Present Value of Annuity Calculator

Present value = payment × [1 − (1 + i)⁻ⁿ] ÷ i × (1 + i if payments are at the beginning of each period) (i = annual rate ÷ payments per year; n = years × payments per year)

Payment

The periodic payment — each equal amount in the stream, whether it is money you will receive or money you will pay out

Annual rate

The nominal annual rate used as the discount rate, divided by the payments per year before it is applied — 7 percent a year becomes 0.5833… percent a month on the default case

Years

How long the payments continue, multiplied by the payments per year to give the number of payments being discounted

i

The discount rate per period — what each payment is divided by, once for every period it is away from today

n

The number of payments, which is also the number of times the discount compounds

Use it when a stream of equal payments exists and you need to know what it is worth right now rather than at the end. The clearest cases are the ones where someone is offering to buy or sell such a stream: a pension buyout, a structured settlement, an annuity contract, a lease with level payments, or a bond's coupon stream. It is also the right page for the reverse of saving — if you are drawing a fixed amount out of a pot each month, this tells you how big the pot has to be today to sustain it, which is the same calculation with the payments running the other way. Two things to get right. The discount rate is the rate you could earn on the money instead, so it is a judgement and the answer moves with it — discounting at a higher rate makes any future stream worth less today. And the timing setting is not a detail: whether each payment arrives at the start or the end of its period is worth one whole period of interest on the entire stream, which on the default case is 376.20.

1. 500 a month for 20 years at 7%, paid at the end of each month

   1. The rate per period: 7% ÷ 12 = 0.5833…% a month
   2. The number of payments: 20 × 12 = 240
   3. The annuity factor: [1 − (1.005833…)⁻²⁴⁰] ÷ 0.005833… = 128.98250649…
   4. Present value: 500 × 128.98250649… = 64,491.25
   5. Total payments 500 × 240 = 120,000, so the discount is 120,000 − 64,491.25 = 55,508.75

   This is the mirror image of the future value page, which runs the identical five inputs and gets 260,463.33. The two answers are the same calculation in opposite directions, and they reconcile exactly: 64,491.2532… grown at 0.5833… percent a month for 240 months is 260,463.3299…. Worth knowing so you do not try to shortcut it: multiplying the printed factor 128.98251 by 500 gives 64,491.255, which rounds to 64,491.26 rather than the correct 64,491.25 — the five-decimal factor is for reading, not for multiplying.

2. The same stream as an annuity due — 64,867.45

   1. Everything is the same except that each 500 arrives at the start of the month instead of the end
   2. Every payment is therefore one period closer to today and is discounted one fewer time
   3. That is the ordinary-annuity answer multiplied by (1 + 0.005833…) : 64,491.25 × 1.005833… = 64,867.45
   4. The discount falls to 120,000 − 64,867.45 = 55,132.55

   376.20 more than the ordinary case for the same 120,000 of payments, which is exactly one month of interest on the whole stream. Getting paid earlier is worth this much, and it is worth more the higher the rate and the longer the term — which is why the setting is on the page rather than buried in the assumptions.

3. 500 a month for 20 years at −5% — the discount becomes a premium

   1. The rate per period: −5% ÷ 12 = −0.4166…% a month
   2. The annuity factor at a negative rate exceeds the number of periods: it comes to 413.752…
   3. Present value: 500 × 413.752… = 206,876
   4. The discount is 120,000 − 206,876 = −86,876, a negative number

   A negative rate means the same money is worth more later than now, so the stream is worth more than the payments add up to and the third cell goes negative. That is the arithmetic being consistent, not a broken output — the negative figure is the premium you would be paying today to receive 120,000 over twenty years in a world where money loses value. Negative real rates are rare but not unheard of, and this is what they do to a stream of payments.

