PVIFA Calculator

One number that prices a whole stream

Suppose someone will pay you ninety thousand a year for the next five years. What is that worth today?

The honest way to answer is to discount each of the five payments back separately and add them up. Five calculations, five exponents, five chances to make a slip.

Or you find one number, multiply your payment by it, and you are done.

That number is PVIFA. Present value interest factor of an annuity. It answers a single question: what is one rupee at the end of every period, for n periods, worth today at rate r?

Feed the calculator a rate and a number of periods and it gives you that factor. Multiply by your payment and you have the value of the entire stream.

Notice the factor is bigger than one, unlike the single-payment factor. Of course it is. You are receiving money many times over, not once.

It is really just the single factors added together

Before the formula, the thing that makes PVIFA click.

Our PVIF calculator gives you the factor for one payment arriving in one specific period. So the value of a stream is obviously the sum of those factors, one for each payment.

Take 8 percent over 10 periods. Work out the single-payment factor for period 1, then period 2, all the way to period 10, and add the ten of them together. That sum comes to 6.7100813989.

Now ask this calculator for PVIFA at 8 percent over 10 periods. It returns 6.7100813989.

The same to every decimal place we display. Not close, identical, because it is the same quantity arrived at two ways.

That is worth holding on to, because it means PVIFA is not a new idea that has to be learned separately. It is ten small discountings stacked up and handed to you as one number.

The shortcut formula, and why it works

Adding ten factors is fine. Adding three hundred and sixty, for a thirty year loan, is not. So there is a closed form:

PVIFA = [ 1 - (1 + r)^-n ] / r

Look closely at the top of that fraction. The bit being subtracted, (1 + r) to the power of minus n, is exactly the single-payment PVIF factor. So the whole thing can be written as:

PVIFA = (1 - PVIF) / r

Which is a lovely way to check either factor against the other. At 8 percent over 10 periods, PVIF is 0.4631934881. One minus that is 0.5368065119. Divide by 0.08 and you get 6.7100813989.

Same answer as the summation, from a formula that takes one line no matter how many payments there are.

Enter your rate as a percentage and the calculator handles the conversion. Both boxes cap at 100, and the rate and period count must describe the same unit of time. A monthly rate needs a count of months. The result comes back to ten decimal places, because the factor gets multiplied by your money and rounding early would scale the error up with it.

The annuity table

These factors were printed at the back of finance textbooks for decades, and they still show up in coursework. Rate across the top, number of payments down the side.

Read down a column and watch the factor grow more and more slowly. At 12 percent, going from 20 payments to 25 adds barely a third of a unit, because those extra five payments arrive so far out that they are worth almost nothing today.

Which points at something genuinely useful. The factor has a ceiling. Keep adding payments forever and PVIFA never exceeds 1 divided by r. At 12 percent that ceiling is 8.333, and the table is already at 7.84 by payment 25. An income stream that lasts forever is worth barely more than one lasting twenty five years, which is the whole idea behind our perpetuity calculator.

Valuing a stream of payments

Present value = Payment × PVIFA

Back to the ninety thousand a year. Five payments, and lets say 7 percent is what you could otherwise earn.

PVIFA at 7 percent over 5 periods is 4.1001974359.

Multiply: 90,000 times 4.1001974359 = ₹3,69,017.77

So five years of ninety thousand, which adds up to four and a half lakh on paper, is worth about three lakh sixty nine today. The missing eighty one thousand is what waiting costs at 7 percent.

Now that number does real work. If someone offers you a lump sum instead of the stream, three lakh sixty nine is your threshold. Above it, take the cash. Below it, take the payments.

The same arithmetic answers a pile of ordinary questions. Whether to commute a pension or draw it monthly. What a structured settlement is actually worth. Whether a lease is cheaper than buying. What a rental income stream is worth to a buyer. All of them are a payment multiplied by a factor.

The same factor, running backwards

Here is the part that surprises people, and it is the reason this factor is worth knowing rather than just looking up.

If present value equals payment times PVIFA, then flip it round:

Payment = Present value / PVIFA

Which is the loan instalment formula. Not similar to it. It is it.

