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Lecture Notes 4
Net Present Value
The single period case
• Recall from the previous set of notes that, when the interest rate per period is
r
, the net present
value of an investment that has a negative cash flow of –
C
0
now and a positive cash flow one
period in the future of
C
1
is
.
(1)
• The amount
C
1
/(1+
r
) is the
present value
of the cash flow that will be earned next period.
• Conversely, we could calculate everything in terms of next period’s values. The
future value
of
the cash investment this period will be –
C
0
(1+
r
).
Multiple periods
• When the interest earned on an investment is reinvested, the interest
compounds
. An investment
of
C
0
becomes
C
0
(1+
r
) after the first year. If this entire sum is reinvested, the value at the end of
two years will be
C
0
(1+
r
)
2
. More, generally the future value of the investment after
T
years of
earning
compound interest
will be
(2)
• Conversely, the present value of a cash flow
C
that is to be received
T
periods into the future is
(3)
The
PV
represents the amount of money that would have to be invested now at the compound
interest rate of
r
per period in order to obtain, in
T
periods, the same cash flow of
C
. The term
multiplying
C
in (3),
NPV
C
0
–
C
1
1
r
+

+
=
FV
C
0
1
r
+
()
T
=
PV
C
1
r
+
T

=
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(4)
is known as the
present value factor
.
• A
stream
of cash flows (–
C
0
,
C
1
,
C
2
, …,
C
T
), representing an investment opportunity that will
cost money immediately but will yield positive amounts in each of
T
future periods, will have a
net present value
(5)
• Often one sees investments where there is a
stated annual interest rate
but compounding occurs
more frequently than once a year. For example, most mortgages have interest paid at monthly in
tervals but the return is given as a stated annual interest rate. The future value of an investment
C
0
with a stated annual interest rate of
r
but compounding for
m
periods within the year will be
(6)
The
effective annual interest rate
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 Spring '06
 Bejan

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