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Section 3.6 Perpetual Annuities

In general, an annuity is a regular set of payments over a set period of time. However, sometimes you might want to set up an investment which will return a set of regular payments forever. This is called a "perpetuity" or a "perpetual annuity". To determine the amount to invest to set up such an instrument, it is best to consider the present value at the beginning of time and require that the investment grow over only one time period to the amount which needs to be dispersed. That is, after one period the investment grows by the amount of the desired payment:

\begin{align*} A & = P + R = P(1+r)\\ R & = P(1+r) - P\\ & = Pr \end{align*}

If you want to consider money compounded more often than the payment period (say yearly payments but investments compounded m times per year

\begin{align*} A & = P + R = P(1+r)^m\\ R & = P(1+r)^m - P\\ & = P((1+r)^m - 1) \end{align*}

So, if you want to have a set of regular payments of \(\$ R\) each year, then set aside an investment of \(\displaystyle P = \frac{R}{(1+r)^m - 1}\)