A loan is an annuity seen from the other side. The bank hands you a lump sum today and receives a stream of equal payments. The lump sum is the present value, the instalments are the annuity, and the factor connects them.

Lets prove it against another tool on this site rather than asserting it.

Take a loan of 4,00,000 over 36 months at 13 percent a year. The monthly rate is 13 divided by 12, which is 1.083333 percent. Ask this calculator for PVIFA at that rate over 36 periods and you get 29.6789168543.

Now divide: 4,00,000 divided by 29.6789168543 = 13,477.58.

Put the same loan into our personal loan calculator and it returns a monthly payment of 13,477.58. To the paisa.

Two tools, two apparently different formulas, one answer. Which is a decent demonstration that none of this is arbitrary, and a decent way to check either tool if you ever doubt one.

End of period or start of period

One assumption sits underneath everything above, and it is the most common thing people get wrong with annuity factors.

This formula assumes each payment arrives at the end of its period. That is called an ordinary annuity, and it is the right assumption for loan instalments, bond coupons, and most pensions.

But plenty of real payments arrive at the start. Rent is the obvious one, and lease payments and insurance premiums usually are too. That is an annuity due, and every payment lands one full period earlier, so the whole stream is worth more.

The correction is small and exact. Multiply the ordinary factor by (1 + r).

At 7 percent over 5 periods, our 4.1001974359 becomes 4.3872112364, and our ninety thousand a year is worth 3,94,849 instead of 3,69,017. Twenty five thousand of difference, purely from when in the month the money moves.

If your payments arrive at the start of the period, our present value annuity due calculator applies that adjustment for you. And if the payments grow each period rather than staying level, the present value of a growing annuity calculator is the one you want.

Questions people ask

What does PVIFA stand for?

Present value interest factor of an annuity. It is what one unit of money, received at the end of every period for n periods, is worth today at rate r.

Why is the factor bigger than 1 when PVIF is always smaller?

Because you are counting many payments rather than one. PVIFA for n periods is the sum of n single-payment factors, so it grows as payments are added, though more and more slowly.

My rate is annual and my payments are monthly.

Divide the annual rate by 12 and use the number of months. The rate and the period count must describe the same unit of time.

Can I use this to find a loan payment?

Yes. Divide the loan amount by the PVIFA for your monthly rate and number of months. That is exactly what a loan calculator does internally, as the worked check above shows.

My payments are at the start of each period, not the end.

Multiply the factor by (1 + r), or use the annuity due calculator. Rent and lease payments are usually start of period.

What if the payments never stop?

The factor converges to 1 divided by r. At 8 percent that is 12.5, so a perpetual stream is worth 12.5 times one payment. The perpetuity calculator handles that case.

My payments are different amounts each period.

Then this factor does not apply, since it assumes every payment is identical. Use the discounted cash flow calculator for uneven streams.

References

A note on the sources. The annuity factor is not merely a textbook convenience: US Treasury regulations require present value factors of exactly this kind when valuing annuities and term interests for tax, and expressly allow exact computation by software in place of the rounded published tables, which is the case for using a calculator rather than reading four decimal places off a page. The rate used in those valuations is published monthly by the Internal Revenue Service. For borrowers, the Reserve Bank of India requires lenders to disclose the amortisation schedule that this same factor generates, which makes it a figure you can hold a lender to rather than an abstraction. The compounding relationship the factor is built on 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 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 annuities and future interests to present value. https://www.irs.gov/businesses/small-businesses-self-employed/section-7520-interest-rates
3. Reserve Bank of India, Key Facts Statement (KFS) for Loans and Advances, circular RBI/2024-25/18 dated 15 April 2024, requiring an amortisation schedule and an all-inclusive annual percentage rate for all retail and MSME term loans. https://rbidocs.rbi.org.in/rdocs/notification/PDFs/CIRCULARKFS1504242AE2500BAF494C2A82442B0B642705C1.PDF
4. 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
5. Brealey, R.A., Myers, S.C., and Allen, F., Principles of Corporate Finance, McGraw-Hill Education, chapters on annuities and the valuation of level payment streams.