An annuity due is a stream of fixed periodic payments in which each payment falls at the beginning of its period, so the first one is made or received immediately rather than one period from now. An annuity is a stream of fixed periodic payments to be paid or received in the future (OpenStax, Principles of Finance, 8.2 Annuities), and due describes only when inside each period the payment lands.
Nothing else about the stream changes. The amount is the same, the number of payments is the same, the rate is the same, and the whole difference is that every payment arrives one period sooner.
The Only Difference Is One Period
The distinction is defined against the default case. 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).
The variation is stated just as plainly. The difference between the two is one period, meaning one additional period of interest. An ordinary annuity assumes a one-period lag between the start of a stream of payments and the actual first payment, while an annuity due assumes that payments begin immediately, and whether the calculation is present value or future value the one-period lag is eliminated (OpenStax, Principles of Finance, 8.2 Annuities).
The Adjustment Is a Single Multiplication
Because the gap is exactly one period, the arithmetic is the ordinary annuity arithmetic with one more period of compounding applied to it. Faced with an annuity due problem, solve as if it were an ordinary annuity and then multiply by one plus the rate, one more time (OpenStax, Principles of Finance, 8.2 Annuities).
The ordinary annuity present value it starts from is the standard closed form, PVa = PYMT × [1 - 1 / (1 + r)^n] / r, where PVa is the present value of the annuity and PYMT is the amount of one payment (OpenStax, Principles of Finance, 8.2 Annuities).
And there is a direction rule that follows from it. Whether one is calculating present value or future value, the result of an annuity due must always be larger than that of an ordinary annuity, all other facts remaining constant (OpenStax, Principles of Finance, 8.2 Annuities).
That is a useful check on a spreadsheet. If a model produces an annuity due value below the ordinary one on identical inputs, the timing switch went the wrong way.
The Same Facts, Both Ways
The textbook runs one set of facts through both conventions. Five annual payments of $25,000, on a fund earning 8% annually, gives a present value of approximately $99,817.81 as an ordinary annuity, and approximately $107,803.24 when the first payment is withdrawn immediately, which makes it an annuity due (OpenStax, Principles of Finance, 8.2 Annuities).
Same payment, same rate, same count
Ordinary annuity, first payment in one year 99,817.81
Annuity due, first payment now 107,803.24The reason the fund has to start larger is worth holding onto. Pulling the first payment out immediately leaves less money behind to compound toward the remaining payments.
Where the Timing Actually Shows Up
Payment at the start of the period is not exotic. Rent taken before the month it covers, an insurance premium paid at the start of the cover period, and a subscription billed in advance are each a level stream whose payments fall at the beginning, which is what makes them annuities due rather than ordinary annuities. The test is where the payment sits inside the period it relates to, not what the payment is called.
The opposite convention has its own name in commercial practice, and the two senses that word carries, a timing convention and an overdue amount, are separated in payable in arrears.
A stream with no end date is a different instrument again, and the closed form that values it is in perpetuity.
Common Questions
A level stream of payments where each payment is made at the beginning of its period. The word due points at the timing inside the period and says nothing about who pays, what the money is for, or how long the stream runs.
One period of interest, and nothing else. The ordinary version lags the start of the stream by one period before the first payment, and the annuity due starts immediately (OpenStax, Principles of Finance, 8.2 Annuities). The ordinary annuity is the default that most formulas and most calculator functions assume.
Run the ordinary annuity calculation on the same payment, rate and number of periods, then multiply the answer by one plus the periodic rate (OpenStax, Principles of Finance, 8.2 Annuities). A spreadsheet’s present value function carries the same switch as an argument for whether the payment falls at the start or the end of the period, so the correction is a setting rather than a second formula.
No, and conflating the two is where most confusion on this subject comes from. Annuity due describes the timing of any level payment stream. An annuity contract sold by an insurer is a product with its own terms, charges and tax treatment, and calling one an annuity due says only when its payments fall.
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