Interest Rates Chapter Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 Types of Rates Treasury rates LIBOR rates Repo rates Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 Measuring Interest Rates The compounding frequency used for an interest rate is the unit of measurement The difference between quarterly and annual compounding is analogous to the difference between miles and kilometers Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 Continuous Compounding (Page 83) In the limit as we compound more and more frequently we obtain continuously compounded interest rates $100 grows to $100eRT when invested at a continuously compounded rate R for time T $100 received at time T discounts to $100e-RT at time zero when the continuously compounded discount rate is R Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 Conversion Formulas (Page 83) Define Rc : continuously compounded rate Rm: same rate with compounding m times per year Rm Rc = m ln1 + m Rc / m Rm = m e −1 ( ) Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 Zero 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 Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 Example (Table 4.2, page 85) Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 Bond Pricing To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate In our example, the theoretical price of a two-year bond providing a 6% coupon semiannually is 3e −0.05×0.5 + 3e −0.058×1.0 + 3e −0.064×1.5 + 103e −0.068×2.0 = 98.39 Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 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 is given by solving to−get y = 0.0676 or 6.76% with cont comp y × 0.5 − y ×1.0 − y ×1.5 − y × 3e + 3e + 3e + 103e = 98.39 Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 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 c −0.05×0.5 c −0.058×1.0 c −0.064×1.5 e + e + e 2 c −0.068×2.0 + 100 + e = 100 2 to get c=6.87 (with s.a compoundin g) Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 10 Par Yield 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 of $1 on each coupon date (100 − 100 × P ) m c= A Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 11 Sample Data (Table 4.3, page 86) Bond Time to Annual Bond Principal Maturity Coupon Price (dollars) (years) (dollars) (dollars) 100 0.25 97.5 100 0.50 94.9 100 1.00 90.0 100 1.50 96.0 100 2.00 12 101.6 Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 12 The Bootstrap Method An amount 2.5 can be earned on 97.5 during months The 3-month rate is times 2.5/97.5 or 10.256% with quarterly compounding This is 10.127% with continuous compounding Similarly the month and year rates are 10.469% and 10.536% with continuous compounding Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 13 The Bootstrap Method continued To calculate the 1.5 year rate we solve −0.10469×0.5 −0.10536×1.0 e + e to get R = 0.10681 or 10.681% + 104e − R×1.5 = 96 Similarly the two-year rate is 10.808% Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 14 Zero Curve Calculated from the Data (Figure 4.1, page 88) 12 Zero Rate (%) 11 10.469 10 10.127 10.53 10.68 10.808 Maturity (yrs) 0.5 1.5 2.5 Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 15 Forward Rates The forward rate is the future zero rate implied by today’s term structure of interest rates Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 16 Calculation of Forward Rates Table 4.5, page 89 Zero Rate for Forward Rate an n -year Investment for n th Year Year (n ) (% per annum) (% per annum) 3.0 4.0 5.0 4.6 5.8 5.0 6.2 5.3 6.5 Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 17 Formula for Forward Rates Suppose that the zero rates for time periods T1 and T2 are R1 and R2 with both rates continuously compounded The forward rate for the period between times T1 and T2 is R2 T2 − R1 T1 T2 − T1 Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 18 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 Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 19 Forward Rate Agreement A forward rate agreement (FRA) is an agreement that a certain rate will apply to a certain principal during a certain future time period Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 20 Forward Rate Agreement continued An FRA is equivalent to an agreement where interest at a predetermined rate, RK 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 Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 21 FRA Example A company has agreed that it will receive 4% on $100 million for months starting in years The forward rate for the period between and 3.25 years is 3% The value of the contract to the company is +$250,000 discounted from time 3.25 years to time zero Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 22 FRA Example Continued Suppose rate proves to be 4.5% (with quarterly compounding The payoff is –$125,000 at the 3.25 year point This is equivalent to a payoff of –$123,609 at the 3-year point Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 23 Theories of the Term Structure Page 93 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 Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 24 Management of Net Interest Income (Table 4.6, page 94) Suppose that the market’s best guess is that future short term rates will equal today’s rates What would happen if a bank posted the following rates? Maturity (yrs) Deposit Rate Mortgage Rate 3% 6% 3% 6% How can the bank manage its risks? Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 25 ... payoff only at time T Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 Example (Table 4. 2, page 85) Fundamentals of Futures and Options Markets, 7th Ed, Ch. .. unit of measurement The difference between quarterly and annual compounding is analogous to the difference between miles and kilometers Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, ... compounding Similarly the month and year rates are 10 .46 9% and 10.536% with continuous compounding Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C Hull 2010 13