## The System of Four Variables for Constant Annuities

As we can see from the annuity formulas we derived, there are four variables involved here: the present value (or equivalently, the future value), the constant annuity amount A, the number of periods n, and the discount rate r. Given any three of these we can calculate the fourth. For example, assume that I have $100,000 today (the present value) and that I invest earning 10% interest per year (the discount rate). I can calculate the maximum amount I can withdraw every year (the annuity amount) if I want to be able to make 10 withdrawals (the number of periods) starting today.

We can rearrange the present value formulas to calculate the annuity amount or the number of periods when the other variables are given. If the discount rate is the unknown, however, we have to calculate the answer by trial and error (iteration). In any case, you will almost always use Excel's built-in functions to do this type of calculations. You only need to recognize that these four variables are tied to one another and once three of them are specified, the other can be calculated.

As we saw, the present value and the future value are not independent of each other, that is, in any problem specifying one automatically specifies the other. We cannot specify both.

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