 4.3, page 93) For an upward sloping yield curve:Fwd Rate Zero Rate Par Yield(See Figure 4.2 on page 94) For a downward sloping yield curvePar Yield Zero Rate Fwd Rate (See Figure 4.3 on page 94)4.18Options, Futures, and Other Derivatives, 4th edition 2000 by John

20、C. Hull Tang Yincai, 2003, Shanghai Normal UniversityForward Rate Agreement A forward rate agreement (FRA,远期利率协议) is an agreement that a certain rate RK will apply to a certain principal during a certain future time period4.19Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull

21、Tang Yincai, 2003, Shanghai Normal UniversityValue of FRA Case I: If RK will be earned/received for the period between T1 and T2 on a principal of L, then its value is Case II: If RK will be paid for the period between T1 and T2 on a principal of L, then its value is Example 4.1 (page 96)22)(12TRFKe

22、TTRRLV22)(12TRKFeTTRRLV4.20Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityForward Rate Agreementcontinued (Page 97) An FRA is equivalent to an agreement where interest at a predetermined rate, RK , is exchanged for interest at t

23、he market rate R (see the two cases on page 97) An FRA can be valued by assuming that the forward interest rate is certain to be realized4.21Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityTheories of the Term Structure(Pages 97-

24、98) 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 (page 98)4.22Options, Futures, and Other Derivatives, 4th

25、 edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityDay Count Conventions in the U.S. (Page 98)Treasury Bonds:Corporate Bonds:Money Market Instruments:Actual/Actual (in period)30/360Actual/360 See the examples on page 99 The day count defines the way that interest accrues ove

26、time. Three day count conventions used in USA4.23Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityTreasury Bond Price Quotesin the U.S The quoted price is not the same as the cash price that is paid by the purchaser. Cash price =

27、Quoted price + Accrued Interest Example on page 1004.24Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityTreasury Bond Price Quotesin the U.S(continued)Example on page 100Consider a 11% coupon bond maturing on July 10, 2001 Coupons

28、 paid semiannuallyActual/actual day count convention Quoted price=95-16, i.e. $(95+16/32)=$95.50Suppose it is March 5, 1999 NOW and the most recent coupon date is Jan 10, 1999July10,2000Case Price=$95.50 + (54/181) x $5.5 = $97.14Jan 10, 99Mar 5July 10541811 year541811 year541811 year4.25Options, Fu

29、tures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityTreasury Bill Quote in the U.S. If Y is the cash price of a Treasury bill that has n days to maturity, the quoted price is360100nY() The quoted price/discount rate is not the same as the rate

30、of return earned on the treasury bill (page100)4.26Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityInterest Rate Futures Two types of contracts1. the underlying is a government bond Treasury bonds futures (CBOT) 长期国债期货 Treasury n

31、otes futures (CBOT) 中期国债期货 Long gilt futures (LIFFE) 金边证券期货 2. the underlying is a shot term Eurodollar or LIBBOR interest rate Eurodollar futures (CME) Euroyen futures (CME and SIMEX) EuroSwiss futures (LIFFE) See the quotes from The Wall Street Journal (page 101 & 102)4.27Options, Futures, and

32、 Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityTreasury Bond FuturesPage 103 Any government bond with more than 15 years to maturity on the first day of the delivery moth and not callable within 15 years from that can be delivered Each contract is f

33、or $100,000 face value of bonds Cash price received by party with short position = Quoted futures price Conversion factor + Accrued interest4.28Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityConversion Factor The conversion fact

34、or for a bond is approximately equal to the value of the bond on the assumption that the yield curve is flat at 8% with semiannual compounding The bond maturity and times to the coupon payment dates are rounded down to the nearest 3 months for the purpose of calculating the conversion factor See exa

35、mple 4.2 & 4.3 on page 104 4.29Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityCBOTT-Bonds & T-NotesFactors that affect the futures price:Delivery can be made any time during the delivery monthAny of a range of eligible b

36、onds can be deliveredThe wild card play: an option of delivery that the party with the short position has 4.30Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityDetermining the Quoted Futures Price Theoretically difficult to determi

37、ne because of the factors above Given that both the cheapest-to-deliver bond and the delivery date are known The treasury bond futures contract is a futures contract on a security providing the holder with known income Equation (3.6) can be used Procedures and illustrative example on page 1064.31Opt

38、ions, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityEurodollar Futures A Eurolldollar is a dollar deposited in a U.S. or foreign bank outside the U.S.A. The 3-month Eurodollar interest rate (or 3-month LIBOR) is the rate of interest ear

39、ned on Eurodollars deposited for 3 months by one bank with another bank The 3-month Eurodollar futures contract (CME) is the most popular futures contract on short-term rates4.32Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University If

40、 Z is the quoted price of a Eurodollar futures contract, the value (contract price) of one contract is 10,000100-0.25(100-Z) A change of one basis point or 0.01 in a Eurodollar futures quote corresponds to a contract price change of $25 Eurodollar Futures (continued)4.33Options, Futures, and Other D

41、erivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityEurodollar Futures (continued) A Eurodollar futures contract is settled in cash When it expires (on the third Wednesday of the delivery month) Z is set equal to 100 minus the 90 day Eurodollar interest rate and

42、 all contracts are closed out4.34Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityForward Rates and Eurodollar Futures (Page 108) Eurodollar futures contracts last out to 10 years For Eurodollar futures lasting beyond two years we

43、 cannot assume that the forward rate equals the futures rate4.35Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal UniversityForward Rates and Eurodollar Futures (continued )A convexity adjustment often made isForward rate = Futures ratewhere is the time to maturity of the futures contract, is the maturity of the rate underlying the futures contract(90 days later than ) and is t