PVIF Calculator

What a discount factor actually is

PVIF stands for present value interest factor, which is a long name for a simple idea.

It is the answer to one question: what is one rupee, arriving n periods from now, worth today at rate r?

That is the whole thing. One rupee. And because it is just one rupee, the answer is always a number between zero and one, and you can multiply it by any amount you like.

So if the factor for 8 percent over 10 periods is 0.4632, then a rupee arriving in ten years is worth about forty six paise now. Ten thousand rupees arriving then is worth 4,632. Five lakh is worth 2,31,597. One factor, unlimited reuse.

This calculator takes a rate and a number of periods and hands you that factor. No amount required, because the amount is your business.

The table this replaces

Before calculators were something everybody carried, these factors lived in printed tables at the back of finance textbooks. You found your rate along the top, your period down the side, and read off the number where they met.

Those tables still turn up in exam halls and coursework, so here is one. Rate across, periods down.

Read down any column and the story of discounting is right there. At 12 percent, money arriving in twenty five years is worth about six paise in the rupee. At 4 percent over the same stretch it is still worth thirty eight. Nothing else on this page will make the effect of a rate choice clearer than reading down those two columns side by side.

Read across any row instead and you see the other half. The further right you go, the harder the discount bites, and the gap widens the further down you are.

The trouble with any printed table is that it only has the rates and periods somebody chose to print. Yours is 9.5 percent over 17 periods, and it is not there. Which is what the calculator is for.

Where the numbers in that table come from

PVIF = 1 / (1 + r)ⁿ

That is compound growth turned upside down. Growing one rupee forward means multiplying by (1 + r) once per period. Coming back means dividing by the same thing, once per period. So the factor is one over that.

Take 8 percent over 10 periods and follow it through. 1.08 raised to the tenth power is 2.158925. One divided by that is 0.4631934881.

Which also tells you something neat. At 8 percent, money roughly doubles in ten years going forward, so it roughly halves coming back. The factor being just under a half is the same fact wearing different clothes.

Enter your rate as a percentage and the calculator handles the division by 100. Rate and periods both cap at 100, and both have to describe the same slice of time. A monthly rate needs a count of months.

Using a factor instead of doing the division

Multiply. That is genuinely it.

Present value = Future amount × PVIF

The reason anybody bothers with the factor rather than dividing straight away is that you often need the same rate and period more than once.

Say you are looking at a project throwing off money at three different points, all discounted at 10 percent.

Ten lakh of promised money, worth a little over seven today. Three lookups and three multiplications, no exponents in sight.

Handy things that fall out of the factor once you have it. One minus the factor is the share of value that waiting costs you, so 1 minus 0.6209 says five periods at 10 percent costs you 38 percent of the money. And factors chain: the factor for 8 periods is the factor for 3 multiplied by the factor for 5, because discounting eight periods is discounting three and then another five.

If you only have one amount and one date, skip the factor entirely and use our present value calculator, which does the whole thing in one step.

Why ten decimal places, and why that is not fussiness

This calculator returns the factor to ten decimal places. Printed tables give you four. That difference is the actual reason to use one over the other, and it is worth a paragraph.

Rounding a factor to four places introduces a small error. Multiply that rounded factor by a large amount and the error scales with it. Do that across a dozen cash flows and the errors accumulate, all in the same direction, because rounding to four places is not symmetric across a set of factors that are all less than one.

On a personal decision this never matters. On a valuation with real money attached, it can. And there is a nice bit of confirmation that this is a real concern rather than a hobby horse.

The US regulations governing how annuities, life estates and remainder interests must be valued for tax purposes publish official factor tables. Then they explicitly permit something else: exact methods of obtaining the factors are allowed, including software using the applicable interest rate and the proper actuarial formula, so long as the method is applied consistently.

Which is a regulator saying, in the driest possible language, that computing the factor precisely beats reading a rounded one off a page. That is the whole argument for this tool.

Small print on the rounding: factors above 1 are shown to ten decimal places, factors below 1 to ten significant figures, and trailing zeros are dropped. So a factor of exactly 0.5 displays as 0.5 rather than 0.5000000000.

PVIF and PVIFA are the same object

You will run into PVIFA next to PVIF, and the extra A stands for annuity. The relationship between them is worth seeing once, because after that neither is mysterious.

PVIF discounts one payment. PVIFA discounts a run of equal payments, one at the end of every period.

And PVIFA is nothing more than the PVIF factors added together. Add up the factors for period 1, period 2, all the way to period 10, and you get the PVIFA for 10 periods. At 8 percent that sum comes to 6.7100813989, which is exactly what our PVIFA calculator returns for the same inputs. Not approximately. To every decimal place we display.

There is a shortcut too, which saves adding ten numbers:

PVIFA = (1 - PVIF) / r

Try it. One minus 0.4631934881 is 0.5368065119. Divide by 0.08 and you get 6.7100813989 again.

So if you have either factor you can get the other in one step, and you can use each to check the other. Which is a genuinely satisfying property for something that started life as a number in the back of a textbook.

Questions people ask

What does PVIF stand for?

Present value interest factor. It is what one unit of money, received n periods from now, is worth today at rate r.

Can the factor ever be more than 1?

Not at any positive rate. At a rate of zero it is exactly 1, because waiting costs you nothing. Above zero it is always less than 1 and shrinks as either the rate or the period count rises.

My rate is annual but my periods are months.

Divide the annual rate by 12 before entering it. The rate and the period count have to describe the same unit of time, or the factor is meaningless.

Should I use the table above or the calculator?

The table if you are checking your work against a textbook that used the same rounding. The calculator for anything real, and for any rate or period the table does not happen to list.

When do I want PVIFA instead?

When the money arrives as a series of equal payments rather than one lump. Loan instalments, rent, a pension. Our PVIFA calculator handles those.

Is there a forward version?

Yes, FVIF, which is just (1 + r) raised to n rather than one over it. The two are reciprocals, so multiply them together and you get exactly 1.

Why so many decimal places?

Because the factor gets multiplied by your money, so rounding error scales with the amount. Ten places means the rounding is yours to do at the end rather than ours to impose at the start.

References

A note on the sources. The claim about precision is not ours: the US Treasury regulations that govern valuing annuities and future interests publish official factor tables and then expressly permit exact computation by software using the proper formula instead, which is about as strong an endorsement of calculating over table-reading as a regulator is likely to give. The rate those factors are built on is published monthly by the Internal Revenue Service. The compounding relationship that the factor inverts is defined by the Securities and Exchange Commission's investor education office.

1. Cornell Law School, Legal Information Institute, 26 CFR § 1.7520-1, Valuation of annuities, unitrust interests, interests for life or terms of years, and remainder or reversionary interests, on actuarial factors and the permissibility of exact computational methods in place of the published tables. https://www.law.cornell.edu/cfr/text/26/1.7520-1
2. Internal Revenue Service, Section 7520 interest rates, the monthly rate prescribed for discounting future interests to present value. https://www.irs.gov/businesses/small-businesses-self-employed/section-7520-interest-rates
3. U.S. Securities and Exchange Commission, Office of Investor Education and Advocacy, Compound Interest, Investor.gov glossary. https://www.investor.gov/introduction-investing/investing-basics/glossary/compound-interest
4. Brealey, R.A., Myers, S.C., and Allen, F., Principles of Corporate Finance, McGraw-Hill Education, chapters on present value and discount factors.