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

2. 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

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 6th Edition, Copyright © John C. Hull 2005

4. 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

5. Conversion Formulas (Page 79)
Define
Rc : continuously compounded rate
Rm: same rate with compounding m times per year
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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
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7. Example (Table 4.2, page 81)
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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 6th Edition, Copyright © John C. Hull 2005

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 6th Edition, Copyright © John C. Hull 2005

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 6th Edition, Copyright © John C. Hull 2005

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
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

12. 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
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13. 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 6th Edition, Copyright © John C. Hull 2005

14. The Bootstrap Method continued
To calculate the 1.5 year rate we solve
This reduces to e-1.5R =
to get R = or %
Similarly the two-year rate is %
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15. Continuously compounded Zero rates
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16. 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

17. 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
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18. Forward Rates
The 3% for 1 year means that in return for an investment of $100 today, an investor receives 100e0.03*1=$ in 1 year
The 4% for 2 years means that in return for an investment of $100 today, an investor receives 100e0.04*2=$ 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

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
Example: T1=3, T2=4, R1=0.046, R2=0.05 and the formula gives RF=0.062
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20. 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

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 6th Edition, Copyright © John C. Hull 2005

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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:
( )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

28. 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

29. 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= and the Duration, D=2.653, so that
ΔB= *2.653*Δy, or
ΔB= *Δy
When the yield on the bond increases by 10 basis points(0.1%) it follows that Δy= The duration relationship predicts that ΔB= *0.001=-0.25, so that the bond price goes down to = 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

30. Duration Continued
When the yield y is expressed with compounding m times per year
The expression
is referred to as the “modified duration”
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31. 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
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32. 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