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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