Interest Rates Chapter 4 Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Types of Rates Interest rate defines the amount of money a borrower promises to pay the lender. Depends on the credit risk. Treasury rates: rates an investor earn on Treasury bills and Treasury bonds LIBOR rates: the rate at which a bank is prepared to make a large wholesale deposit with other banks (AA credit rating is needed) Repo rates: The difference between the price at which securities are sold and the (higher) price at which they are repurchased Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

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 Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Continuous Compounding (Page 79) In the limit as we compound more and more frequently we obtain continuously compounded interest rates $100 grows to $100(1+R)T when invested at an annually compounded rate R for time T $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 Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Conversion Formulas (Page 79) Define Rc : continuously compounded rate Rm: same rate with compounding m times per year Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

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 Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Example (Table 4.2, page 81) Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

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 Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

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%. Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

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 Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

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 In the example, m=2,P=e-0.68*2=0.8728, and Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Determining Treasury Zero Rates Sample Data (Table 4.3, page 82) Bond Time to Annual Bond Cash 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 8 96.0 100 2.00 12 101.6 Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

The Bootstrap Method 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 10.127% with continuous compounding Similarly the 6 month and 1 year rates are 10.469% and 10.536% with continuous compounding Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

The Bootstrap Method continued To calculate the 1.5 year rate we solve This reduces to e-1.5R =0.85196 to get R = 0.10681 or 10.681% Similarly the two-year rate is 10.808% Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Continuously compounded Zero rates Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Forward Rates The forward rate is the future zero rate implied by today’s term structure of interest rates Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Calculation of Forward Rates Table 4.5, page 85 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 Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Forward Rates The 3% for 1 year means that in return for an investment of $100 today, an investor receives 100e0.03*1=$103.05 in 1 year The 4% for 2 years means that in return for an investment of $100 today, an investor receives 100e0.04*2=$108.33 in 2 years The forward interest rate for year 2, is 5%. Is the rate that if applied to the second year, combined with 3% in the first year, gives 4% overall for 2 years 100e0.03*1* e0.05*1=$108.33= 100e0.04*2 Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

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 Example: T1=3, T2=4, R1=0.046, R2=0.05 and the formula gives RF=0.062 Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

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 (T2 approaches T1). It is where R is the T-year zero rate Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

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 Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Forward Rate Agreement 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 The assumption is that the borrowing or lending would normally be done at LIBOR Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Definitions 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 time T1 for the period between times T1 and T2 L: The principal underlying the contract Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

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 Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Forward Rate Agreements Consider a FRA where company X is agreeing to lend company Y for the period between T1 and T2 Normally company X would earn RM, from the LIBOR loan. The extra interest it earns as a result of entering into a FRA is (RK - RM) (could be negative). The extra interest rate leads to a cash flow to company X at time T2 of L(RK - RM)(T2 - T1) Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Valuation Formulas (equations 4.9 and 4.10 page 88) Value of FRA where a fixed rate RK will be received on a principal L between times T1 and T2 is Value of FRA where a fixed rate is paid is RF is the forward rate for the period and R2 is the zero rate for maturity T2 What compounding frequencies are used in these formulas for RK, RM, RF and R2? Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Example Consider an FRA where we will receive a rate of 6%, measured on annual compounding, on a principal of 1$ million between the end of year 1 and the end of year 2 The forward rate is 5% with cont. comp. or 5.127% with annual comp. The value of the FRA is: 1.000.000(0.06-0.05127)e-0.04*2=$8,058 We always assume that the forward rates are realized, which means that RM = RF Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Duration (page 89) Duration is a measure of how long on average the holder of the bond has to wait before receiving cash payments 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) When a small change Δy in the yield is considered: Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Example Consider a 3-year 10% coupon bond with a face value of 100. B=5*e-0.12*0.5+ 5*e-0.12*1.0+ 5*e-0.12*1.5+ 5*e-0.12*2.0+ 5*e-0.12*2.5+ 105*e-0.12*3.0=94.213 The bond price is B=94.213 and the Duration, D=2.653, so that ΔB=-94.213*2.653*Δy, or ΔB=-249.95*Δy When the yield on the bond increases by 10 basis points(0.1%) it follows that Δy=0.001. The duration relationship predicts that ΔB=-249.95*0.001=-0.25, so that the bond price goes down to 94.213-0.25=93.963. How accurate is this? B=5*e-0.121*0.5+ 5*e-0.121*1.0+ 5*e-0.121*1.5+ 5*e-0.121*2.0+ 5*e-0.121*2.5+ 105*e-0.121*3.0=93.963 Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Duration Continued When the yield y is expressed with compounding m times per year The expression is referred to as the “modified duration” Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Convexity For large yield changes, the convexity is needed The convexity measures the curvature of the term structure.The conv. of a bond is defined as Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

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 Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005