Future Value of an Annuity

In one sentence: The future value of an annuity says how much a row of equal payments adds up to by the end, once the interest earned on each payment is included.

$FV = P \cdot \frac{(1 + r)^{n} - 1}{r}$

Read it out loud: “FV equals P times a fraction. The top is one plus r, raised to the power n, minus one. The bottom is r. FV is the future value, the total at the end. P is the payment made each period. r is the interest rate per period, written as a decimal. n is the number of periods, which is the number of payments.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Annuity and the ordinary annuity: An annuity is a set of payments of the same size, made at the same regular interval, such as 100 dollars every month. Savings plans, loan repayments and pension payments are all built from annuities. An ordinary annuity is one where each payment is made at the end of its period, which is the case this formula describes.
Exponent (power) and the growth factor: An exponent, also called a power, says how many times to multiply a number by itself: $(1.005)^{36}$ means 1.005 multiplied by itself 36 times. The result, $(1 + r)^{n}$ , is the growth factor: the number that one dollar becomes after $n$ periods of interest. A growth factor above 1 means the money gets bigger.
Percent, decimal and period: A percent is a number out of 100, so 6 percent is 6 out of 100, or 0.06 as a decimal. A period is one equal step of time that the rate and the payments both refer to, such as one month. A common way to get a monthly rate is to divide the yearly rate by 12, and the count $n$ must then be the number of those same months.
The sum of growth factors: The formula is a short way to write a long sum. The sum $1 + (1 + r) + (1 + r)^{2} + \cdots + (1 + r)^{n-1}$ adds up the growth factor of every payment. The last payment has factor 1, and the first payment has the largest factor, because it has had the most periods to grow. The fraction in the formula equals this whole sum.

What each symbol means

Symbol	Meaning
$FV$	Future value of the whole savings plan: the total in the account after the last payment, measured in dollars. It includes every payment you made plus all the interest those payments earned. It is the answer the formula gives.
$P$	Payment per period, the fixed amount added at the end of each period, measured in dollars. The formula assumes the same amount every time. In the example P is 100 dollars a month.
$r$	Interest rate per period, written as a decimal and matched to the payment period. For monthly payments it is a monthly rate, so 0.005 means half a percent each month. It has no unit beyond the period it refers to.
$n$	Number of payments, counted in the same periods as the rate. Three years of monthly payments gives n = 36. It is a count of periods, so it has no unit beyond the period it counts.

The formula piece by piece

$P$: $P$ is the payment made at the end of each period, a fixed amount in dollars. Every payment has the same size, so the future value is a count of these equal payments, each one grown by its own factor, and $P$ is the building block.
$1 + r$: The growth of one dollar over one period. If $r = 0.005$ , the bracket is 1.005, so one dollar becomes 1.005 dollars after a month.
$(1 + r)^{n}$: The growth factor over all $n$ periods, the number that one dollar becomes after the whole savings plan. For 36 months at 0.5 percent it is about 1.1966805.
$(1 + r)^{n} - 1$: Subtract one to leave only the growth, the part of the money that is interest. For 36 months this is about 0.1966805, the extra money earned on each dollar.
$\frac{(1 + r)^{n} - 1}{r}$: Divide that growth by the rate. The result is the total of the growth factors of all the payments, a count of how many payments’ worth of money the account holds at the end. For the example it is about 39.3361.
$FV = P \cdot \frac{(1 + r)^{n} - 1}{r}$: The whole formula in one sentence: the future value equals the payment per period times the total of the growth factors of all the payments, which is the balance at the end of the savings plan.

What the formula is doing, and why

An annuity is a series of equal payments made at regular times, such as the same amount put aside at the end of every month. The future value formula asks how much those payments add up to at the end, once interest has been added along the way. This version assumes each payment is made at the end of its period, which is called an ordinary annuity. The first payment therefore earns interest for $n - 1$ periods, and the last payment earns none.

Why the fraction? Each payment grows by a different amount, because the earlier payments have had more time to earn interest. The last payment has no growth at all, the one before it grows by one period, and the first grows by $n - 1$ periods. Adding up all these growth factors gives the sum $1 + (1 + r) + (1 + r)^{2} + \cdots + (1 + r)^{n-1}$ , and this sum can be written in the short form $\frac{(1 + r)^{n} - 1}{r}$ . The fraction is that whole sum in one line.

Why multiply by $P$ ? Every payment has the same size, so the total is $P$ times that sum. Doubling the payment doubles the future value, and the fraction tells you how many payments’ worth of money the account holds at the end.

The rate does much of the work. Doubling the time does not double the answer, because each extra period adds interest on top of everything saved before. The formula needs a rate above zero, because the fraction divides by $r$ . With a rate of zero the account only collects the payments, so the future value is $P n$ , which is the plain total paid in.

