Future Value Annuity Problems: Find Payments or Periods

A future value annuity problem gives you a savings goal and asks for a missing piece, usually the deposit amount or the number of deposits. For an ordinary annuity, payment = goal ÷ annuity factor. To find the number of deposits, divide the goal by the payment, then find that factor in your interest rate’s column.

An annuity is a series of equal payments made at regular intervals. A future value annuity factor is a multiplier that converts the amount of each equal payment into the accumulated value of the entire payment series, including interest. You can get it from a table or a formula. When a business saves this way toward a future debt or purchase, the savings are called a sinking fund.

Start by spotting the unknown, because that tells you which calculation to run. Unless a section says otherwise, this lesson uses equal deposits, a constant rate, matching payment and compounding intervals, and no opening balance, fees, or taxes. All examples are hypothetical.

How do you set up future value annuity problems?

Write down the future balance, payment amount, periodic interest rate, and number of payment periods. Mark the missing item. Also record whether deposits occur at the beginning or end. A missing payment and a missing period count require different uses of the same relationship.

The starting relationship is future value of an ordinary annuity: FV = R × [(1 + i)ⁿ − 1] ÷ i. Here R means one deposit, not total contributions. The factor includes both contributed money and its accumulated interest.

Rearrange that formula and you get the payment directly: R = FV × i ÷ [(1 + i)ⁿ − 1].

Exam tip: if a question gives you a factor table, use that table’s factor. Your answer will then match the question’s rounding.

How do you find the required payment?

Hypothetical example. Oakbridge Studio wants $30,000 at the end of four years. It will make equal deposits at each half-year end. The fund earns a 6% nominal annual rate, which is the stated yearly rate before compounding. Interest is compounded semiannually, or twice a year. How large must each deposit be?

Without interest, eight deposits would need to be $3,750 each. The calculated payment is smaller because the fund also earns interest. Dividing $30,000 by eight ignores growth; multiplying the goal by the annuity factor moves in the wrong direction.

What changes if deposits occur at the beginning?

Keep the same $30,000 goal, eight deposits, 3% periodic rate, and end-of-four-year valuation date. Beginning-of-period deposits form an annuity due. Its factor is 8.892336046 × 1.03 = 9.159106127.

The required deposit decreases because every payment has one more interval to earn interest. Divide the ordinary payment by 1.03; do not multiply it. The comparison of ordinary annuity vs annuity due explains the timing behind this adjustment.

How do you find the number of payments?

Hypothetical example. Birch Lab deposits $2,500 at each year-end and earns 5% compounded annually. Its goal is $18,000. Divide the goal by one payment: $18,000 ÷ $2,500 = 7.2. That is the required annuity factor, not the number of deposits.

The required factor of 7.2 falls between the six- and seven-period factors, so six deposits aren’t enough. The fund first reaches the goal at the end of year seven, right after the seventh deposit.

You can also solve for the count directly: n = ln(1 + FV × i ÷ R) ÷ ln(1 + i). Here, ln means the natural logarithm, which is the LN key on a calculator or =LN() in Excel. For Birch Lab, that’s ln(1.36) ÷ ln(1.05) ≈ 6.302 periods.

That 6.302 is a math result, not a date. Birch Lab only deposits at year-end, so there’s no such thing as 0.302 of a deposit, and the balance doesn’t reach $18,000 at 6.3 years.

Does the seventh deposit need to be a full $2,500? Not really. After six deposits, Birch Lab has $17,004.78. One more year at 5% brings that to $17,855.02, so a final deposit of $144.98 lands exactly on $18,000. If every deposit must stay at $2,500, the fund ends year seven at $20,355.02.

When a table gives an exact factor match, read its period count directly. When the result is extremely close to a whole number, check whether a rounded target or table factor caused the small difference. Do not turn 5.000001 into six deposits without checking the original assumptions.

What if Birch Lab deposits at the beginning of each year?

Beginning-of-period deposits earn one extra year of interest, so adjust the required factor before you use a standard ordinary annuity table. Divide it by 1.05: 7.2 ÷ 1.05 = 6.857143.

That still falls between 6.801913 (six periods) and 8.142008 (seven periods), so Birch Lab needs seven deposits. Six beginning-of-year deposits grow to only $17,855.02 by the end of year six. The seventh deposit, made at the start of year seven, pushes the fund past the goal.

In Excel, =NPER(5%,-2500,0,18000,1) returns about 6.042 periods. The final 1 tells Excel that deposits come at the beginning.

The timing can change your answer. With a $6,900 goal, $1,000 deposits, and a 5% rate, end-of-year deposits take seven payments, but beginning-of-year deposits take only six.

What if the fund already has money in it?

Hypothetical example. Suppose Oakbridge Studio already has $5,000 in the fund today. Grow that money first: $5,000 × 1.03⁸ = $6,333.85.

