PVIFA Calculator

PVIFA = [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)

i

The discount rate per period — the annual rate divided by the payments per year, which is the rate each payment is discounted at

n

The number of periods — years multiplied by the payments per year, and the exponent that makes distant payments count for less

PVIFA

The present value interest factor of an annuity: what one unit of currency paid every period for n periods is worth today, with no amount of money in it at all

PVIFA due

The same factor when each payment arrives at the beginning of its period instead of the end, which is the ordinary factor multiplied by one plus the periodic rate

Use it when you need the factor rather than an amount: checking a figure against a published table, working a problem where the payment is not given, preparing a spreadsheet where the payment will be a separate input, or comparing how the factor moves as the rate and the term change without the arithmetic being obscured by an amount of money. The factor is also the honest way to see the shape of discounting, because it isolates it: 6.71008 at 8 percent over ten years says that ten annual units are worth under seven today, and the same factor at 50 percent is 1.96532, which says that at that rate almost nothing beyond the first couple of years survives the discount. Two things are worth knowing before you use a published table. The tables are almost always built for payments at the end of each period, so if you are valuing an annuity due you need the second column rather than the first. And the periodic rate and the period count have to match the payment frequency: a ten-year monthly annuity has 120 periods at the monthly rate, not 10 periods at the annual one, and looking up the annual row would overstate the factor by a factor of about twelve.

1. 8% for 10 years, paid annually — the textbook cell

   1. The rate per period: 8% ÷ 1 = 8% a year
   2. The number of periods: 10 × 1 = 10
   3. PVIFA: [1 − 1.08⁻¹⁰] ÷ 0.08 = 6.7100814…, printed as 6.71008
   4. Annuity due: 6.7100814… × 1.08 = 7.2468879…, printed as 7.24689

   This is the default case and the reason the page defaults to an annual, ten-year setup rather than the monthly, twenty-year one the other annuity pages use. Look up 8 percent in any standard annuity table and the tenth column reads 6.71008, because this is not an approximation of that number, it is that number. The due column beside it is the same table's beginning-of-period entry, and the 8.0 percent gap between them is exactly one period of interest.

2. 7% over 20 years paid monthly — 128.98251

   1. The rate per period: 7% ÷ 12 = 0.5833…% a month
   2. The number of periods: 20 × 12 = 240
   3. PVIFA: [1 − (1.005833…)⁻²⁴⁰] ÷ 0.005833… = 128.98250649…, printed as 128.98251

   The same inputs the present value of an annuity page uses for its default, where a payment of 500 gives 64,491.25. Multiply 128.98250649… by 500 and you get exactly that, which is the whole relationship between the two pages: one prints the factor, the other multiplies it by an amount. Note that multiplying the printed 128.98251 by 500 gives 64,491.255 and rounds to 64,491.26, so the five-decimal factor is for reading rather than for chaining into more arithmetic.

3. 5% for a single year, paid annually — 0.95238, and the due factor is exactly 1

   1. One period at 5%: PVIFA = 1 ÷ 1.05 = 0.952380…, printed as 0.95238
   2. Annuity due: the single payment moves to today, where the discount factor is (1.05)⁰ = 1
   3. So the due factor is exactly 1, with no rounding in it at all

   The cleanest boundary on the page. One payment, made a year from now, is worth 0.95238 of its face value at 5 percent; move that same payment to today and the factor is 1, because nothing has had time to discount it. The gap between the two columns is at its largest here relative to the factor — about 5 percent — and it narrows as the term lengthens, which is the opposite of what most people expect.

