Annuity Calculator

Present value = payment × [1 − (1 + i)⁻ⁿ] ÷ i Future value = payment × [(1 + i)ⁿ − 1] ÷ i (× (1 + i) for both, when the payments come at the start of the period — the ordinary annuity case pays at the end of the period and the annuity due case pays at the start) where i = annual rate ÷ payments per year, n = years × payments per year

Payment

The equal amount in each period — the payment per period, whether it is money coming in to you or going out from you

Annual rate

The nominal annual rate used for both directions at once: divided down to a per-period rate to discount the stream back, and compounded up to carry it forward — the default 7% a year becomes 0.5833% a month

Years

How long the payments run for, multiplied by the payments per year to give the number of periods

i

The per-period rate — the annual rate divided by the number of payments a year

n

The total number of periods — the years times the payments a year

Present value

What the whole stream is worth today, with every future payment discounted back one period at a time

Future value

What the same stream accumulates to by the end, with every payment compounded forward to the final date instead

Use it when you have a stream of equal payments and want both of its values in one view rather than one at a time — a pension being weighed against a lump sum, a lease or rental figure being judged against a purchase price, a savings plan where you want to see the money you put in next to what it is worth at either end. It is also the fastest way to see the compounding gap directly: the same 120,000 is 64,491 today and 260,463 at the end, and the difference between those two is not a fee or an error but twenty years of compound growth. It is the wrong page when you need to go the other way and solve for the payment itself from a lump sum, which is a different question with its own page, and it is the wrong page when the payments are not equal — a stream that steps up with inflation or varies with a market return is not an annuity, and neither formula applies.

1. The default case, and the cross-check against both single-sided pages

   1. Periodic rate: 7% ÷ 12 = 0.58333% a month. Number of periods: 20 × 12 = 240.
   2. Present value factor: [1 − 1.0058333⁻²⁴⁰] ÷ 0.0058333 = 128.98, so 500 × 128.98 = 64,491.25.
   3. Future value factor: [1.0058333²⁴⁰ − 1] ÷ 0.0058333 = 520.93, so 500 × 520.93 = 260,463.33.
   4. Payments: 500 × 240 = 120,000, which needs no rate at all.
   5. Check: 64,491.25 × 1.0058333²⁴⁰ = 64,491.25 × 4.0387 = 260,463.33.

   The two figures in the first two steps are the default results of the present value and future value pages respectively — three pages, built separately, agreeing to the cent on the same five inputs. The last step is the one this page exists for: the future value is the present value multiplied by (1 + i) once per period, which is 4.0387 over 240 months. Worth noticing that 4.0387 is not 20 × 7% = 1.40 and not 1.07²⁰ = 3.87 either: once the rate is converted monthly, the compounded factor is (1 + 0.0058333)²⁴⁰, and no shortcut with the annual figure reproduces it.

2. Payments at the beginning of each period

   1. Both factors are the ones from the default case.
   2. Shift every payment one period earlier and the whole stream moves: multiply both results by 1.0058333.
   3. 64,491.25 × 1.0058333 = 64,867.45.
   4. 260,463.33 × 1.0058333 = 261,982.70.

   Both values rise by exactly the same 0.583% — one period's interest — which is the cleanest way to see that this page's two figures are two views of one object. If the timing adjustment moved them by different ratios, they would not be the same stream. Payments at the beginning of the period are worth more in both directions because every payment is one period closer to you, and the total, 120,000, does not move at all: timing changes what the stream is worth, never how many dollars it contains.

3. A zero rate, where all three figures converge

   1. With no rate there is nothing to discount and nothing to compound.
   2. Both closed forms are 0 ÷ 0 in this case, so both fall back to the number of periods: 240.
   3. 500 × 240 = 120,000 in all three places.

   All three figures collapsing onto the same number is the clearest statement of what the other two mean: the present value and the future value are both the total, adjusted for time, and with no rate there is no adjustment. This is also the case that breaks the naive implementation — written as a closed form, the factor is 0 ÷ 0 and the page would print NaN — which is why the shared kernel handles a zero rate as its own branch rather than as a small number.

4. Paid once a year instead of monthly, at the same total

   1. Periodic rate: 7% a year, used as-is. Number of periods: 20.
   2. Present value factor: [1 − 1.07⁻²⁰] ÷ 0.07 = 10.59, so 500 × 10.59 = 5,297.01.
   3. Future value factor: [1.07²⁰ − 1] ÷ 0.07 = 40.995, so 500 × 40.995 = 20,497.75.
   4. Payments: 500 × 20 = 10,000.

   The ratio here is 3.8697, against 4.0387 for monthly — the same nominal 7% compounds to less when it is applied once a year instead of twelve times. That is the frequency convention showing up in the answer rather than in the rate, and it matters more than it looks: the annual figures differ from the monthly ones by more than a fifth, and the only thing that changed is how often the interest is credited. The same comparison at 0% would show no difference at all, which is a useful reminder that this is a compounding effect and not a fee.

