Interest Rate Markets

Transcription

1 Interest Rate Markets 5. Chapter 5 5. Types of Rates Treasury rates LIBOR rates Repo rates 5.3 Zero Rates A zero rate (or spot rate) for maturity T is the rate of interest earned on an investment with a payoff only at time T

2 5.4 Example (Table 5., page 0) Maturity Zero Rate (years) (% cont. comp.) Notice in this example that Zero rates rise with duration Going from 5% pa to 6.8% pa. 5.6 Bond Pricing Bond cash price calculation To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate Example In our example above, the theoretical price of a two-year bond providing a 6% annual coupon paid semi-annually is Zero rate for mths Zero rate for 6 mths 6% x 00/ 3e e + 3e e =

3 Bond Yield The bond yield 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 Example Suppose that the market price of the bond in our example equals its theoretical price of The bond yield is given by solving 6% x 00/ y 0. 5 y. 0 y. 5 y. 0 3e + 3e + 3e + 03e = to get y = or 6.76%. Yield Coupon + par value Bond price 5.7 Par Yield The par yield Par yield for a certain maturity is the coupon rate (in % terms) that causes the bond price to equal its face or par value. Example In the example above we solve for c: c c c e + e + e c e = 00 toget c= (with s.a. compounding) In general if Par Yield cont d m is the number of coupon payments per year, P is the present value of $ received at maturity and A is the present value of an annuity of $ on each coupon date then ( 00 00P ) m c = A

4 5.0 Par Yield cont d For example, in the previous formula we had A = e P = e m= x + e e e Sample Data (Table 5., page 0) 5. Bond Time to Annual Bond Principal Maturity Coupon Price (dollars) (years) (dollars) (dollars) Note: Half the stated coupon is paid every 6 months Bond price calculation (table 5.) 5. Bond principal ($) Time to maturity (years) Annual coupon ($) Bond price ($) Zero rate (%) Bond : Get back $00 in 3 months for which you paid only $97.5 gives profit of.5 Bond 3: Get back $00 in months for which you paid only $90 gives profit of $ Bond : Convert to rate pa with continuous. compounding: R C = 4ln(+.5/97.5) Bond 3: Convert to rate pa with continuous. compounding: R C = ln(+0/90)

5 Calculating zero rates: The Bootstrap Method From the table, if I buy at $97.5 and receive $00 at the end of a quarter I receive $.5 (= ) gain on $97.5 investment over one quarter. The quarterly rate annualises this and expresses it as a percentage: It is.5/97.5 =.56% per quarter or.56% x 4 = 0.56% p.a. In continuous terms this is equivalent to 0.7% p.a. See first row, last column 5.3 Bootstrapping cont d For the.5 year rate (row 4,T5. ) we have to take into account intermediate cash flows We: Calculate the first two 6-monthly zeros and Zero rate for 6 mths Solve the following equation for the final zero, R: 4 e Coupon 8% x 00/ e Zero rate for mths + 04 e R. 5 = 96 to get.5 year rate R = or 0.68% Note that the coupon differs from the.5 and year bonds Similarly the two-year rate is 0.808% 5.4 Final zero solved for Bond price Zero Curve Calculated from the Data (Figure 5., page 04) Zero Rate (%) Maturity (yrs)

6 5.6 Forward Rates The forward rate is the future zero rate implied by today s term structure of interest rates given certain arbitrage relationships 5.7 Arbitrage Example We have the following no-arbitrage relationships between the periods and zeros and the forward rate: e R T F [ T T ] R T e RT + R [ T T ] = R T R R F F = e F = ( R T RT ) /( T T ) = R R,for T =, T = Where R and R are the zero rates for T and T is the forward rate between them R F Calculation of Forward Rates Table 5.4, page Formula : R = (R T R T )/(T T ) F Zero Rate for Forward Rate an n -year Investment for n th Year Year (n ) (% per annum) (% per annum) R F =[(0.5)-(0)]/(-) R F =[3(0.8)-(0.5)] /(3-)

