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 .