Future Value of an Ordinary Annuity

The future value of an ordinary annuity is the amount that a series of equal payments, made at the end of each period for a set number of periods, will have grown to by the end of the last period, once compound interest has been earned on each payment. An ordinary annuity, also called an annuity in arrears, is the most common payment pattern in finance: loan repayments, bond coupons and regular savings deposits made at month end all follow it.

The calculation is one of the core tools of the time value of money. It answers questions such as how much a business will have in a fund if it sets aside $50,000 at the end of every year for ten years. Its close relative, used when payments are made at the start of each period, is the future value of an annuity due. The general concept of an annuity is covered in its own entry.

The future value of an ordinary annuity is:

• $FV$ is the future value at the end of the final period.
• $P$ is the payment made at the end of each period.
• $r$ is the interest rate per period, written as a decimal.
• $n$ is the number of payments, or periods.

The fraction is called the future value annuity factor. Older textbooks print tables of these factors, but spreadsheets now do the work with the FV function. The formula assumes the interest rate stays the same throughout and that every payment is the same size.

Suppose a company deposits $1,000 at the end of each year for five years in an account paying 5 percent a year, compounded annually. The factor is $(1.05^{5} - 1) \div 0.05$, which is 5.5256, so the future value is about $5,525.63. Of that amount, $5,000 is the deposits themselves and $525.63 is compound interest. The first deposit earns interest for four years, the second for three, and so on; the last deposit, made on the final day, earns nothing.

For monthly payments, the rate and number of periods must match the payment frequency. Someone saving $200 at the end of every month for ten years at 6 percent a year compounded monthly uses $r = 0.005$ and $n = 120$. The future value is about $32,776, of which $24,000 is contributions and about $8,776 is interest.

Accountants use the future value of an ordinary annuity in several situations:

• Planning a bond sinking fund, where a company makes regular deposits to accumulate enough cash to repay bonds at maturity; the formula can be rearranged to solve for the payment needed.
• Estimating the value of retirement contributions, such as monthly payments into a pension or retirement annuity.
• Setting aside funds for planned equipment replacement or other future obligations.
• Checking the build-up of balances in an amortization schedule or savings plan.

In financial reporting, present values are used far more often than future values, because standards measure assets and liabilities in today's money. The present value of an ordinary annuity, for example, is used to value bonds, leases and notes. The two are related: the future value equals the present value multiplied by $(1 + r)^{n}$.

The most frequent error is mixing periods, for example using an annual interest rate with monthly payments. Others include using the ordinary annuity formula when payments are actually made at the start of each period, which understates the result, and confusing the nominal annual rate with the effective annual rate. A simpler mistake is to forget that the formula gives the value at the moment of the last payment; if the money stays invested longer, it must be compounded forward further. Finally, since real returns vary, the result should be treated as an estimate when used for planning rather than a guaranteed outcome, and a sensitivity check at a few different rates is good practice.