2. 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. 4.3
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. 4.4
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. 4.5
Conversion Formulas
Define
Rc : continuously compounded rate
Rm: same rate with compounding m times
per year
R m
R
m
R m e
c
m
m
R m
c
ln
/
1
1
6. 4.6
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. 4.7
Example: Use zero (or spot) rates
to price a bond
Maturity
(years)
Zero Rate
(% cont. comp.)
0.5 5.0
1.0 5.8
1.5 6.4
2.0 6.8
8. 4.8
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
3 3 3
103 9839
0 05 0 5 0 058 1 0 0 064 1 5
0 068 2 0
e e e
e
. . . . . .
. .
.
9. 4.9
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 = 0.0676 or 6.76%.
3 3 3 103 9839
0 5 1 0 1 5 2 0
e e e e
y y y y
. . . .
.
10. 4.10
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
g)
compoundin
s.a.
(with
get
to 87
6
100
2
100
2
2
2
0
.
2
068
.
0
5
.
1
064
.
0
0
.
1
058
.
0
5
.
0
05
.
0
.
c=
e
c
e
c
e
c
e
c
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. 4.12
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
c
P m
A
( )
100 100
13. 4.13
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 0 97.5
100 0.50 0 94.9
100 1.00 0 90.0
100 1.50 8 96.0
100 2.00 12 101.6
14. 4.14
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 10.127% with continuous compounding:
97.5 e^(.25R)= 100 implies R = 10.127%
Similarly the 6 month and 1 year rates are
10.469% and 10.536% with continuous
compounding
15. 4.15
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%
96
104
4
4 5
.
1
0
.
1
10536
.
0
5
.
0
10469
.
0
R
e
e
e
16. 4.16
Zero Curve Calculated from the
Data
9
10
11
12
0 0.5 1 1.5 2 2.5
Zero
Rate (%)
Maturity (yrs)
10.127
10.469 10.536
10.681 10.808
17. 4.17
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. 4.18
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. 4.19
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
1
2
1
1
2
2
T
T
T
R
T
R
F
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. 4.21
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. 4.22
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. 4.23
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. 4.25
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. 4.26
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 = - 0.125M
This is equivalent to a payoff of –$123,609 at the
3-year point: -0.125M{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: 4.10. 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. 4.28
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
4.29
Maturity (yrs) Deposit Rate Mortgage
Rate
1 3% 6%
5 3% 6%
