This preview shows pages 1–2. Sign up to view the full content.
AAS
S
CHOOL
OF
B
USINESS
U
NIVERSITY
OF
C
ALIFORNIA
AT
B
ERKELEY
UGBA 103 S
UMMER
2008
© A
VINASH
V
ERMA
T
EACHING
N
OTE
:
P
ERPETUITIES
, A
NNUITIES
,
AND
G
ROWING
A
NNUITIES
Please make sure you read Homework 2 at the end of this note.
A
perpetuity
is a stream of constant cash flows every period that
goes on for ever. Suppose you have $50,000 and you deposit it in a
bank account that pays interest at 6% a year for all time to come.
Then you, and, after you, your heirs, will get $3000 [$50,000 times
6%] every year in perpetuity. You have therefore exchanged
$50,000 now at
t=0
for $3000 every year in perpetuity staring at
t=
1. Since both you and the bank entered into the transaction
voluntarily, it must be the case that the value today of that
perpetual stream of payments is $50,000. Suppose we now want to
find out what amount of deposit would generate annual cash flows
of $12,000 in perpetuity. We shall arrive at the answer by
following in reverse the steps by which we arrived at the amount of
annual interest. That is, we shall divide the targeted cash flow of
$12,000 by the annual interest rate of 6%, and get the answer as
$200,000.
To recapitulate, when we want to find out the annual (per period)
interest payment, denoted
C
, we multiply the amount deposited at
t=
0 by the interest rate per year (period). Therefore, when we are
given the annual cash flow and the annual interest rate and want to
work out the amount of deposit that would generate those cash
flows, or, in other words,
the present value
of those cash flows,
we shall of course have to reverse the process. It thus follows that:
(
29
r
C
C
PV
=
perpetuity
in
period
every
$
Notice that we have used no algebra for arriving at this result.
1
The
important thing to remember about this very simple result is that
the point of valuation is one period before the point in time when
the first cash flow occurs.
The economic intuition is obvious: our
1
The result does have both an origin and a foundation in algebra:
(
29
(
29
(
29
(
29
∞
+
+
+
+
+
+
=
...
1
1
1
perpetuity
in
period
every
$
3
2
r
C
r
C
r
C
C
PV
is an
infinite
geometric progression
that
converges
for
0
r
. The formula for
sum of an infinite geometric progression is
Σ
k
a
*
f
k
=
a
/(1 –
f
) where
a
is the
first term of the series,
f
(<1) is the factor by which terms in the series increase,
and
k
is an index that runs from 0 to ∞. In our case, the first term is
a
= C/(1+r),
and the factor by which the terms increase is
f
= 1/(1+r). Substituting these in
the formula, we get the same result.
Page 1 of 6
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 02/12/2012 for the course UGBA 101A taught by Professor Mccullough during the Spring '08 term at University of California, Berkeley.
 Spring '08
 MCCULLOUGH

Click to edit the document details