Inv Lecture 4.pdf-Investments Prof. Andr...
Inv_Lecture_4.pdf-Investments Prof. Andrea Buraschi Lecture 4
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Inv Lecture 4.pdf-Investments Prof....
Inv_Lecture_4.pdf-Investments Prof. Andrea Buraschi Lecture 4
##### Page 10
What is a forward rate?
Simplest example: suppose you want to borrow or lend \$50 million in 6M for
a period of 6M. You can do this transaction
The “in 6M for 6M” forward rate from now is the rate at which you can borrow
or lend in 6
months for a period of 6
months.
It’s not an option. You are locked into borrowing or lending at the forward rate.
Denote the forward rate as
0.5
f
0.5
: 0.5 refers to a 6
M period.
More generally: the forward rate is the interest rate on a loan that begins in
m
years and lasts for n
years, the “in m
years for n
years forward rate”
Notation:
m
f
.n
is the rate for borrowing and lending that starts in m
years and
lasts for n
years (in
m
years
f
or
n
years)
Examples:
1
f
.5
: rate on a 6
month loan, 1 year from now
5
f
.5
: rate on a 6
month loan, 5 years from now
7
f
3
: rate on a 3
year loan, 7 years from now

##### Page 11
Goal: how to price a forward contract
“Price” a forward? find the forward rate.
How? No
arbitrage arguments. Take two alternative
investments:
1.
Invest \$1 at the 1
year spot rate today
2.
Invest \$1 at the 6
M spot rate. In 6 months, re
invest the proceeds at
the forward rate for the next 6 months.
Since the forward rate is contracted today, both strategies are
risk
less and have the same cost today. By the principle of
no
arbitrage
,” they must have the same value in 1 year! Solve
for the forward rate.

##### Page 12
Spot and forward rates
Scenario 1
: Invest \$1 in at 1
year spot
In 1 year, it is worth
Scenario 2
: ½ year spot rate and forward
Invest \$1 for 6M at 6M spot rate. In 6M it is worth:
At 6M, re
invest at
.5
f
.5
for 6M when it is worth (note compounding of
forward rates is the same as spot rates)
2
1
1
)
2
/
r
1
(
V
2
/
r
1
.5
r
1
r
.5
.5
f
.5
Scenario 1
Scenario 2
0
1
2
/2)
f
/2)(1
r
1
(
V
.5
.5
.5
2

##### Page 13
Both scenarios have the same investment and are both riskless.
The terminal values must be the same! Thus, by no arbitrage
Solve for
.5
f
.5
:
If not? Arbitrage
Note: forward rates are ‘derivatives’ on the spot rates
2
.5
.5
.5
2
1
1
V
/2)
f
/2)(1
r
(1
/2)
r
(1
V
No
Arbitrage
1
/2)
r
1
(
/2)
r
1
(
2
f
.5
2
1
.5
.5

##### Page 14
‘Arbitrages’
Forward rates generate additional restrictions between
Spot rates and forward rates
Bond prices and forward rates
On your homework: you will work out some ‘arbitrage’
examples.

##### Page 15
Example 1: Current rates
If the spot rates are 6
month: r
.5
=0.16% and 1
year: r
1
=0.33%.
What is the “in 6M for 6M” forward rate?
Solve the following equation:
To get
Interpretation? Why is it higher? What does it mean?
)
2
/
f
2)(1
/
0016
.
0
1
(
2)
/
0033
.
0
(1
.5
5
.
2
%
50
.
0
1
2
/
0016
.
0
1
/2)
0033
.
0
(1
2
f
2
.5
.5

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