Chapter 5 NEW Slides Interest Rate Markets2

Derivative Securities
Chapter 5
Interest Rate Markets
FIN 480 ( Instructor- SfR) Chapter 5 2
Types of Rates
Treasury rates
LIBOR
Repo rates
Treasury rates
Rates on Treasury securities
Most recently auctioned issues of a given maturity called
On the run issues exist for following maturities:
1m, 3m, 6m, 12 m, 2 yr, 5yr, 10yr
Those with maturities 3 yrs, 7 yrs, 15 yrs, 20 yrs and
30 yrs have been discontinued
Par values are used to figure out the underlying yields
LIBOR- London Interbank offered rate
Determined based on the quotations of 16 major banks; Rate at
which banks and FIs transact in the London Interbank market
The lender bank invests cash in a CD issued by borrower bank.
Maturity of the CD short tem i.e. overnight to 1 year
Credit rating of the borrower is AA ( LIBOR Is not completely a risk
free rate)
Currencies supported: Pound, Euro, US $, CAD, AUD, Yen, SW
Francs
Called Euro currency market; outside the control of a single
government
Borrowing USD in LIBOR market would be a Eurodollar loan
LIBOR vs. LIBID
LIBOR:
London Interbank offered rate
rate at which a bank makes deposits
LIBID:
London Interbank bid rate
rate at which a bank accepts deposits
LIBOR>LIBID
Offer rate>bid rate
Repurchase agreement (Repo) rates
Borrower deposits securities with custodian and
borrows money from a Lender
At maturity the buyer buys back the securities at a
pre-agreed price (includes a premium)
The implied rate is the repo rate
Essentially a Forward contract
Maturity:
Overnight repo and term repo
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
Spot rates Example (Table 4.2)
Maturity
(years)
Zero Rate
(% cont. comp.)
0.5 5.0
1.0 5.8
1.5 6.4
2.0 6.8

0
1
2
3
4
5
6
7
8
0.5 1 1.5 2
spot rates
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
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

+ +
+ =
. . . . . .
. .
.
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 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
+ + + =
. . . .
.
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 87 6 get to
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
=
|
.
|
\
|
+ +
+ +

Bootstrapping to get spot curve
Sample Data (Table 4.3)
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
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 10.256%
with quarterly compounding
This is 10.127% with CC
Similarly the 6 month and 1 year rates are
10.469% and 10.536% with CC
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
Zero Curve Calculated from the
Data (Figure 4.1)
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
Forward Rates: The forward rate is the
future zero rate implied by todays term
structure of interest rates
Suppose that the zero rates for time periods T
1
and T
2
are R
1
and R
2
with both rates CC and let
R
c
be the CC forward rate for the period
between times T
1
and T
2
is
( )
1 2
1 1 2 2
1 2 2 2 2 2
: for solve
) (
1 2 1 1 2 2
T T
T R T R
R
T T R T R T R
e e e
F
F
T T R T R T R
F
+ =
=

Calculation of Forward Rates
Table 4.5
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
Theories of the 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 because of
expected risk premium for longer term rates
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
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
T
1 T
2
Receive R
k
: FRA rate
FRA contract done at time t
To receive R
k
: FRA rate for a
loan amount $L
at time T
1
for a maturity of
T2-T1
Forward Rate Agreement
continued
An FRA is equivalent to an
agreement where interest at a
predetermined rate, R
K
is exchanged
at time T
1
for interest at the time T
1
market rate
An FRA can be valued by assuming
that the market rate at time T
1
= forward
interest rate today
Valuation FRA
Assume that forward rates are realized
Calculate the terminal payoffs
Discount using current spot rate
T
1 T
2
Receive R
k
: FRA rate
Receive R
k
: FRA rate
Pay R
F
: LIBOR rate for T
1
T
2
period
( )( )
2 2
1 2
rate LIBOR - rate FRA $L
T R
e T T


Valuation FRA: example
Assume that forward rates are realized
Calculate the terminal payoffs
Discount using current spot rate
T1: 3 yrs
T2: 3 yrs 3 mts
Receive R
k
: FRA rate
Receive 4: FRA rate
Pay 3: LIBOR rate for T
1
T
2
period
L= $100 mi
( )( )
2 2
12
3
3% - % 4 $100mi
T R
e

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)
This leads to
(
B
e c
t
i
yt
i
n
i
i
1
y D
B
B
A =
A
Duration
Duration Continued
When the yield y is expressed with
compounding m times per year
The expression
is referred to as the modified duration
m y
y BD
B
+
A
= A
1
D
y m 1+
Duration Matching
This involves hedging against interest
rate risk by matching the durations of
assets and liabilities
It provides protection against small
parallel shifts in the zero curve
Duration-Based Hedge Ratio
F C
P
D F
PD
F
C
Contract Price for Interest Rate Futures
D
F
Duration of Asset Underlying Futures at
Maturity
P Value of portfolio being Hedged
D
P
Duration of Portfolio at Hedge Maturity
Example (page 144-145)
Three month hedge is required for a $10 million
portfolio. Duration of the portfolio in 3 months will
be 6.8 years.
3-month T-bond futures price is 93-02 so that
contract price is $93,062.50
Duration of cheapest to deliver bond in 3 months is
9.2 years
Number of contracts for a 3-month hedge is
42 . 79
2 . 9 50 . 062 , 93
8 . 6 000 , 000 , 10
=

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Derivative Securities

FIN 480 ( Instructor- SfR) Chapter 5 23

FIN 480 ( Instructor- SfR) Chapter 5 24

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