7 5.9 Formula for Forward Rates Generalising, suppose that the zero rates for time maturities T and T are R and R and both rates continuously compounded. Then the forward rate for the period T to T can be shown to be: R F RT RT = T T 5.0 Thus, in Table 5.4 the forward rate for year 3 of.4% is implied by the zero rates for years and 3 of 0.8% and 0.5% since R F R T RT 0.8(3) 0.5() = = T T 3 =.4 5. Upward vs. Downward-Sloping Yield Curve For an upward-sloping yield curve ( R / T > 0) : Fwd Rate > Zero Rate > Par Yield (see Fig 5.) For a downward-sloping yield curve ( R / T < 0) : Par Yield > Zero Rate > Fwd Rate

8 5. The first and third inequalities can be verified by rearranging the formula for R F to get: R F = R + ( R R ) T T T If R><R then RF><R I.e. up/downwardsloping YC 5.3 Also, taking limits as T approaches T we find: R( T ) R( T) RF = R + T RF R = T T T where R(T) = the spot rate at time T T = the maturity of the bond Forward Rate Agreement A forward rate agreement (FRA) is an agreement that a fixed rate will apply to a certain principal for a certain future time period 5.4

9 Forward Rate Agreement continued (Page 06) 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 interest rate is certain to be realized 5.5 Theories of the Term Structure Pages 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 5.6 Day Count Conventions in the U.S. (Page 08) 5.7 Treasury Bonds: Corporate Bonds: Money Market Instruments: Actual/Actual (in period) 30/360 Actual/360

10 Treasury Bond Price Quotes in the U.S 5.8 Cash price = Quoted price + Accrued Interest 5.9 Treasury 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 is 360 ( 00 Y ) n Treasury Bond Futures Page Cash price received by party with short position = Quoted futures price Conversion factor + Accrued interest

11 5.3 Conversion Factor The conversion factor for a bond is approximately equal to the value of the bond on the assumption that the yield curve is flat at 6% compounding is semiannual CBOT T-Bonds & T-Notes 5.3 Factors that affect the futures price: Delivery can be made any time during the delivery month Any of a range of eligible bonds can be delivered The wild card play 5.33 Eurodollar Futures (Page 6) Let Z be the quoted price of a Eurodollar futures contract Then the value of one contract is 0,000[00-0.5(00-Z)] A change of one basis point or 0.0 in a Eurodollar futures quote corresponds to a contract price change of $5

12 5.34 Exercise Verify the last statement on the previous slide Eurodollar Futures continued A Eurodollar futures contract is settled in cash It expires on the third Wednesday of the delivery month, when Z is set equal to 00 minus the 90 day Eurodollar interest rate (actual/360) and all contracts are closed out Forward Rates and Eurodollar Futures (Page 7) Duration of Eurodollar futures contracts last up to 0 years Longer duration Eurodollar futures contracts Contracts lasting beyond two years: we cannot assume that the forward rate equals the futures rate

13 5.37 Forward Rates and Eurodollar Futures continued A "convexity adjustment "often madeis Forwardrate= Futuresrate σ t t where t isthetime tomaturity of thefuturescontract, t isthematurity oftherateunderlyingthefuturescontract (90dayslater thant ) and σ isthestandarddeviation of theshort ratechanges per year (typicallyσ isabout 0.0) Duration 5.38 Duration of a bond that provides cash flow c i at time t i is n yt c e i i t i i = B where B is its price and y is its yield (continuously compounded) This leads to δ B B = D δ y Duration Continued When the yield y is expressed with compounding m times per year BDδy δb = + y m The expression D + y m is referred to as the modified duration 5.39

14 5.40 Duration Matching Duration-matching involves hedging against interest rate risk by matching the durations of assets with the durations of liabilities Function of-duration matching It provides protection against small parallel shifts in the zero curve