Future Value of an Annuity Due

The future value of an annuity due is the amount a series of equal payments, each made at the beginning of a period, will be worth at the end of the final period after compound interest has been earned. Because every payment is made one period earlier than in an ordinary annuity, each one earns interest for one extra period. As a result, the future value of an annuity due is always greater than the future value of an ordinary annuity with the same payment, rate and number of periods, provided the interest rate is positive.

An annuity due is common wherever payments are made in advance. Rent is usually paid at the start of the month, insurance premiums are paid before coverage begins, and many savers make deposits at the start of each month or year. The calculation is part of the wider time value of money toolkit and is closely tied to the general idea of an annuity.

The future value of an annuity due is the ordinary annuity formula multiplied by one extra period of growth:

• $FV_{\text{due}}$ is the future value at the end of the last period.
• $P$ is the payment made at the start of each period.
• $r$ is the interest rate per period, as a decimal.
• $n$ is the number of payments.

The factor $(1 + r)$ is the only difference from the ordinary annuity formula. In a spreadsheet, the FV function handles an annuity due by setting its "type" argument to 1 instead of 0.

A small business owner deposits $1,000 at the start of each year for five years into an account paying 5 percent a year. The ordinary annuity factor is 5.5256, so the ordinary annuity value would be $5,525.63. Multiplying by 1.05 gives a future value for the annuity due of about $5,801.91. The extra $276.28 is the additional year of compound interest each deposit earns because it is made at the start of the year.

A second example shows the effect over a longer period. Depositing $500 at the start of each year for ten years at 4 percent grows to about $6,243.18, compared with about $6,003.05 if the same deposits were made at the end of each year. The gap widens as the rate or the number of periods rises.

Practical uses include:

• Projecting the value of savings or retirement plans where contributions are made at the start of each period, such as a retirement annuity funded in advance.
• Planning a fund to replace equipment where deposits are made on the first day of each year.
• Comparing payment options, since paying in advance has a cost to the payer equal to the interest that could have been earned.
• Understanding lease arrangements where payments are due at the start of each period; for valuing the lease liability, accountants use the present value of an annuity due, which is the future value discounted back.

The two patterns differ only in timing, but timing changes the numbers. An ordinary annuity's last payment earns no interest, while an annuity due's last payment earns one period's interest. For present values the same logic applies in reverse: the present value of an annuity due is higher than that of an ordinary annuity because each payment is received sooner and so is discounted less.

A common error is to apply the ordinary annuity formula to payments that are really made in advance, which understates the future value. Another is mixing a yearly interest rate with monthly payments without adjusting both $r$ and $n$. Accountants should also check whether interest is compounded at the same frequency as payments are made; if not, the effective rate per payment period must be calculated first, using the method shown in the entry on nominal interest rates. Where a precise figure matters, a schedule showing each payment and its interest is a simple way to check the formula's answer.