Interest Rates Outline 1. Types of rates 2. Measuring interest rates 3. Zero rates 4. Bond pricing 5. Determining Treasury zero rates 6. Forward rates 7. Forward rate agreements 8. Duration 9. Convexity 10. Theories of the term structure of interest rates Types of Rates(1/2) (1) Treasury Rates A government to borrow in its own currency. An investor earns on Treasury bills and Treasury bonds. It is usually assumed “risk-free rates”. (2) LIBOR LIBOR is short for London Interbank Offered Rate. Derivatives traders regard LIBOR rates as a better indication of the “true” risk-free rate than Treasury rates. Large banks also quote LIBID rates ( London Interbank Bid Rate ) and is the rate at which they will accept deposits from other banks. Types of Rates(2/2) (3) Repo Rates An investment dealer who owns securities agrees to sell them to another company now and buy them back later at a slightly higher price. The difference between the price is the interest it earns. We called the interest rate is repo rate. The most common type of repo is an overnight repo. Measuring interest rates(1/3) Suppose that an amount A is invested for n years at an interest rate of R per annum. If the rate is compounded m times per annum, the terminal value of the investment is mn A (1+ R/m) (1) • With continuous compounding, it can be shown that an amount A invested for n years at rate R grows to Ae Rn (2) Compounding a sum of money at a continuously compounded rate R for Rn n years involves multiplying it by e . Discounting it at a continuously compounded rate R for n years involves multiplying e-Rn. Measuring interest rates(2/3) • Suppose that Rc is a rate of interest with continuous compounding and Rm is the equivalent rate with compounding m times per annum. From the results in equations (1) and (2), we have Rc m e = ( 1+ Rm/m ) Rc = m ln(1+ Rm/m) (3) Rc/m Rm=m (e -1) (4) These equations can be used to convert a rate with a compounding frequency of m times per annum to a continuously compounded rate and vice versa. Measuring interest rates(3/3) or Rm mn A(1 ) Ae Rc n Rm m Rc n mn ln(1 ) m Rm mn e (1 ) Rc n Rc n ln(1 Rm ) m mn m Rm Rc Rm Rc n mn ln(1 ) e m e ln(1 m ) m Rc Rm Rm e m 1 Rc m ln(1 )(3) m m Rc Rm m(e m 1)(4) Zero rates Definition: 1. The n-year zero-coupon interest rate is the rate of interest earned on an investment that starts today and lasts for n years. All the interest and principal is realized at the end of n years. 2. Sometimes also referred to as the n-year spot rate. Bond Pricing(1/4) Example (Table 2) Suppose that a 2-year Treasury bond with a principal of $100 provides coupons at the rate of 6% per annum semiannually. Maturity Zero Rate (years) (% cont comp) 0.5 5.0 1.0 5.8 1.5 6.4 2.0 6.8 Bond Pricing(2/4) (1)Theoretical price of the bond 3e-0.05*0.5 ＋ 3e-0.058*1.0 ＋ 3e-0.064*1.5 ＋ 103e-0.068*2.0＝98.39 Maturity Zero Rate (years) (% cont comp) 0.5 5.0 1.0 5.8 Continuous 1.5 6.4 Compounding 2.0 6.8 Ae-Rn Bond Pricing(3/4) (2) Bond Yield A bond’s yield is the single discount rate that, when applied to all cash flows, gives a bond price equal to its market price. Assumed: y is the yield on the bond. -y*0.5 -y*1.0 -y*1.5 -y*2.0 3e ＋ 3e ＋3e ＋ 103e ＝98.39 y＝6.76% (trial and error ) Bond Pricing(4/4) (3) Par Yield The par yield for a certain bond maturity is the coupon rate that causes the bond price to equal its par value. Suppose that the coupon on a 2-year bond in our example is c per annum. c/2e-0.05*0.5＋c/2e-0.058*1.0＋c/2e-0.064*1.5＋(100+c/2)e-0.068*2.0＝100 c＝6.87 The 2-year par yield is 6.87% per annum with semiannual compounding. Determining Treasury zero rates(1/5) The most popular approach is known as the bootstrap method. Table 3 Date for bootstrap method Bond Time to Annual Bond Cash Principal Maturity Coupon Price (dollars) (years) (dollars) (dollars) 100 0.25 0 97.5 100 0.50 0 94.9 100 1.00 0 90.0 100 1.50 8 96.0 100 2.00 12 101.6 ＊Half the stated coupon is assumed to be paid every 6 months. Determining Treasury zero rates(2/5) The bootstrap method The 3-month bond provides a return of 2.5 in 3 months on an initial investment of 97.5. With quarterly compounding, the 3-month zero rate is (4×2.5)/97.5＝10.256％(per annum). The rate is expressed with continuous compounding, it becomes 4 ln （1＋0.10256/4）＝0.10127 Similarly the 6 month and 1 year rates are 10.469％ and 10.536 ％ with continuous compounding. Determining Treasury zero rates(3/5) The bootstrap method To calculate the 1.5 year zero rate, the payments are as follows: 6 months : ＄4 ; 1 year:＄4 ; 1.5 years :＄104 Suppose the 1.5-year zero rate is denoted by R. 0.104690.5 0.105361.0 R1.5 4e 4e 104e 96 R＝10.681％ Similarly the two-year rate is 10.808％ Determining Treasury zero rates(4/5) Zero Curve Calculated