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Ordinary annuity vs annuity due

An ordinary annuity pays at the end of each period. An annuity due pays at the start. The due total is the ordinary total grown one extra period. $500 a month for 20 years at 6 percent compounded monthly is $231,020.45 ordinary and $232,175.55 due.

 Ordinary annuityAnnuity due
When the payment landsThe end of each period.The start of each period.
FormulaPMT times ((1+i)^n - 1) / i.The ordinary total, times (1+i) once.
Teaching sheet, 20 years$231,020.45 from $500 a month at 6 percent monthly.$232,175.55, which is 0.5 percent more.
Interest on that sheet$111,020.45. You paid in $120,000.$112,175.55. Same $120,000 paid in. The extra is all interest.
Where you meet itMonth-end deductions, most savings plans, the default in a textbook.First-of-the-month rent, insurance premiums, many pension contributions.
What the gap isThe base. The due version is this number grown one period.Exactly the period rate, for any horizon. At 0.5 percent a month it is always 0.5 percent.
On this page

One extra period, applied once

The two versions are the same stream of money. Every payment in an annuity due sits in the account one period longer, so the ordinary total is grown one more time:

FVdue=FVordinary×(1+i)FV_{\text{due}} = FV_{\text{ordinary}} \times (1 + i)

On the teaching sheet, i=0.06/12=0.005i = 0.06/12 = 0.005 and n=240n = 240. The ordinary factor is 462.040895, so $500 a month reaches $231,020.45. Multiply by 1.005 and the due total is $232,175.55.

The money paid in has not moved: 500×240500 \times 240 is $120,000 either way. Interest rises from $111,020.45 to $112,175.55. The gap is 232,175.55−231,020.45=1,155.10232{,}175.55 - 231{,}020.45 = 1{,}155.10, which is 0.5 percent of the ordinary total, one month of interest on the whole balance.

That share does not grow with the years. Thirty years of the same ordinary stream reaches $502,257.52. The due version of that run would still sit 0.5 percent above it, because the extra factor is still one month, not thirty years.

How annuity future value works is the long form, with the future value of an annuity calculator under the answer.

Read the payment date, not the sales copy

Which label applies is a fact about when the cash moves. A payroll deduction taken on the last working day is ordinary. A premium taken on the first of the month is due. Calling a product an annuity does not settle it: annuities explained is an insurance contract, and this table is a savings-stream identity.

Mixing the two is a quiet error. The streams look identical and differ by exactly one period of growth, so multiplying by (1+i)(1+i) when the payments land at the end, or forgetting it when they land at the start, moves the answer by the full period rate.

The annuity timing explorer draws both endings on one picture. This is educational material, not financial advice.

Worked examples

\$500 a month for 20 years at 6 percent

You pay $500 into an account at the end of every month for 20 years, and it earns 6 percent a year compounded monthly. What is it worth at the end?

  1. Find the period rate: i=0.06/12=0.005i = 0.06/12 = 0.005, so 0.5 percent a month.
  2. Count the payments: n=12×20=240n = 12 \times 20 = 240.
  3. Build the annuity factor: 1.005240−10.005=3.310204476−10.005=462.040895\frac{1.005^{240} - 1}{0.005} = \frac{3.310204476 - 1}{0.005} = 462.040895.
  4. Multiply by the payment: 500×462.040895=500 \times 462.040895 = $231,020.45.
  5. Add up what you paid in: 500×240=500 \times 240 = $120,000.
  6. Take one from the other: $231,020.45 minus $120,000 leaves $111,020.45.

The account reaches $231,020.45. You paid in $120,000, so $111,020.45 of it is interest, which is a shade under half the total.

The same payments made at the start of each month

Same $500, same 6 percent compounded monthly, same 20 years, but every payment lands on the first of the month instead of the last. That is an annuity due. What changes?

  1. The ordinary annuity answer is the starting point: $231,020.45.
  2. Every payment now sits in the account one extra month, so the total is grown one more time: 231,020.45×1.005231{,}020.45 \times 1.005.
  3. That comes to $232,175.55.
  4. What you paid in has not moved: 500×240=500 \times 240 = $120,000.
  5. So the interest is $232,175.55 minus $120,000, which is $112,175.55.

Paying at the start reaches $232,175.55 against $231,020.45 at the end, and the interest rises from $111,020.45 to $112,175.55. The gap is exactly 0.5 percent, one month of interest on the whole balance, because each of the 240 payments gets one extra month of growth.

The same \$500 a month, run for 30 years

Keep the payment, the rate and the end-of-month timing, but let the payments run for 30 years instead of 20.

  1. The period rate is still 0.005, but now n=12×30=360n = 12 \times 30 = 360.
  2. The annuity factor grows to 1.005360−10.005=6.022575212−10.005=1004.515042\frac{1.005^{360} - 1}{0.005} = \frac{6.022575212 - 1}{0.005} = 1004.515042.
  3. Multiply by the payment: 500×1004.515042=500 \times 1004.515042 = $502,257.52.
  4. What you paid in: 500×360=500 \times 360 = $180,000.
  5. Interest: $502,257.52 minus $180,000 leaves $322,257.52.

Thirty years reaches $502,257.52 with $322,257.52 of it interest. Paying in half as much again, $180,000 rather than $120,000, more than doubles the ending value, because the extra ten years also give every earlier payment ten more years to compound.

Common questions

Is an annuity due always larger?

At a positive rate, yes, by exactly one period of growth. At a zero rate the two totals match, because there is nothing to earn in the extra period. At a negative rate the due total would be smaller, which is not how a savings stream is usually written.

Does the gap grow with the number of years?

The dollar gap grows because it is a percentage of a larger total. The percentage gap does not: it stays equal to the period rate. Twenty years or thirty years, 0.5 percent a month is still 0.5 percent of the ordinary ending value.

Keep reading

This page is educational material, not financial advice. The figures come from the formula shown and assume the inputs you enter hold for the whole term. Your own rate, fees, taxes and timing will differ, so treat the output as arithmetic to check a decision against, not as a recommendation.