5. A large payment and a long term, where compounding dominates

   1. Periodic rate 0.58333% a month, 600 periods.
   2. Present value: 100,000 × 166.20 = 16,619,896.78.
   3. Future value: 100,000 × 5,448.07 = 544,807,091.51.
   4. Payments: 100,000 × 600 = 60,000,000.
   5. Ratio: 544,807,091.51 ÷ 16,619,896.78 = 32.78, which is 1.0058333⁶⁰⁰.

   Sixty million paid in becomes five hundred and forty-four million at the end, over a horizon where the ratio reaches 32.78. The present value column tells the other half of the story: at 16.6 million, the promise is worth barely more than a quarter of the money that will eventually change hands, because the payments near the end of a fifty-year schedule are discounted almost to nothing. Both figures are correct, and the reason a single page showing both is more useful than either alone is that the gap between them is the entire subject.

The payments are equal and the rate is fixed for the whole term. Neither formula applies to a stream that steps up with inflation, varies with a market return, or stops early; those need a year-by-year schedule rather than a closed form. The rate is nominal and compounded at the payment frequency. Changing only the frequency, with the rate left alone, changes the answer by more than a fifth on the default inputs — the annual case gives 20,497.75 where the monthly one gives 260,463.33. A rate quoted with a different compounding convention is a different rate, and this page will not convert it for you. No fees, taxes, or inflation are modelled. The payment stream itself is exact: 120,000 is 120,000. What it is worth is a nominal figure, so the 64,491 today is 64,491 in today's money only in the sense that it is not adjusted for what money will buy later. There is no mode selector, so this page always takes the payment as an input and reports both values as results. Solving for the payment from a lump sum is a different question and lives on its own page rather than as a setting here, because a mode that silently ignores one of the input boxes is worse than an extra visit. A rate at or below −100% is rejected; the range runs down to −99.9%, where the present value of the same stream becomes astronomically large. That is the mathematics of discounting at a negative rate rather than a bug, but it is not a realistic input and the present value column is best read as a warning sign when it explodes. The two values are not a comparison against each other in any evaluative sense. A present value of 64,491 and a future value of 260,463 for the same stream are not two offers to choose between; they are the same offer measured at two different dates.

What does this annuity calculator show that the single-sided ones do not?

Both answers at once. The present value of an annuity and the future value of an annuity are the same stream measured at two different dates, and each has its own page that reports one of them. This page takes the same five inputs and reports both, along with the total of the payments, which is the figure that does not depend on the rate at all. Seeing 120,000, 64,491.25 and 260,463.33 in one view is what makes the relationship between them concrete rather than something to be taken on trust.

Why is the future value so much larger than the present value?

Because they are the same money at two different dates, and time is the whole difference. Over 20 years at 7% compounded monthly, one dollar grows to 4.0387 dollars, so the amount that is worth 64,491 today is worth 260,463 at the end. Neither number is the fee, the profit, or the error — they are two valid prices for one promise, and quoting either without the date it applies to is meaningless.

Can I solve for the payment instead, if I know the lump sum?

Not on this page, and deliberately so: the payment is an input here rather than an unknown. A page that let you switch which field is being solved would have to show all the inputs at once, and whichever one was currently being computed would sit there as a box that ignores what you type into it. The payment from a lump sum has its own page, and it goes further than a mode switch would — it also reports the payout rate and how much of the total comes from growth.

Does it matter whether payments arrive at the start or the end of the period?

Both values change by exactly one period's interest, 0.583% on the default inputs, and they change by the same ratio in both directions. That is the sign that the two figures are views of one object: an adjustment that moved them by different amounts would mean they were describing different streams. The total paid, 120,000, does not move at all — timing changes what the stream is worth, not how many dollars it contains.

Why does changing the payment frequency change the answer so much?

Because the rate is nominal and gets compounded at whatever frequency you choose. A 7% annual rate applied once a year compounds to less over twenty years than the same 7% applied twelve times a year, and the difference is large — 20,497.75 against 260,463.33 on 500 a month, though the two differ in total paid as well. The honest way to read any quoted annual rate is to ask what it is compounded at, because a nominal rate without its frequency is only half a rate.

What happens at a zero rate?

All three figures become the same number — on the default inputs, 120,000. With nothing to discount and nothing to compound, the present value and the future value are both just the sum of the payments. It is worth knowing that this case is also where a closed-form implementation breaks: the factor is 0 ÷ 0 when the rate is zero, so a page that did not special-case it would print NaN. The kernel behind this page handles it as its own branch.

• 26 CFR 20.2031-7(d)(2)(iv)(A) and (B) — the United States regulation defining the annuity factor: subtract the term-certain remainder factor from 1.000000 and divide by the interest rate, with a further adjustment factor when payments fall at the end of monthly or quarterly periods — Internal Revenue Service regulation, via the Electronic Code of Federal Regulations (United States, current as of 2026)
• Annuities — a single payment or a series of premiums buying periodic payments, and how the timing of those payments is set out in the contract — FINRA (Financial Industry Regulatory Authority), United States
• Planning for retirement — comparing a lump sum with a stream of income, and why the two cannot be compared without picking a rate to discount at — Consumer Financial Protection Bureau, United States