An ordinary annuity is a stream of fixed periodic payments in which each payment falls at the end of its period, so the first payment arrives one full period after the stream begins. Where the first payment will be received one year from now, that stream is specifically called an ordinary annuity (OpenStax, Principles of Finance, 8.2 Annuities).
It is the default convention, which is why it usually goes unnamed. A formula or a calculator function that says nothing about timing is running this one, and the variation that moves every payment one period earlier is set out in annuity due.
The Two Ends That Earn Nothing
The end-of-period convention costs a compounding period twice over, and the textbook is blunt about it on a five-year investing plan. The payment for year 1 is not invested until the last day of that year, so year 1 is wasted as a compounding opportunity and the amount only compounds for four years rather than five, and the fifth payment is not made until the last day of the contract in year 5, so it has no chance to earn a compounded future value (OpenStax, Principles of Finance, 8.2 Annuities).
The same source states the conclusion plainly, that the investor has lost on both ends (OpenStax, Principles of Finance, 8.2 Annuities).
That is the whole reason the timing switch matters. Nothing about the payments changes, and the outcome does.
The Future Value Formula
Because every payment is identical, the stream collapses into one step rather than a row per period. The future value of an annuity is FVa = PYMT × [(1 + r)^N - 1] / r, where FVa is the future value of the annuity, PYMT is a one-time payment or receipt in the series, r is the interest rate and n is the number of periods (OpenStax, Principles of Finance, 8.2 Annuities).
The textbook runs it on one set of facts. Investing $3,000 at the end of each year for five years at 4% per year produces a future value of $16,248.98 (OpenStax, Principles of Finance, 8.2 Annuities).
Five payments of 3,000, at the end of each year, earning 4%
Total paid in 15,000.00
Future value of the stream 16,248.98The gap between those two lines is all the interest the arrangement earned across five years, on a stream in which two of the five payments never compound for a full period.
The present value side of the same stream, and the closed form that discounts it back, is worked through in annuity due, which starts from the ordinary annuity result and adjusts it.
Where the Convention Is Already Assumed
A loan repaid in level instalments after each period of use, and a bond paying its coupon at the end of each interest period, are ordinary annuities without anybody labelling them. That is why the convention is usually assumed rather than stated, and why a schedule that departs from it has to say so.
Commercial practice has its own word for that timing, and the two senses it carries, a timing convention and an amount that is overdue, are separated in payable in arrears.
A stream with no end date is not an annuity at all in this sense, and the closed form that values it is in perpetuity. Where the flows are uneven rather than level, no annuity formula applies and each period is discounted on its own, which is the summation in net present value.
Common Questions
For a future value, multiply the payment by the accumulation factor [(1 + r)^N - 1] / r (OpenStax, Principles of Finance, 8.2 Annuities). For a present value, discount the same level payment over the same periods. Whichever direction, the rate and the payment interval have to describe the same length of period, and mixing an annual rate with monthly payments gives an answer wrong by a multiple rather than by rounding.
One period, and the cost of it falls on the compounding rather than on the payments, which is what the two wasted ends above are. The adjustment between the two conventions, and the rule that the annuity due value is always the larger of the pair on identical facts, are both in annuity due.
Where the instalments are level and each one falls at the end of its period, yes. Where a payment is taken in advance of the period it covers, it is an annuity due instead, so the schedule decides it and the label does not.
No. Ordinary annuity describes the timing of any level payment stream. An annuity contract bought from an insurer is a product with its own terms, charges and tax treatment, and describing its payments as an ordinary annuity says only when in each period they fall.
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