Present Value of an Annuity

In one sentence: 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.

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

Read it out loud: “PV equals P times a fraction. The top is one minus one plus r, raised to the power minus n. The bottom is r. PV is the present value, the amount needed today. P is the payment made each period. r is the interest rate per period, written as a decimal. n is the number of periods.”

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 series of equal payments made at regular intervals, such as a scholarship paid every year. An ordinary annuity is one where each payment is made at the end of its period. The present value of an annuity answers the question: how much money today has the same value as all of those future payments together?
Negative exponent and the discount factor: A negative exponent means divide: $(1 + r)^{-n}$ equals $\frac{1}{(1 + r)^{n}}$ . For example, $(1.05)^{-4} = \frac{1}{(1.05)^{4}} \approx 0.8227025$ . This number is a discount factor, the value today of one dollar paid after four years, here about 0.8227025 dollars. It is below 1 because waiting for money means it is worth less.
Discounting, the reverse of interest: Discounting means working out the value today of money you will receive later. Interest runs forward: money today becomes more money later. Discounting runs backward: a payment due later is turned into a smaller amount today, using the interest rate for each period it has to wait.
Percent, decimal and period: A percent is a number out of 100, so 5 percent is 0.05 as a decimal. The rate and the payments must share one period, such as a year. If the payments are monthly, the rate must be a monthly rate, and $n$ must count months.

What each symbol means

Symbol	Meaning
$PV$	Present value of the annuity, the amount needed today so that the fund can pay every scheduled payment and end with nothing left. Measured in dollars. This is the number the formula gives you.
$P$	Payment per period, the fixed amount paid at the end of each period, measured in dollars. The formula assumes every payment is the same size. In the example P is 2000 dollars a year.
$r$	Interest rate per period, written as a decimal and matched to the payment period. Five percent a year is r = 0.05 when the payments are yearly. It has no unit beyond the period it refers to.
$n$	Number of payments, counted in the same periods as the rate. Four yearly payments give n = 4. A count of periods has no unit beyond the period it counts.

The formula piece by piece

$P$: $P$ is the payment at the end of each period, a fixed amount in dollars. Every payment has the same size, so the present value is a count of these equal payments, each one scaled by its own discount factor.
$r$: $r$ is the interest rate per period as a decimal. It sets how fast the money grows and therefore how strongly each later payment is discounted. With $r = 0.05$ , each year of waiting shrinks the value by the factor $\frac{1}{1 + r}$ , which is about 0.952381.
$(1 + r)^{-n}$: The discount factor for the last payment, the number that turns one dollar paid after $n$ periods into its value today. For four years at 5 percent it is about 0.8227025.
$1 - (1 + r)^{-n}$: Subtract that factor from one. The result, about 0.1772975 here, equals the rate times the total of the discount factors, so the next step can recover that total by dividing by the rate.
$\frac{1 - (1 + r)^{-n}}{r}$: Divide by the rate to get the total of all the discount factors, the number of payments’ worth of money needed today. In the example it is about 3.54595.
$PV = P \cdot \frac{1 - (1 + r)^{-n}}{r}$: The whole formula in one sentence: the present value of an annuity equals the payment per period times the total of the discount factors for all of its payments, which is the money needed today to pay every one of them.

What the formula is doing, and why

An annuity is a fixed stream of equal payments, such as a scholarship paid every year for four years. Money set aside today, earning interest, can pay those payments out one by one. The present value of the annuity is the starting amount that does exactly that: after each payment, whatever is left keeps earning interest, and nothing remains at the end.

Why the fraction? Each future payment is worth less today than its face value, because it has to be discounted for the time it waits. The payment at the end of year 1 is discounted by one period, the payment at the end of year 4 by all four periods. Adding up all those discounted payments gives the fraction $\frac{1 - (1 + r)^{-n}}{r}$ , which is one short expression instead of $n$ separate divisions.

Why the negative exponent? A negative power means divide, so $(1 + r)^{-n}$ is the same as $\frac{1}{(1 + r)^{n}}$ . It is the discount factor for the last payment, the number that turns one dollar paid after $n$ periods into its value today. Subtracting it from one and dividing by $r$ gives the total of all the discount factors in a single step.

Why multiply by $P$ ? Every payment is the same size, so the present value is $P$ times the total of the discount factors. A bigger payment or a longer series needs more money today. A higher rate needs less money today, because each future dollar is discounted more heavily.