from the Data (Figure 1) 12 Zero Rate (%) 11 10.68 10.808 10.469 10.53 1 10 6 10.127 Maturity (yrs) 9 0 0.5 1 1.5 2 2.5 Determining Treasury zero rates(5/5) A common assumption is that the zero curve is linear between the points determined using the bootstrap method. The zero curve is horizontal prior to the first point and horizontal beyond the last point. By using longer maturity bonds, the zero curve would be more accurately determined beyond 2 years. Forward rates(1/5) Definition: the rates of interest implied by current zero rates for periods of time in the future. Forward rates(2/5) Calculation of Forward Rates (Table 5) Zero Rate for Forward Rate an n-year Investment for n th Year Year (n) (% per annum) (% per annum) 1 3.0 2 4.0 5.0 3 4.6 5.8 4 5.0 6.2 5 5.3 6.5 Forward rates(3/5) Calculation of Forward Rates (Table 5) • The 4％ per annum rate for 2 years mean that, in return for an investment of ＄100 today, the receive 100e0.04*2＝＄108.33 . • Suppose that ＄100 is invested. A rate of 3％ for the first year and 5％ for the second year gives at the end of the second year. 100e0.03*1e0.05*1＝＄108.33 When interest rates are continuously compounded and in successive time periods are combined, the overall equivalent rate is simply the average rate during the whole period. Forward rates(4/5) Formula of Forward Rates The forward rate for year 3 is the rate of interest that is implied by a 4％ per annum 2-year zero rate and a 4.6％ per annum 3-year zero rate. If R1and R2 are the zero rates for maturities T1 and T2 ,respectively, and RF is the forward interest rate for the period of time between T1 and T2 ,then R2T2 R1T1 RF T2 T1 (5) T1 RF R2 ( R 2 R1 ) T2 T1 (6) Forward rates(5/5) 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 R RF R T T where R is the T-year rate Forward rate agreements(1/3) Definition: FRA is an over-the-counter agreement that a certain interest rate will apply to either borrowing or lending a certain principal during a specified future period of time. Forward rate agreements(2/3) Valuation Formulas Define Rk: The rate of interest agreed to in the FRA RF: The forward LIBOR interest rate for the period between times T1 and T2 calculated today. RM: The actual LIBOR interest rate observed in the market at times T1 for the period between times T1 and T2 L: The principal underlying the contract. Forward rate agreements(3/3) Valuation Formulas Value of FRA where a fixed rate RK will be received on a principal L between times T1 and T2 is V L( RK RF )(T2 T1 )e R T 2 2 Value of FRA where a fixed rate is paid is V L( RF RK )(T2 T1 )e R2T2 RF is the forward rate for the period and R2 is the zero rate for maturity T2 Duration(1/3) Definition: A measure of how long on average the holder of the bond has to wait before receiving cash payment. A zero-coupon bond that lasts n years has a duration of n years. A coupon-bearing bond lasting n years has a duration of less than n years. Duration(2/3) • Duration of a bond that provides cash flow c i at time t i is n ci e yti D ti n i 1 B (12) B Ci e yti i 1 where B is its price and y is its yield (continuously compounded) • This leads to B Dy B Duration(3/3) Modified Duration • The preceding analysis is based on the assumption that y is expressed with continuous compounding. If y is expressed with a compounding frequency of m times per year, then BDy B 1 y / m A variable D*, defined by D D* 1 y / m Convexity(1/2) The duration relationship applies only to small changes in yields. It shows the relationship between the percentage change in value and change in yield for two bond portfolios having the same duration. ( Figure 2) Convexity(2/2) (Figure 2) P3 P4 A P0 P1 P2 Y2 Y0 Y1 For large yield changes, the portfolio X has more curvature in its relationship with yields than portfolio Y. Theories of the term structure of interest rates •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. Summary(1/2) Treasury rates are the rates paid by a government on borrowings in its own currency. LIBOR rates are short-term lending rates offered by banks in the interbank market. The n-year zero or spot rate is the applicable to an investment lasting for n years when all of the return is realized at the end. Forward rates are the rates applicable to future periods of time implied by today’s zero rates. Summary(2/2) FRA is an over-the-counter agreement that a certain interest rate will apply to either borrowing or lending a certain principal at LIBOR during a specified future period of time. Duration measures the sensitivity of the value of a bond portfolio to a small parallel shift in the zero-coupon yield curve. Liquidity preference theory can be used to explain the interest rate term structures that are observed in practice.