An everyday picture

Think of a row of seeds, one planted at the end of every season, where each seed grows by the same fraction every season after it is planted. The seed planted first has had the most seasons to grow, and the last one has had none. The future value formula adds up the grown size of every seed in one step, instead of tracking each seed separately.

Worked example: Saving 100 dollars a month for a laptop over 3 years

A student saves 100 dollars at the end of every month for 3 years. Suppose the savings account pays 6 percent per year, which is a rate of 0.5 percent per month, and that interest is added each month. How much will the account hold after the last payment? In the formula, $P = 100\ \text{dollars}$ , $r = 0.005$ and $n = 36$ months. The rate is an illustration, not a real offer.

Write down the inputs and match the periods. The payment is 100 dollars each month, so the period is one month. The yearly rate of 6 percent must become a monthly rate, so we divide 6 percent by 12 to get 0.5 percent, written as $r = 0.06 \div 12 = 0.005$ . The 3 years become $n = 36$ months, so the rate and the count use the same period.
Work out the growth factor. One dollar grows by a factor of $1 + r = 1.005$ each month. Over 36 months the factor is $(1.005)^{36}$ , which is 1.005 multiplied by itself 36 times. A calculator’s power key gives $(1.005)^{36} \approx 1.1966805$ , so a dollar left in the account for the whole time grows to about 1.1966805 dollars.
Subtract one and divide by the rate. The top of the fraction is $(1.005)^{36} - 1 = 0.1966805$ , the growth above the original dollar. Dividing it by the monthly rate, $\frac{0.1966805}{0.005} \approx 39.3361$ , gives the total of the growth factors of all 36 payments. In other words, the account holds about 39.34 payments’ worth of money at the end.
Multiply by the payment. Each payment is 100 dollars, so the future value is $FV = 100\ \text{dollars} \times 39.3361 \approx 3933.6\ \text{dollars}$ . This is the balance after the last payment, with all the interest included.
Find the interest earned. The student paid in $36 \times 100 = 3600$ dollars. The balance is 3933.6 dollars, so the interest earned is $3933.6 - 3600 = 333.6$ dollars. Over three years that is about 9 percent of the money paid in, and the interest is the reason the balance is larger than the total of the payments.

Answer: After 3 years the account holds about 3933.6 dollars. The student paid in 3600 dollars, so the account earned about 333.6 dollars of interest. A real account may pay a different rate or add interest on a different schedule, and the figures would change with those terms.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Using the yearly rate with monthly payments. A yearly rate of 6 percent is 0.06, but for monthly payments the rate per period is 0.06 divided by 12, which is 0.005.
Writing the number of years instead of the number of periods. Three years of monthly payments is $n = 36$ , not $n = 3$ .
Forgetting that the formula assumes each payment is made at the end of its period. If each payment is made at the start, every payment earns one more period of interest, so the future value is bigger.
Confusing the payment $P$ with the future value $FV$ . $P$ is the amount added each period, and $FV$ is the total at the end, which includes interest and so is larger than the sum of the payments paid in.

Where you will meet it

Planning how much to save each month to reach a goal, such as a deposit, a trip or a school fund.
Working out how a retirement savings plan grows when the same contribution is made every period.
Comparing savings accounts that pay different rates for the same regular monthly contribution.
Sinking funds, where a business sets aside equal amounts each period to pay for a large expense later.

Questions people ask

What is an annuity?

An annuity is a series of equal payments made at regular intervals, such as monthly deposits or yearly payments. The word covers any fixed stream of payments, whether you are saving the money or receiving it.

Why is the future value bigger than the total I paid in?

Each payment earns interest from the day it is made until the end. The gap between the future value and the sum of the payments is the total interest earned. In the example that gap is $FV - nP = 333.6$ dollars.

What is the difference between an ordinary annuity and an annuity due?

An ordinary annuity makes each payment at the end of the period, which is what this formula assumes. An annuity due makes each payment at the start, so every payment earns one more period of interest, and the future value is multiplied by $(1 + r)$ .

Should the rate be per month or per year?

Use the rate for the same period as the payments. For monthly deposits, divide the yearly rate by 12 and use the number of months for $n$ . Mixing a yearly rate with a count of months gives an answer that is far off.

Present Value: Present value says how much money you must put aside today, at a steady interest rate, so that it grows to a known amount by a later date.
Present Value of an Annuity: The present value of an annuity says how much money you need today to pay a fixed amount every period for a set number of periods, with interest working in the background.
Compound Interest: Compound interest says a savings balance grows faster over time, because each round of interest is added to the balance and then earns interest of its own.
Loan Payment: The loan payment formula gives the fixed amount you pay each month so that a loan is fully repaid, interest included, by the last payment.