What is the formula for the present value of an annuity due?

The present value of an annuity due is used to derive the current value of a series of cash payments that are expected to be made on predetermined future dates and in predetermined amounts. The calculation is usually made to decide if you should take a lump sum payment now, or to instead receive a series of cash payments in the future (as may be offered if you win a lottery).

The present value calculation is made with a discount rate, which roughly equates to the current rate of return on an investment. The higher the discount rate, the lower the present value of an annuity will be. Conversely, a low discount rate equates to a higher present value for an annuity.

The formula for calculating the present value of an annuity due (where payments occur at the beginning of a period) is:

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

Where:

P = The present 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 made

This is the same formula as for the present value of an ordinary annuity (where payments occur at the end of a period), except that the far right side of the formula adds an extra payment; this accounts for the fact that each payment essentially occurs one period sooner than under the ordinary annuity model.

For example, ABC International is paying a third party $100,000 at the beginning of each year for the next eight years in exchange for the rights to a key patent.  What would it cost ABC if it were to pay the entire amount immediately, assuming an interest rate of 5%? The calculation is:

P = ($100,000 [(1 - (1 / (1 + .05)⁸)) / .05]) x (1+.05)

P = $678,637

The factor used for the present value of an annuity due can be derived from a standard table of present value factors that lays out the applicable factors in a matrix by time period and interest rate. For a greater level of precision, you can use the preceding formula within an electronic spreadsheet.

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