The deposits now only need to cover the rest: $30,000 − $6,333.85 = $23,666.15. Divide by the same factor: $23,666.15 ÷ 8.892336046 ≈ $2,661.41 each half-year.

In Excel, =PMT(3%,8,-5000,30000,0) returns about −$2,661.41. The −5000 is the money already in the fund.

Want to see a similar setup? The lump sum chapter of the video combines a starting amount with regular deposits.

How do you solve future value annuity problems in Excel?

For Oakbridge, =PMT(6%/2,4*2,0,30000,0) returns approximately −$3,373.69. The negative sign identifies a cash deposit paid out. Change the final argument to 1 for beginning-of-period payments, giving approximately −$3,275.43. See Microsoft’s PMT function documentation.

For Birch Lab, =NPER(5%,-2500,0,18000,0) returns approximately 6.302 periods. Microsoft’s NPER reference explains the constant-payment and timing inputs. NPER reports periods; convert to years only after identifying the length of each period.

Need the rate instead? If four $5,000 year-end deposits grow to $21,232.32, =RATE(4,-5000,0,21232.32,0) returns 4%. With a table, divide the goal by the payment to get 4.246464, then look across the four-period row. Microsoft’s RATE reference explains the inputs, and the rate chapter of the video shows the table method.

Common mistakes when solving for unknowns

• Reading the table the wrong way. With a known rate, scan down that rate’s column for the factor.
• Confusing periods with years. Eight semiannual deposits cover four years.
• Rounding too early. Keep all your factor decimals, then round the final payment or period count.
• Mixing rates and periods. With semiannual compounding, use the half-year rate and half-year periods: 3% and 8, not 6% and 4.
• Skipping the timing check. Beginning-of-period deposits need the 1 + i adjustment before you read the table.

Finish every future value annuity problem by plugging your answer back into the formula. You should land on the goal, give or take rounding. If the goal falls between deposit dates, a full-deposit schedule will finish above it.

Key takeaways

• Find the unknown first: the payment, the number of deposits, or the rate.
• For an ordinary annuity, payment = goal ÷ annuity factor.
• For the number of deposits, divide the goal by the payment, then find that factor in your rate’s column.
• For beginning-of-period deposits, divide the ordinary payment by 1 + i.
• For a fixed deposit schedule, check whether rounding caused a fractional count. If the count is still not a whole number, use the next whole deposit count and identify its scheduled date.
• In Excel, PMT finds the payment, NPER finds the number of periods, and RATE finds the rate. The timing input is 0 for end-of-period deposits and 1 for beginning-of-period deposits.

Three practice questions with answers

These questions are hypothetical. Assume constant rates, retained interest, and no fees or taxes.

1. Find a payment

A fund needs $12,610 after three equal year-end deposits at 5% annual interest. How much is each deposit?

2. Read the number of payments

The goal is $21,232.32, the annual year-end payment is $5,000, and the rate is 4%. A supplied table shows factors 3.121600 for three periods and 4.246464 for four. How many payments are needed?

3. Adjust the required deposit

An ordinary-annuity plan needs $1,200 each period. The rate is 2% per period. What payment achieves the same goal with the same count of beginning-of-period deposits?

Frequently asked questions

What does “solve for the rent” mean?

It means find the recurring payment amount. In annuity terminology, rent is a general label for the payment.

Is the required annuity factor the number of payments?

No. The factor includes accumulated interest. Match it to the known rate’s table column to find the payment count.

Should a fractional NPER result always be rounded up?

Usually, yes, when deposits follow a fixed schedule. Birch Lab’s 6.302 means seven deposits, because no deposit date falls at 6.302 years. First check that the decimal isn’t just rounding. A result like 5.000001 may really be five.

What is a sinking fund?

A sinking fund is money set aside through regular deposits to reach a future amount, such as paying off a debt. Finding the required deposit is a future value annuity problem: divide the goal by the annuity factor.

How do you find the payment for an annuity due?

Find the ordinary annuity payment, then divide it by 1 + the periodic rate. For Oakbridge Studio, $3,373.69 ÷ 1.03 ≈ $3,275.43.

What if deposits and compounding happen on different schedules?

Convert the rate so it matches the deposit schedule before you use a factor. For annual deposits at 6% compounded semiannually, the effective annual rate is 1.03² − 1 = 6.09%. Use 6.09% as the rate for each annual deposit. The compounding periods lesson explains nominal and effective rates.

What is the difference between future value and present value annuity problems?

A future value problem builds toward an amount at a later date, like a savings goal. A present value problem starts from an amount today, like a loan balance. The time value of money guide covers both.

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

About the author

Mansour Farhat, CPA, CIA, DBA is the founder of Farhat Lectures and a veteran accounting educator. He holds a DBA from Temple University and a Master of Accountancy from Bloomsburg University, and he helps students from their first accounting course through the CPA, CMA, and EA exams.