Finance

Perpetuities

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 + ...

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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