A perpetuity is similar to an annuity, but it involves a never ending series of equal payments, effectively going on until infinity. As a result, the present value of a perpetuity is represented as an infinite time series:
PV = C / ( 1 + i ) + C / ( 1 + i ) ^ 2 + C / ( 1 + i ) ^ 3 + C / ( 1 + i ) ^ 4 + ...
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Whilst it may seem that this would give an infinite PV, as the terms continually reduce in size they will ultimately converge on a finite value. As with the annuity, this value can be found by dividing by (1 + i):
PV / ( 1 + i ) = C / ( 1 + i ) ^ 2 + C / ( 1 + i ) ^ 3 + C / ( 1 + i ) ^ 4 + ...
Subtracting the second equation from the first will eliminate all except the first term of the equation to give:
PV - PV / ( 1 + i ) = C / ( 1 + i )
Multiplying by 1 + i and simplifying give:
i * PV = C
Hence PV = C / i
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