Calculating Future Value of Annuities: Formulas and Examples

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The future value of an annuity is the amount a series of equal, evenly spaced payments will be worth at a specified future date, once each payment has earned compound interest at a given rate. Three inputs drive the result: the payment amount, the interest rate per period and the number of periods. Timing also matters. Payments at the end of each period form an ordinary annuity, and payments at the beginning form an annuity due.

How the future value of an annuity works

The calculation rests on the time value of money: a dollar today can be invested and earn a return, so it is worth more than a dollar received later. Each payment in an annuity starts compounding only when it arrives.

In an ordinary annuity, payments land at the end of each period, so they compound for different lengths of time. With n payments, the first one compounds for n − 1 periods, the second for n − 2, and so on. The final payment arrives on the valuation date and earns no interest. Adding them up gives:

FV = PMT(1 + r)^(n−1) + PMT(1 + r)^(n−2) + … + PMT(1 + r) + PMT

This is a geometric series with ratio (1 + r), and summing it produces the closed-form formula below.

Future value of an ordinary annuity formula

FV = PMT × [(1 + r)^n − 1] / r

where:

• FV = future value of the payment stream
• PMT = amount of each payment
• r = interest rate per period, as a decimal
• n = number of payments

The (1 + r)^n term is the compound growth of one dollar over the full term. Subtracting 1 and dividing by r converts that single-sum growth into the accumulated value of one dollar deposited every period.

Future value of an annuity due

When each payment is made at the start of the period, every deposit gets one more period of compounding than in an ordinary annuity, so the formula adds one factor of (1 + r):

FV(due) = PMT × [(1 + r)^n − 1] / r × (1 + r)

Rent paid at the beginning of the month and savings deposited on the first of each month follow this timing. Loan repayments and many bond coupons, paid at the end of the period, match an ordinary annuity. With the same payment, rate and term, an annuity due always has the higher future value whenever r is positive.

Worked example: ordinary annuity vs. annuity due

Hypothetical example. An investor contributes $125,000 a year for five years and assumes the money compounds at 8% a year.

Ordinary annuity (payments at year end):

1. Growth factor: (1.08)^5 = 1.469328
2. Subtract 1 and divide by the rate: 0.469328 / 0.08 = 5.866601
3. Multiply by the payment: $125,000 × 5.866601 = $733,325

Annuity due (payments at year start):

1. Start from the ordinary result: $733,325
2. Multiply by (1 + r): $733,325 × 1.08 = $791,991

The difference, $58,666, is exactly one year of 8% growth on the ordinary annuity’s value. On $625,000 in total contributions, the ordinary annuity earns about $108,325 of interest and the annuity due about $166,991.

Matching the rate to the payment period

The most common error is mixing annual rates with monthly or quarterly payments. When payments and compounding happen at the same frequency (a simple annuity), divide the annual nominal rate by the number of periods per year and multiply the years by the same number.

Hypothetical example. Someone saves $200 at the end of every month for 10 years at a 6% annual rate compounded monthly.

• r = 0.06 / 12 = 0.005
• n = 10 × 12 = 120
• FV = $200 × [(1.005)^120 − 1] / 0.005 = $200 × 163.879 ≈ $32,776

Contributions total $24,000, so roughly $8,776 comes from interest.

When payment frequency and compounding frequency differ (a general annuity), first convert the nominal rate into an effective rate per payment period:

i = (1 + j/m)^(m/p) − 1

Here j is the annual nominal rate, m is the number of compounding periods per year and p is the number of payments per year. Use i as r in the standard formula. For quarterly payments under monthly compounding, the effective quarterly rate is (1 + j/12)^3 − 1.

The future value annuity factor (FVIFA)

The bracketed term, [(1 + r)^n − 1] / r, is the future value interest factor of an annuity, or FVIFA: the future value of $1 per period. Multiplying it by any payment gives that payment stream’s future value. The 5.866601 in step 2 of the example above is the FVIFA for 8% over five periods.

Before spreadsheets, analysts read these factors from a future value annuity table organized by rate and number of periods. For an annuity due, multiply the table factor by (1 + r).

The FVIFA differs from the future value factor of a single sum, (1 + r)^n, which measures the growth of one lump sum. An annuity factor is always at least n, the number of payments, because it adds up n deposits before counting any interest.

Rearranging and extending the formula

Solving for the payment. Savers often know the target and need the required contribution:

PMT = FV × r / [(1 + r)^n − 1]

Hypothetical example: to reach $50,000 in 10 years at 5% a year with year-end deposits, PMT = $50,000 / 12.578 ≈ $3,975 a year.

Growing annuity. If payments rise at a constant rate g each period (for example, contributions that increase with salary), the future value of a growing annuity with the first payment at the end of period one is:

FV = PMT × [(1 + r)^n − (1 + g)^n] / (r − g)

This version requires r and g to be different. When they are equal, the result simplifies to n × PMT × (1 + r)^(n−1).

Zero interest. If r = 0, the standard formula cannot be used because it divides by zero. In that case the future value is simply PMT × n.

Present value vs. future value

Present value and future value describe the same cash flows from opposite ends of the timeline. Future value compounds payments forward to a later date. Present value discounts them back to today, using the formula PV = PMT × [1 − (1 + r)^−n] / r for an ordinary annuity. The two are linked by FV = PV × (1 + r)^n.

In the $125,000 example, the present value at 8% is about $499,089, and compounding it for five years at 8% returns the $733,325 future value. Analysts usually know one figure and solve for the other: a saver with a fixed monthly budget solves for future value, while a buyer planning a down payment in a set number of years starts from the future target.

The word “annuity” also has a second meaning. In time-value math, an annuity is any series of equal periodic payments. An annuity sold by an insurer is a contract whose value depends on its own terms, fees and payout options, not just on this formula.

Limitations of the future value of annuity calculation

• Constant rate. Actual returns vary, so the result is a projection under an assumed rate, not a forecast.
• Equal payments and timing. Skipped, irregular or variable contributions need period-by-period calculation, or the growing-annuity version if the increases are regular.
• Nominal dollars. The result ignores purchasing power. Use a real rate of return or deflate the result with a measure such as the U.S. inflation rate by year.
• No fees or taxes. Account charges and taxes on growth lower the effective rate, so deduct them from r before calculating.
• Sensitivity to inputs. Because r appears as an exponent, small changes in the assumed rate produce large differences over long terms.

FAQs

The Bottom Line

Savers projecting regular contributions and anyone planning toward a fixed target can use these formulas to estimate an ending balance or the payment needed to reach it. Match the rate to the payment period, note whether deposits come at the start or end of each period, and treat the result as a projection, since it assumes a constant rate and ignores inflation, fees and taxes.