The factor is only as useful as the rate and period count behind it, and both have to be per the payment frequency. A published table is almost always annual, so using one for a monthly annuity requires converting the rate and multiplying the periods, and getting that wrong is the commonest way to be off by an order of magnitude. The factor assumes the same amount is paid every period for a known number of periods, so a stream that grows, steps up, or continues for an uncertain length is outside it. It assumes the discount rate is constant throughout, which no real rate is. And it is a present value, not a price: it contains no allowance for credit risk, tax, fees, or the possibility that the payments stop. Where an official table is being checked, note that the government's own figures are assembled differently. Under 26 CFR 20.2031-7, Table B gives a once-a-year annuity factor — 4.6325 at 2.6 percent for a five-year term — and Table K then multiplies that by a frequency adjustment for payments made at the end of each interval, 1.0097 for quarterly payments at 2.6 percent, so the regulation's own worked example gets 46,774.35 from a 10,000-a-year annuity paid quarterly. This page reaches the same present value a different way, compounding the nominal rate directly, and gets 46,744.27 — about 30 less, or 0.06 percent. Neither is wrong; they are two conventions, and a figure that has to match a filed actuarial valuation has to use the regulation's.

What is PVIFA?

It stands for present value interest factor of an annuity, and it is what one unit of currency paid every period for a given number of periods is worth today at a given discount rate. It is a bare factor with no money in it, so you multiply it by your own payment to get a present value. At 8 percent for ten annual periods it is 6.71008, which means ten annual payments of one are worth 6.71008 today.

How do I calculate PVIFA?

Use [1 − (1 + i)⁻ⁿ] ÷ i, where i is the rate per period and n is the number of periods. At 8 percent a year for ten years that is [1 − 1.08⁻¹⁰] ÷ 0.08 = 6.7100814…, printed as 6.71008. For a monthly annuity, divide the annual rate by twelve and multiply the years by twelve first, then use the same formula — the rate and the period count must both be per period.

Why is the annuity due factor larger?

Because every payment arrives one period earlier, so each is discounted one fewer time. The relationship is exact: the due factor is the ordinary factor multiplied by (1 + i). At 8 percent over ten years that is 6.71008 × 1.08 = 7.24689, about 8 percent more. The page prints both so you do not have to decide which one you need before seeing the numbers.

Does the default setup match a published table?

Yes, and that is deliberate. The default is 8 percent for ten years paid annually, which gives 6.71008 — the figure in the 8 percent row, tenth column of a standard annuity table. The reference table runs from 4 percent to 12 percent at the same term, and the six factors 8.1109, 7.72173, 7.36009, 6.71008, 6.14457 and 5.65022 match the published tables digit for digit, so the page can be used to check one.

What happens at a zero interest rate?

The factor degenerates to the number of periods, because with no discounting the payments are simply worth their sum. Ten annual payments at 0 percent give a factor of 10, and ten years paid monthly give 120. The due factor is identical at 0 percent, since multiplying by (1 + 0) changes nothing. It is a useful check on what the factor means: all of the excess over the period count is discounting.

Is PVIFA the same as the present value of an annuity?

It is one step short of it. The present value is the factor multiplied by the payment, so the factor is what you have when the payment is unknown, unimportant or already accounted for elsewhere. On this site the present value of an annuity page prints the amount and this page prints the factor, from the identical five inputs — at 7 percent over twenty years paid monthly the factor is 128.98250649… and a payment of 500 makes the present value 64,491.25.

• 26 CFR 20.2031-7(d)(6), Table B — the official annuity factor table for a term certain; at 2.6 percent for a five-year term the annuity factor is 4.6325, and the regulation's worked examples use it — Internal Revenue Service regulation, via the Electronic Code of Federal Regulations (United States)
• 26 CFR 20.2031-7(d)(2)(iv), Table K — adjustment factors for annuities payable at the end of each interval, used to convert an annual annuity factor to a more frequent payment schedule (Table J does the same for payments at the beginning of each interval) — Internal Revenue Service regulation, via the Electronic Code of Federal Regulations (United States)
• Annuities — Investor.gov glossary, for the stream of fixed-interval payments the factor is pricing — U.S. Securities and Exchange Commission, Investor.gov (United States)