Finance > Perpetuities

### 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 .

Finance > Perpetuities

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