The formula for the future value of an annuity due

Future value is the value of a sum of cash to be paid on a specific date in the future. An annuity due is a series of payments made at the beginning of each period in the series. Therefore, the formula for the future value of an annuity due refers to the value on a specific future date of a series of periodic payments, where each payment is made at the beginning of a period. Such a stream of payments is a common characteristic of payments made to the beneficiary of a pension plan.

The formula for calculating the future value of an annuity due (where a series of equal payments are made at the beginning of each of multiple consecutive periods) is:

P = (PMT [((1 + r)n - 1) / r])(1 + r)


P = The future value of the annuity stream to be paid in the future
PMT = The amount of each annuity payment
r = The interest rate
n = The number of periods over which payments are to be made

This value is the amount that a stream of future payments will grow to, assuming that a certain amount of compounded interest earnings gradually accrue over the measurement period. The calculation is identical to the one used for the future value of an ordinary annuity, except that we add an extra period to account for payments being made at the beginning of each period, rather than the end.

For example, the treasurer of ABC Imports expects to invest $50,000 of the firm's funds in a long-term investment vehicle at the beginning of each year for the next five years. He expects that the company will earn 6% interest that will compound annually. The value that these payments should have at the end of the five-year period is calculated as:

P = ($50,000 [((1 + .06)5 - 1) / .06])(1 + .06)

P = $298,765.90

As another example, what if the interest on the investment compounded monthly instead of annually, and the amount invested were $4,000 at the end of each month? The calculation is:

P = ($4,000 [((1 + .005)60 - 1) / .06])(1 + .005)

P = $280,475.50

The .005 interest rate used in the last example is 1/12th of the full 6% annual interest rate.

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