Discounting Future Cash Flows
Daniel K. Saunders
1
Geometric Series
1.1
Perpetuity (Infinitely Many Terms)
For a
geometric series
, let 0
< x <
1. Then the following is true
∞
X
n
=0
x
n
=
1
1

x
With this fact we can show that
.
9999
...
= 1
.
First, notice that
0
.
9999
....
=
9
10
+
9
100
+
9
1000
+
...
=
9
10
1 +
1
10
+
1
100
+
...
Then the series inside of the paranthesis is a
geometric series
, and we know that
∞
X
n
=0
1
10
n
=
1
1

1
10
=
1
9
10
=
10
9
Therefore, we are left with
.
9999
...
=
9
10
×
10
9
= 1
1.2
Annuity (Finitely Many Terms)
With finitely many terms (0
< x <
1) the adjusted result of a
geometric series
is as follows
N

1
X
n
=0
x
n
=
1

x
N
1

x
Notice that this result is consistent with the infinite case, since
lim
N
→∞
N

1
X
n
=0
x
n
= lim
N
→∞
1

x
N
1

x
=
1
1

x
for 0
< x <
1
1
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2
Application to Finance
2.1
A Technical Note
For this course, we consider the current period (
t
= 0) as a separate issue from discounting
future values. For example, if you look at the first set of lecture notes regarding the present
value of an
annuity
, you will notice that the index,
n
, begins with
n
= 1.
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 Fall '09
 m.
 Geometric Series

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