Present Value Annuity Due Calculator

Everything arrives one period earlier

An annuity is a run of equal payments at regular intervals. The only question this page is about is whether each one lands at the start of its period or the end.

End of period is called an ordinary annuity. Start of period is an annuity due. Same amounts, same count, same rate, and a different answer, because money that arrives sooner is worth more.

How much more? Exactly one period's worth of interest on the whole thing. Not roughly. Exactly, and the worked example below demonstrates it to the paisa.

This calculator does the start-of-period version. Three inputs, one answer.

Which payments are which

The distinction sounds academic until you try to classify a real payment, so here is the practical version.

The pattern underneath is simple enough. If you pay for the thing before you use it, that is an annuity due. If you pay after you have had it, that is ordinary.

So rent is due on the first of the month for the month ahead, which is start of period. A loan instalment is paid at the end of the month for the month of borrowing you have just had, which is end of period. Almost everything else follows from that test.

If your payments land at the end of the period, this is the wrong tool and our PVIFA calculator is the right one.

The formula, and the bit on the end

PV = (PMT / r) × [ 1 - 1 / (1 + r)ⁿ ] × (1 + r)

Look at the first two thirds of that expression. It is the ordinary annuity formula, the same one behind our PVIFA calculator, with the payment already multiplied in.

Then there is (1 + r) hanging off the end. That is the entire adjustment.

Why one multiplication is enough: every payment in the stream moves forward by exactly one period, so every payment gets discounted one period less, so every payment is worth (1 + r) times what it was. Multiply the whole present value by (1 + r) and you are done.

Which means you never strictly need this calculator. Work out the ordinary annuity and multiply by one plus the rate. It is here because doing it in one step is less error prone than doing it in two, and because forgetting the adjustment entirely is the more common mistake.

Note the caps: the rate and the period count both stop at 100, and the rate must be at least 1 percent.

Ninety thousand a year, both ways

Five payments of 90,000, at 7 percent a period.

Run as an ordinary annuity, with payments at the end of each year, the present value is 369,017.77. That is 90,000 multiplied by the annuity factor of 4.1001974359.

Run as an annuity due, with payments at the start, this calculator returns 394,849.01.

The difference is 25,831.24.

Now check where that number came from. Seven percent of the ordinary present value, 369,017.77, is 25,831.24.

Identical. The whole gap is one period of interest on the ordinary value, which is what the (1 + r) term is doing.

Check it the other way too. Divide 394,849.01 by 369,017.77 and you get 1.07 exactly, to as many decimal places as you care to carry. Not approximately 1.07. Precisely.

In percentage terms, moving five payments forward by one year each is worth 7 percent of the whole stream. Which is a useful thing to hold in your head: the annuity due premium is always exactly the interest rate, whatever the payment size and however many periods there are.

Where the difference is worth real money

Seven percent of a stream is not a rounding error, and there are situations where getting it wrong changes a decision.

Lease versus buy. Lease payments are almost always in advance. Loan instalments are almost always in arrears. Compare a lease priced as an ordinary annuity against a loan and the lease looks cheaper than it is by roughly one period of interest, every time.

Valuing a rental income stream. A buyer pricing a tenanted property is buying an annuity due, since rent arrives at the start of each month. Treating it as ordinary undervalues the property.

Pension choices. Some pensions pay at the beginning of the period and some at the end, and the paperwork does not always make it obvious. On a long stream the difference compounds into a meaningful sum.

Settlement offers. Anyone offering you a stream of payments in place of a lump sum has decided when those payments land, and the answer moves the value by the full rate.

The general habit worth forming: before valuing any stream, find out when the first payment arrives. If it arrives today, this is the tool. If it arrives one period from today, it is not.

A related point on the same theme. If the payments grow rather than staying level, neither tool fits, and our present value of a growing annuity calculator handles the escalating case. If they never stop, the perpetuity calculator is the one you want.

Questions people ask

How do I know which type I have?

Ask when the first payment falls due. Today, or at the start of the first period, means an annuity due. One full period from now means an ordinary annuity. Rent and leases are usually due. Loans and bonds are usually ordinary.

How much larger is an annuity due?

Larger by exactly the interest rate. At 7 percent it is 7 percent larger, regardless of the payment size or the number of periods.

Can I convert one to the other?

Yes, and it is one multiplication. Ordinary times (1 + r) gives the due value. Due divided by (1 + r) gives the ordinary value.

My payments are monthly but my rate is annual.

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

Why will it not accept a rate below 1 percent?

The rate field currently requires at least 1. For a monthly rate below that, work out the ordinary annuity with our PVIFA calculator and multiply the result by one plus your rate.

Does this give the future value?

No, the present value, meaning what the whole stream is worth today. Future value is a different calculation.

What if the payments increase each period?

Then it is a growing annuity and this formula does not apply, since it assumes every payment is identical. Use the growing annuity calculator.

References

A note on the sources. The distinction between payments at the start and end of a period is not merely a textbook nicety: United States Treasury regulations governing the valuation of annuities and term interests turn on exactly these mechanics and require the correct actuarial factors to be applied, while expressly permitting exact computation by software rather than reliance on rounded published tables. The compounding relationship that makes an earlier payment worth (1 + r) times a later one is defined by the Securities and Exchange Commission's investor education office, whose free calculator can verify any figure here.

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. 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 term 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 annuities and the valuation of level payment streams.