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 = 

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 = 

+ 

+ 

+ . . . 
To simplify this expression, first multiply each side by (1 + g) / (1 + i):

= 

+ 

+ . . . 
Then subtract the second equation from the first:
PV  

= 

Finally, solving for PV yields the expression for the present value of a growing perpetuity:
PV = 

For this expression to be valid, the growth rate must be less than the interest rate, that is, g < i .
Finance > Perpetuities