1. Interest Rates
Chapter 4

2. Types of Rates Treasury rates
LIBOR rates: London Interbank Offered Rate (Large banks willing to lend to other large banks)
Repo rates: Repurchase Agreement; sell with commitment to buy back (Selling company borrows, buying company lends)

3. Interest Rates: Compounding Period
Default assumption of textbook: interest rates quoted assuming continuous compounding
The difference between quarterly and annual compounding is analogous to the difference between miles and kilometers, i.e. can convert

4. 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

5. Conversion Formulas Define Rc : continuously compounded rate
Rm: same rate with compounding m times per year

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
Default assumption: interest rates quoted assuming continuous compounding. But FRA contracts are quoted assuming compounding period is length of FRA period.

7. Example: Use zero (or spot) rates to price a bond

8. Bond Pricing
To calculate the cash price of a bond discount each cash flow at the appropriate zero rate
The theoretical price of a two-year bond providing a 6% coupon semiannually (default assumption: semiannual payment of coupons) is

9. Bond Yield
The bond yield is the (single) 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 = or 6.76%.

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

11. When is par yield used?
Investment bankers set the coupon rate on a new bond issue to equal the par yield. Result: new bonds initially trade at par.
Invoked in interest rate swaps: the swap rate is the par yield. (To be discussed in the swaps chapter.)

12. Par Yield (bond cash flows split into annuity and lump sum components)
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

13. Sample Data Bootstrap Method Table 4.3, p. 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 8
96.0
100
2.00 12
101.6

14. The Bootstrap Method
First line: an amount 2.5 can be earned on 97.5 during 3 months, i.e. future value of 97.5 is 100
This is % with continuous compounding: 97.5 e^(.25R)= 100 implies R = %
Similarly the 6 month and 1 year rates are % and % with continuous compounding

15. The Bootstrap Method continued
To calculate the 1.5 year rate we solve
to get R = or %
Similarly the two-year rate is %

16. Zero Curve Calculated from the Data
Zero Rate (%)
10.808
10.681
10.469
10.536
10.127
Maturity (yrs)

17. Forward Rates (assumption: continuously compounded)
The forward rate is the future zero (or spot) rate implied by today’s zero or spot curve of interest rates.
The forward rate is inferred (from observed zero rates) now but pertains to a future time period.
Observed: R1 R2 ; Inferred: F
R2 T2 = R1 T1 + F(T2 – T1 )

18. 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) 1
3.0 2
4.0
5.0 3
4.6
5.8 4
5.0
6.2 5
5.3
6.5

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, F, for the period between times T1 and T2 is

20. Nexus: F, R1 , and R2
R2 is a weighted average of (“is between”) R1 and F
R2T2= R1T1 + F (T1 – T2 )
Yield is a weighted average of (“is between”) R1 and R2

21. Upward vs Downward Sloping Zero (or Spot) Curve
For upward sloping zero curve, R2> R1 :
Fwd Rate or F > R2 > Yield > R1
For downward sloping zero curve, R2< R1 :
Fwd Rate or F < R2 < Yield < R1

22. Forward Rate Agreement, FRA
FRA is an agreement that a certain rate will apply to a certain principal during a certain future time period
FRA (interest) rate or RK is quoted assuming a compounding period equal to the FRA period.
FRA buyer pays the FRA rate, hedges a liability
FRA seller receives the FRA rate, hedges a deposit

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, i.e. employ the prevailing forward rate to value a forward contract.

24. FRA at inception V of FRA = zero
Generic requirement: At inception the value of a forward contract is zero
Contractual rate specified RK = adjusted F
F is adjusted so that its compounding period equals that of the FRA period; must shift from continuous compounding to, say quarterly or semi-annual compounding

25. FRA seller: post-inception valuation
A company had agreed that it would receive 4% on $100 million for 3 months starting in 3 years
The forward rate for the period between 3 and 3.25 years is 3%, assuming quarterly compound.
3.25-year zero rate is 2% assuming CC
The value now of the contract to the company is +$234,267 or $250,000 discounted from time 3.25 years to now: 100M(4%-3%).25
e^[-2%(3.25)]

26. FRA seller continued: Settlement at start of forward period (always)
Suppose rate proves to be 4.5% (quarterly compounding)
The payoff is –$125,000 at the 3.25 year point; contract assumes value conveyance at end of forward period: 100M(4%-4.5%).25 = M
This is equivalent to a payoff of –$123,609 at the 3-year point: M{1/[1 +(4.5%/4)]}= -123,609
Settlement occurs at start of forward period: FRA seller pays $123,609 to FRA buyer

27. FRA problems (Test Bank)
Determine FRA contractual rate: 4.3; 4.6
Valuation post-inception: 4.8
Determine settlement amount: Note: Settlement is always assumed to occur at the start of the FRA period in this course. This is true of the overwhelming majority of cases in financial markets.

28. Theories of the Zero (Spot) Curve Page 87
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

29. 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?
Depositors prefer short-term; borrowers prefer long-term
How can the bank manage its risks? Raise 5-year deposit/mortgage rates to mitigate asset/liability mismatch
Maturity (yrs)
Deposit Rate
Mortgage Rate 1 3% 6% 5