An everyday picture

Think of a water tank that refills by a fixed fraction of whatever water is still in it each month. You want to draw the same cup of water every month for a set number of months, until the tank is empty. The present value is the amount of water you must start with. The fuller the tank, the faster it refills, so you need less at the start than the total of the cups you draw.

Worked example: Funding a 2000 dollar a year scholarship for 4 years

A scholarship fund wants to pay a student 2000 dollars at the end of each year for four years. The fund’s money can earn 5 percent per year, with interest added once a year. How much must the fund hold today? In the formula, $P = 2000\ \text{dollars}$ , $r = 0.05$ and $n = 4$ years. The rate is an illustration, not a real offer.

Write down the inputs. The payment is 2000 dollars at the end of each year, so the period is one year. The rate of 5 percent per year is $r = 0.05$ as a decimal, and there are four payments, so $n = 4$ . Every number uses years, which keeps the rate and the count in step.
Work out the discount factors. A payment at the end of year $k$ is discounted by $(1.05)^{-k} = \frac{1}{(1.05)^{k}}$ . The four discount factors are $0.9523810$ for year 1, $0.9070295$ for year 2, $0.8638376$ for year 3 and $0.8227025$ for year 4. The last one comes from $(1.05)^{4} \approx 1.2155063$ and $\frac{1}{1.2155063} \approx 0.8227025$ . Each dollar paid later is worth less than a dollar today.
Add the discount factors into one total. Adding the four factors gives $0.9523810 + 0.9070295 + 0.8638376 + 0.8227025 = 3.54595$ . The formula reaches the same total in one step: $\frac{1 - 0.8227025}{0.05} = \frac{0.1772975}{0.05} \approx 3.54595$ . This is the number of payments’ worth of money needed today.
Multiply by the payment. Each payment is 2000 dollars, so the present value is $PV = 2000\ \text{dollars} \times 3.54595 \approx 7091.90\ \text{dollars}$ . Multiplying the total of the discount factors by the payment turns that count into dollars.
Run the fund forward to check it. Start with 7091.90 dollars. After the first year, 5 percent interest adds about 354.60 dollars, and the 2000 dollar payment leaves about 5446.50 dollars. After year two the balance is about 3718.82 dollars, after year three about 1904.76 dollars, and after the fourth payment it is back to zero, apart from a rounding difference of about a tenth of a cent. The fund pays every scholarship and ends empty, so the starting amount is right.

Answer: The fund must hold about 7091.90 dollars today. Earning 5 percent a year, that starting amount pays 2000 dollars at the end of each of four years and is used up exactly at the last payment. A higher rate would need less money today, and a longer scholarship would need more.

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

Common mistakes

Using $(1 + r)^{n}$ where the formula needs $(1 + r)^{-n}$ . Without the minus sign the later payments are multiplied up instead of discounted down, and the result comes out negative.
Discounting every payment by the same number of periods. The first payment is discounted by one period, the second by two, and so on up to $n$ .
Using the yearly rate with monthly payments. Divide the yearly rate by 12 and use the number of months for $n$ , so that the rate and the count match.
Treating the present value as the total of the payments. Because of interest, the amount needed today is smaller than the sum of the payments: four payments of 2000 dollars add up to 8000 dollars, but only about 7091.90 dollars is needed today.

Where you will meet it

Pricing a bond’s regular coupon payments, which form an annuity, before adding the value of the final repayment of face value.
Valuing a fixed rent stream, such as the total value today of a series of equal rent payments under a lease.
Working out what a series of fixed annual payments, such as a prize paid over several years, is worth as a single lump sum today.
Estimating the amount a loan covers, since the loan amount equals the present value of its scheduled payments.

Questions people ask

How is this different from the present value formula?

The present value formula discounts one future amount. This formula adds up a series of equal payments, each discounted for its own number of periods, and gives their total present value in one step.

Why does the first payment count as one period away?

The formula assumes each payment is made at the end of its period, which is the ordinary annuity. A payment at the end of year 1 is one year away from today, so it is discounted by one year.

Does the fund really end with nothing left?

Yes, for this formula. The starting amount is set so that interest plus the payments use it up exactly at the last payment. A real fund with changing rates or fees would need a different calculation, which this formula does not include.

What if each payment is made at the start of each period?

That is called an annuity due. Each payment comes one period earlier, so the present value is multiplied by $(1 + r)$ . The amount needed today is larger, because the fund has less time to earn interest before it pays.

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.
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.
Perpetuity: The perpetuity formula gives the amount of money needed today to pay a fixed amount every period forever, using only the payment and the interest rate.
Future Value of an Annuity: 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.