Jun 12, 2010

MBA Finance Topics-PERPETUITIES

A perpetuity is a series of equal payments over an infinite time period into the future. Consider the case of a cash payment C made at the end of each year at interest rate i, as shown in the following time line:




Perpetuity Time Line

0

1

2

3


PV
C
C
C

Because this cash flow continues forever, the present value is given by an infinite series:

PV = C / ( 1 + i ) + C / ( 1 + i )2 + C / ( 1 + i )3 + . . .

From this infinite series, a usable present value formula can be derived by first dividing each side by ( 1 + i ).

PV / ( 1 + i ) = C / ( 1 + i )2 + C / ( 1 + i )3 + C / ( 1 + i )4 + . . .

In order to eliminate most of the terms in the series, subtract the second equation from the first equation:

PV - PV / ( 1 + i ) = C / ( 1 + i )

Solving for PV, the present value of a perpetuity is given by:
PV =
C
i

Growing Perpetuities

Sometimes the payments in a perpetuity are not constant but rather, increase at a certain growth rate g as depicted in the following time line:

Growing Perpetuity Time Line

0

1

2

3


PV
C
C(1+g)
C(1+g)2



The present value of a growing perpetuity can be written as the following infinite series:
PV =
C
( 1 + i )
+
C ( 1 + g )
( 1 + i )2
+
C ( 1 + g )2
( 1 + i )3
+ . . .


To simplify this expression, first multiply each side by (1 + g) / (1 + i):
PV ( 1 + g)
( 1 + i )
=
C ( 1 + g )
( 1 + i )2
+
C ( 1 + g )2
( 1 + i )3
+ . . .


Then subtract the second equation from the first:
PV -
PV ( 1 + g)
( 1 + i )
=
C
( 1 + i )
Finally, solving for PV yields the expression for the present value of a growing perpetuity:
PV =
C
i - g
For this expression to be valid, the growth rate must be less than the interest rate, that is, g < i .

Source: www.netmba.com

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