Present Value Annuity Problems: Find Payments and Rates

A present value annuity problem asks how a lump sum today relates to equal payments over time. The core equation is PV = payment × annuity factor. Divide PV by the factor to find the payment. Divide PV by the payment to find the required factor, then find the rate that produces it.

An annuity is a series of equal payments made at regular intervals. Here’s the good news: once you spot which number is missing, these problems all follow the same few steps. Start by marking the payment dates and the rate per payment interval. Unless otherwise stated, examples assume equal scheduled payments, a constant rate, no fees or taxes, and a zero balance after the last payment. The final payment may be adjusted to clear the remaining balance.

How do you set up a present value annuity problem?

Write down the present value, one payment amount, the periodic rate, and the payment count. Mark the unknown. Confirm whether the first payment occurs one period from now or immediately; that choice determines the factor.

The annuity factor is the present value of $1 paid each period.

Ordinary annuity factor (payments at the end of each period) = [1 − (1 + i)⁻ⁿ] ÷ i

Annuity due factor (payments at the beginning of each period) = ordinary annuity factor × (1 + i)

Here, i is the rate per payment period and n is the number of payments.

For end-of-period payments, use the ordinary-annuity present value formula. For beginning-of-period payments, use the annuity-due factor. Do not switch between the two after calculating the required factor.

Not sure which one you have? Compare ordinary annuities and annuities due.

Teaching tip: Before you touch the numbers, draw a quick timeline. Mark today, each payment date, and the rate for one period. That one small sketch catches most timing and unit mistakes.

How do you find the payment on an annuity?

Hypothetical example. A training fund contains $24,000 today. It earns a nominal annual rate of 6%, compounded semiannually. The fund will make six equal withdrawals at the end of each half-year over three years. How much can it withdraw each time, exhausting the fund with the final withdrawal?

The half-year rate is 6% ÷ 2 = 3%. The payment count is 3 × 2 = 6. The first withdrawal occurs six months from now, so the ordinary-annuity factor applies.

Dividing $24,000 by six would give only $4,000. That approach ignores interest earned on the money still in the account. The larger calculated withdrawal includes both the original fund and its earnings.

Check the answer with a schedule and Excel

After the first six months, the fund grows to $24,720. Subtracting the $4,430.34 withdrawal leaves $20,289.66. Repeat that growth-and-withdrawal step for each half-year, and the balance lands exactly on zero:

This schedule rounds interest to the nearest cent each half-year. The cents work out perfectly here. In other problems, rounding can leave a few cents, so the final payment needs a small adjustment. Total withdrawals are $26,582.04, which is the $24,000 fund plus $2,582.04 of interest.

In Excel, =-PMT(6%/2,3*2,24000,0,0) displays a positive withdrawal amount. PMT normally returns a negative amount against a positive present value because it represents the opposite cash-flow direction. Microsoft’s PMT documentation explains these arguments.

If the same six withdrawals begin today, divide $24,000 by the due factor, 5.417191444 × 1.03. The payment becomes $4,301.30. Earlier withdrawals leave less time for the fund to earn interest, so a fixed starting fund supports a smaller equal withdrawal.

How do you find the interest rate on an annuity?

Hypothetical example. A balance of $2,350.88 can be repaid with 18 monthly payments of $150, beginning one month from today. What monthly interest rate is implied?

Find the 18-payment row in an ordinary-annuity present value table and compare the factor with the rate columns. These factors come straight from the formula, so you can check any of them yourself:

The implied rate is approximately 1.5% per month. The tiny difference arises because the loan balance is stated to the nearest cent. A higher rate produces a lower factor for the same payment count.

If the required factor lies between two table columns, do not claim an exact match. You can estimate the rate with interpolation, shown below, or use a calculator or Excel. In Excel, enter =RATE(18,-150,2350.88,0,0). The result is about 1.50002% per month. Excel may show a rounded percent, such as 2%, until you add decimal places, so check the format before you copy your answer. Microsoft’s RATE documentation explains its iterative method.

What if the factor falls between two table columns?

Hypothetical example. A $10,000 loan is repaid with 10 year-end payments of $1,400. What annual rate is implied?

Required factor = $10,000 ÷ $1,400 = 7.142857

The 10-payment factors are 7.360087 at 6% and 7.023582 at 7%, so the rate sits between 6% and 7%. Interpolation means estimating a value between two known points by assuming a straight line between them.

Estimated rate = 6% + [(7.360087 − 7.142857) ÷ (7.360087 − 7.023582)] × 1% ≈ 6.65%

Excel’s =RATE(10,-1400,10000,0,0) returns about 6.64%. The small gap is normal because factors don’t change in a perfectly straight line, so report an interpolated rate as an estimate.

