This preview shows pages 1–3. Sign up to view the full content.
S
CHOOL
OF
B
USINESS
A
DMINISTRATION
U
NIVERSITY
OF
C
ALIFORNIA
AT
R
IVERSIDE
B
US
106 S
PRING
2011
A
VINASH
V
ERMA
T
EACHING
N
OTE
:
P
ERPETUITIES
, A
NNUITIES
,
AND
G
ROWING
A
NNUITIES
Please make sure you read Homework 3 at the end of this note. The
homework is due on April 25, 2011.
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
Teaching Note and Homework 3
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document S
CHOOL
OF
B
USINESS
A
DMINISTRATION
U
NIVERSITY
OF
C
ALIFORNIA
AT
R
IVERSIDE
B
US
106 S
PRING
2011
A
VINASH
V
ERMA
deposit has to have been in the bank for one period before any
interest on it can be accrued and paid.
Now, let us define an
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 04/23/2011 for the course BUSINESS 106 taught by Professor Verma during the Spring '11 term at UC Riverside.
 Spring '11
 verma
 Business, Management

Click to edit the document details