The discount rate is a single constant for the whole term, and it is the input the answer is most sensitive to, which is a problem because it is also the one input nobody can measure in advance. It is a nominal rate with no inflation adjustment, so the answer is in the currency of the payments rather than in today's purchasing power — over twenty years that is a large difference. The payments are equal and evenly spaced, so a stream that grows with inflation, steps up on a schedule, or skips a period is outside this page; a growing stream needs a different formula rather than a different input. It assumes the stream is certain to be paid in full and on time. Real annuities and settlements carry the credit risk of whoever owes them, and a pension depends on a sponsor that may or may not still be there in twenty years — discounting a risky stream at a risk-free rate overstates what it is worth, which is exactly why a buyout offer usually prices it above the risk-free present value. It ignores tax, and the tax treatment of an annuity or a settlement can differ from that of the underlying investment. It ignores fees and any cost of administering the arrangement. Where the stream is a loan you are paying off rather than an income you are receiving, the same number is the amount you could borrow against it, not the amount you will actually pay. Finally, the negative rate case is arithmetically correct but unusual enough to be worth stating plainly: money losing value at a constant 5 percent a year for twenty years is not a scenario anyone should plan around, and the output is there to show what the formula does rather than to model a forecast.

What is the present value of an annuity?

It is what a stream of equal payments is worth today, after discounting each one back from the date it arrives. The payments are not worth their total, because money arriving in twenty years buys less than money arriving next month. On the default case 500 a month for twenty years adds up to 120,000 but is worth 64,491.25 today at 7 percent — so 55,508.75 of the face value is the discount, not a fee anyone charges, just the effect of time.

How do I calculate the present value of an annuity?

Divide the annual rate by the payments per year to get the rate per period, multiply the years by the payments per year to get the number of payments, then use payment × [1 − (1 + i)⁻ⁿ] ÷ i. On the default numbers i is 0.5833… percent and n is 240, so the factor is 128.98250649… and 500 × that is 64,491.25. If the payments come at the beginning of each period rather than the end, multiply by one more (1 + i) and you have the annuity due.

What is the difference between an ordinary annuity and an annuity due?

When each payment lands inside its period. An ordinary annuity pays at the end, an annuity due at the beginning, and a beginning payment is one period closer to today so it is discounted one fewer time. On the default stream that is the difference between 64,491.25 and 64,867.45 — 376.20, which is one month of interest on the whole stream. Rent and most loan payments are ordinary; a lease paid in advance or an insurance premium at the start of the year is an annuity due.

What rate should I discount at?

The rate you could earn on the money if you did not commit it, because that is what receiving the payments later costs you — the same discount rate used anywhere else, and sometimes called the hurdle rate. It is a judgement rather than a measurement, and the answer is highly sensitive to it: a higher rate makes any future stream worth less today. It should reflect the risk as well as the time, so discounting a stream that might not be paid at the rate on a government bond will overstate what it is worth.

Can the present value be higher than the total of the payments?

Yes, whenever the rate is negative. At −5 percent the same 500 a month for twenty years is worth 206,876 against 120,000 of payments, and the discount cell shows −86,876 — a premium rather than a discount. That is the formula working correctly on a rate where money is worth more later than now, not a broken output. Negative rates are rare, and a constant −5 percent for twenty years is not a forecast, but the case is left in because a formula that silently special-cased it would be hiding an assumption.

Does the payment frequency matter?

It changes two things at once. The annual rate is divided by the payments per year, so paying monthly discounts at a different per-period rate than paying annually does, and the number of payments being discounted changes with it. At 1,000 a year for ten years at 6 percent the present value is 7,360.09; the same annual total paid as 500 twice a year is worth slightly more, because each half arrives sooner and is discounted less. If you hold the payment size fixed instead of the annual total, the difference is much larger, because you are comparing different amounts of money.

Is this the same as the future value of an annuity?

It is the same calculation pointing the other way. Both use the identical annuity factor; this page multiplies it by the payment to get what the stream is worth today, and the future value page multiplies it by a growth factor to get what the account holds at the end. On the default inputs the two answers reconcile exactly: 64,491.2532 grown at 0.5833 percent a month for 240 months is 260,463.33. Use this page when you are valuing a stream that exists now, and the other when you are projecting one you will pay into.

• 26 CFR 20.2031-7(d)(6), Table B — the United States government's own published annuity factor table, the official present-value-of-an-annuity figures used for valuing annuities in estates — Internal Revenue Service regulation, via the Electronic Code of Federal Regulations (United States)
• Annuities — Investor.gov glossary (a series of payments made at fixed intervals, and the contract built on one), the definition this page discounts — U.S. Securities and Exchange Commission, Investor.gov (United States)
• Interest — Investor.gov glossary, for the rate each payment in the stream is discounted at — U.S. Securities and Exchange Commission, Investor.gov (United States)