Interest Rates Chapter 4 Options, Futures, and Other Derivatives 7th International Edition, Copyright © John C. Hull 2008

Types of Rates Treasury rates: Rates an investor earns on T-bills and T-bonds. LIBOR rates: is the rate of interest at which the bank is prepared to make large wholesale deposit with other banks. The bank must have AA credit rating. In derivatives market risk-free rate is LIBOR Repo rates: It is a contract where an investor dealer who owns securities agrees to sell them to another company now and buy them back late at a slighly higher price. Options, Futures, and Other Derivatives 7th International Edition, Copyright © John C. Hull 2008

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 7th International Edition, Copyright © John C. Hull 2008

Continuous Compounding 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 Options, Futures, and Other Derivatives 7th International Edition, Copyright © John C. Hull 2008

Conversion Formulas In pricing derivatives,continuously compounding interest rates are used. Define Rc : continuously compounded rate Rm: same rate with compounding m times per year Options, Futures, and Other Derivatives 7th International Edition, Copyright © John C. Hull 2008

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. All coupon and principal payments are realized at the end of maturity. Suppose a 5-year zero rate with continuous compaunding is quoted as 5% per annum. $100e0.05x5 = $128.4 Options, Futures, and Other Derivatives 7th International Edition, Copyright © John C. Hull 2008

Example: Suppose a 2-year T-bond with a principal of $100 provides coupons at the rate of 6% per annum semiannulaly. Options, Futures, and Other Derivatives 7th International Edition, Copyright © John C. Hull 2008

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 7th International Edition, Copyright © John C. Hull 2008

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 7th International Edition, Copyright © John C. Hull 2008

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 7th International Edition, Copyright © John C. Hull 2008

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 Options, Futures, and Other Derivatives 7th International Edition, Copyright © John C. Hull 2008

m=2 d=e-0.068x2 = 0.87238 A= e-0.05x0.5 + e-0.058X1 +e-0.064x1.5 +e-0.068x2 =3.70027

Determing Treasury Zero Rates Sample Data (Table 4.3, page 86) 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 7th International Edition, Copyright © John C. Hull 2008

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 7th International Edition, Copyright © John C. Hull 2008

The Bootstrap Method continued To calculate the 1.5 year rate we solve to get R = 0.10681 or 10.681% Similarly the two-year rate is 10.808% Options, Futures, and Other Derivatives 7th International Edition, Copyright © John C. Hull 2008

Zero Curve Calculated from the Data (Figure 4.1, page 82) Zero Rate (%) 10.808 10.681 10.469 10.536 10.127 Maturity (yrs) Options, Futures, and Other Derivatives 7th International Edition, Copyright © John C. Hull 2008

Forward Rates - The forward rate is the rate of interest implied by current zero rates for periods of time in the future. Similar to Treasury zero rates, the rates are assumed to be continuesly compounding. Options, Futures, and Other Derivatives 7th International Edition, Copyright © John C. Hull 2008

Calculation of Forward Rates Table 4.5, page 83 n-year Forward Rate zero rate 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 7th International Edition, Copyright © John C. Hull 2008

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 Options, Futures, and Other Derivatives 7th International Edition, Copyright © John C. Hull 2008

If T1 = 3 , T2 = 4 , R1 = 0.046 and R2 =0.05 RF = (0.05x4-0.046x3)/ (4-3)= 0.062

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 7th International Edition, Copyright © John C. Hull 2008

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. Options, Futures, and Other Derivatives 7th International Edition, Copyright © John C. Hull 2008

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 Assuming the borrowing and lending would normally be done at LIBOR. Options, Futures, and Other Derivatives 7th International Edition, Copyright © John C. Hull 2008

RK= rate of interest agreed to in the FRA RF= forward LIBOR rate for the period between T1 and T2 calculated today. RM= ACTUAL lıbor rate at time T1 for the period between T1 and T2 L=principal underlying the contract

Consider a forward rate agreement where company X is agreeing to lend at LIBOR to company Y for the period between T1 and T2 The extra interest that will be earned in entering into FRA is L(RK−RM)(T2−T1) CF to Y is L(RM −RK )(T2−T1)

Suppose that a company enters into a FRA that specifies it will receive a fixed rate of 4% on a principal of $100 million for a 3-month period starting in 3 years. If 3-month LIBOR proves to be 4.5% for the 3-month period. CF to lender is; 100,000,000 x (0.04-0.045)x0.25 = -$125,000 at the 3.25 year point. (-125,000)/(1+0.045x0.25) = -$123,609 at the 3 year point.

Valuation Formulas (equations 4.9 and 4.10, pages 86-87) Value of FRA where a fixed rate RK will be received on a principal L between times T1 and T2 is L(RK−RF)(T2−T1)exp(-R2T2) Value of FRA where a fixed rate is paid is L(RF−RK)(T2−T1)exp(-R2T2) RF is the forward rate for the period and R2 is the zero rate for maturity T2 Options, Futures, and Other Derivatives 7th International Edition, Copyright © John C. Hull 2008

Suppose that LIBOR zero and forward rates are as Table 4. 5 (page 89) Suppose that LIBOR zero and forward rates are as Table 4.5 (page 89). Consider an FRA where a company will receive a rate of 6% measured with annual compounding, on a principal of $100 million between the end of year 1 and year 2. Forward rate is 5% with continues compounding or 5.127% with annual compounding. VFRA= 100,000,000(0.06-0.05127)e-0.04x2 = $805,800