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Interest Rates
Finance (Derivative Securities) 312
Tuesday, 8 August 2006
Readings: Chapter 4
Types of Rates
Treasury Rates
• Short-term government securities
LIBOR
• London Interbank Offer Rate
• Rate applicable to wholesale deposits
between banks
Repo Rates
• Repurchase agreements
Measuring Rates
Compounding frequency is unit of
measurement
Increased frequency leads to continuous
compounding
• $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 continuously compounded
discount rate is R
Conversion Formula
Rc : continuously compounded rate
Rm: same rate with compounding m times
per year
Rm 

Rc  m ln1 


m

Rm  m e
Rc / m

1
Bond Pricing
 Relies on interest
rates on zero-coupon
bonds (zero rates)
• Interest is realised
only at maturity date
Maturity
Zero Rate
(years) (% cont comp)
0.5
5.0
1.0
5.8
1.5
6.4
2.0
6.8
To calculate the price of a two year coupon bond
paying 6% semi-annually:
3e
0.05 0.5
 3e
 103e
0.0581.0
 0.068 2.0
 3e
 98.39
0.064 1.5
Bond Yield
Single interest rate that discounts
remaining CFs to equal the price
Using the previous example, solve the
following equation for y:
3e
 y  0.5
 3e
 y 1.0
 3e
 y 1.5
y = 0.0676 or 6.76%
 103e
 y  2.0
 98.39
Par Yield
 Coupon rate that equates a bond’s price to its
face value
 Using previous example:
c 0.050.5 c 0.0581.0 c 0.0641.5
e
 e
 e
2
2
2
c  0.0682.0

 100  e
 100
2

to get c=6.87 (w iths.a. compoundin g)
Par Yield
If: m = no. of coupon payments per year
P = present value of $1 received at
maturity
A = present value of an annuity of $1
on each coupon date
then:
(100  100 P )m
c
A
Calculating Zero Rates
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
Bootstrap Method
2.5 can be earned on 97.5 after three
months
3-month rate is 4 times 2.5/97.5 or
10.256% with quarterly compounding, and
10.127% with continuous compounding
Similarly the 6-month and 1-year rates are
10.469% and 10.536% with continuous
compounding
Bootstrap Method
To calculate 1.5-year rate, solve:
4e
0.104690.5
 4e
0.105361.0
 104e
 R1.5
 96
to get R = 0.10681 or 10.681%
Similarly the two-year rate is 10.808%
Zero Curve
12
Zero
Rate (%)
11
10.681
10.469
10
10.808
10.536
10.127
Maturity (yrs)
9
0
0.5
1
1.5
2
2.5
Forward Rates
Future zero rates implied by the current
term structure
Zero Rate for
Forward Rate
an n -year Investment for n th Year
Year (n )
(% per annum)
(% per annum)
1
2
3
4
5
10.0
10.5
10.8
11.0
11.1
11.0
11.4
11.6
11.5
Calculating 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:
R2 T2  R1 T1
T2  T1
Slope of 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
Forward Rate Agreements
Agreement that a fixed rate will apply to a
certain principal during a specified future
time period
Equivalent to agreement where interest at
a predetermined rate, RK , is exchanged for
interest at the market rate
Can be valued by assuming that the
forward interest rate will be realised
Forward Rate Agreements
Let:
• RK = interest rate agreed to in FRA
• RF = forward LIBOR rate for period T1 to T2
calculated today
• RM = actual LIBOR rate for period T1 to T2
observed at T1
• L = principal underlying the contract
Forward Rate Agreements
If X lends to Y under the FRA, then:
• Cashflow to X at T2 = L(RK – RM)(T2 – T1)
• Cashflow to Y at T2 = L(RM – RK)(T2 – T1)
Since FRAs are settled at T1, payoffs must
be discounted at [1 + RM (T2 – T1)]
Value of FRA is the payoff, based on
forward rates, discounted at R2T2
–R T
• ValueX = L(RK – RF)(T2 – T1)e 2 2
–R T
• ValueY = L(RF – RK)(T2 – T1)e 2 2
Theories of Term Structure
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