How do you find the number of payments?

Sometimes you know the payment and the rate, but not how long the payoff will take. Divide present value by the payment to get the required factor, then find the payment count that produces it.

Hypothetical example. A $10,000 balance charges 1% per month, and you pay $200 at each month-end. The required factor is $10,000 ÷ $200 = 50.

In Excel, =NPER(1%,-200,10000,0,0) returns about 69.66. Because payments occur at month-end, the balance is repaid with 69 full payments of $200 and a smaller 70th payment of about $132.37. This amount retains full precision in the monthly balance calculations and rounds only the final payment. Rounding monthly interest to cents instead gives a final payment of $132.36.

What is the difference between periodic, nominal, and effective annual rates?

A monthly result is not an annual rate. Using the approximate 1.5% monthly rate, the nominal annual rate is 1.5% × 12 = 18%. The effective annual rate includes monthly compounding:

These rates describe the same assumed monthly pattern in different ways. When calculating the monthly payment stream, continue to use 1.5% per month and 18 payments. Do not insert 18% or 19.56% as the monthly input.

What are the most common mistakes in present value annuity problems?

• Multiplying when solving for a payment. Rearranging PV = R × factor requires division.
• Treating the factor as a percentage. PV ÷ R gives a factor to match, not the interest rate itself.
• Mixing units. A six-month rate must be paired with a half-year payment count.
• Using the wrong timing mode. Excel’s type argument tells it when payments happen: 0 means end-of-period, and 1 means beginning-of-period.
• Rounding too early. Retain precision in factors and rates, then round the reported payment. A small final adjustment is different from choosing an incorrect factor.

Where will you see these problems in accounting?

These aren’t just textbook puzzles. Bonds, notes payable, and leases all rest on the same math. Under ASC 842, the Financial Accounting Standards Board (FASB) lease standard, a lessee measures its lease liability at the present value of the lease payments it hasn’t paid yet.

Here’s how a rate you found gets used. Say a company owes $2,350.88 on a note at 1.5% per month. For the first $150 payment, interest is $2,350.88 × 1.5% = $35.26. The other $114.74 reduces the note to $2,236.14.

Three practice questions with answers

These practice questions use made-up numbers. Assume no remaining balance after the final payment.

1. Find a payment

A $12,000 fund earns 2% per quarter and supports four equal quarter-end withdrawals. Find one withdrawal.

2. Infer a rate

A balance of $2,813.77 is repaid by twelve $250 month-end payments. The 12-payment factors are 11.255077 at 1% monthly and 10.575341 at 2% monthly. Which rate fits?

3. Interpret an earlier withdrawal

For the same starting fund, positive rate, and payment count, will beginning-of-period withdrawals be larger or smaller than end-of-period withdrawals?

Key takeaways

• PV = payment × annuity factor.
• To find a payment, divide present value by the correct factor.
• To find a rate or a payment count, divide present value by the payment, then match the factor.
• Keep the rate and the payment count in the same time unit.
• A higher rate means a lower factor.
• Payments at the beginning of each period mean a larger factor and a smaller payment.

Frequently asked questions

What is a present value annuity problem?

It connects a lump sum today with a series of equal payments over time. You know most of the pieces and solve for the missing one.

How do you find the payment in a present value annuity problem?

Divide the present value by the correct annuity factor. Use the ordinary annuity factor for end-of-period payments and the annuity due factor for beginning-of-period payments.

How do you find an interest rate that isn’t in the table?

Interpolate between the two closest rates for an estimate, or use a financial calculator or Excel’s RATE function for a closer answer.

How does the annuity due factor differ from the ordinary annuity factor?

The annuity due factor equals the ordinary annuity factor × (1 + i). It’s larger because each payment happens one period sooner.

What does “rent” mean in an annuity problem?

Some accounting texts use rent to mean the recurring payment. It need not be a payment for property use.

Why can Excel return a negative payment?

Cash received and cash paid use opposite signs. A negative payment can correctly represent a withdrawal or repayment; it is not automatically an error.

Can I find the rate by dividing total interest by the opening balance?

That shortcut ignores the changing balance and payment timing. Solve the annuity equation using the dated payment stream.

Saving toward a target instead of spending down a fund? Try these future value annuity problems.

For more worked examples and practice, explore the Intermediate Accounting course.

About the author: Mansour Farhat, CPA, CIA, DBA, teaches accounting and exam prep for CPA, CMA, and EA candidates at Farhat Lectures.