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Inv_Lecture_4.pdf
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Inv Lecture 4.pdfInvestments Prof. Andrea Burasch...
Inv_Lecture_4.pdfInvestments Prof. Andrea Buraschi Lecture 4
Inv Lecture 4.pdfInvestments Prof....
Inv_Lecture_4.pdfInvestments 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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