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
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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 = |
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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
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The present value of a growing perpetuity can be written as the following infinite series:
PV = |
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To simplify this expression, first multiply each side by (1 + g) / (1 + i):
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Then subtract the second equation from the first:
PV - |
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Finally, solving for PV yields the expression for the present value of a growing perpetuity:
PV = |
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For this expression to be valid, the growth rate must be less than the interest rate, that is, g < i .
Finance > Perpetuities
