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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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= or 6.76%.
Options, Futures, and Other Derivatives 7th International Edition, Copyright © John C. Hull 2008

10. 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

11. 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

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

13. 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

14. 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 % with quarterly compounding
This is % with continuous compounding
Similarly the 6 month and 1 year rates are % and % with continuous compounding
Options, Futures, and Other Derivatives 7th International Edition, Copyright © John C. Hull 2008

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

16. 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

17. 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

18. 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

19. 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

20. If T1 = 3 , T2 = 4 , R1 = and R2 =0.05
RF = (0.05x x3)/ (4-3)= 0.062

21. 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

22. 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

23. 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

24. 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

25. 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)

26. 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 ( )x0.25 =
-$125,000 at the 3.25 year point.
(-125,000)/( x0.25) = -$123,609 at the 3 year point.

27. 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

28. 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( )e-0.04x2 = $805,800