Present Value

In one sentence: 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.

$PV = \frac{FV}{(1 + r)^{n}}$

Read it out loud: “PV equals FV divided by one plus r, all raised to the power n. PV is the present value, the amount you need today. FV is the future value, the amount you want later. r is the interest rate for each period, written as a decimal. n is the number of periods, such as years. So the present value is the future amount divided by one plus the rate, multiplied by itself n times.”

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.

Interest rate, percent and decimal: Interest is the money paid for using money over time, given as a rate such as 4 percent per year. A percent is parts per hundred, so 4 percent is 4 out of 100. In the formula the rate is written as a decimal: divide the percent by 100, so 4 percent becomes 0.04 and 5 percent becomes 0.05.
Exponent (power): multiplying a number by itself: An exponent, also called a power, is a small raised number that says how many times to multiply a number by itself. $(1.04)^{5}$ means $1.04 \times 1.04 \times 1.04 \times 1.04 \times 1.04$ , five copies multiplied together. Any number to the power 1 is itself, and any number to the power 0 is 1.
Period: the step of time the rate refers to: A period is one equal step of time that the rate refers to, most often one year or one month. If the rate is 4 percent per year, a period is a year. The count $n$ must use the same period, so a yearly rate needs the number of years, and a monthly rate needs the number of months.
Growth factor: The growth factor is the number that an amount is multiplied by to grow it over a whole span of time. For one period at 4 percent it is 1.04, and over five periods it is $(1.04)^{5}$ , about 1.216653. Present value divides by this factor, because it reverses the growth from the future amount back to today.

What each symbol means

Symbol	Meaning
$PV$	Present value, the amount of money you need today to reach the target. Measured in dollars. This is the number the formula gives you. A smaller present value means you can start with less money and let interest do more of the work.
$FV$	Future value, the amount you want to have, or are promised, on a later date. Measured in dollars, such as the 2000 dollars in the example. It is the known target, and the formula works backward from it to find the starting amount.
$r$	Interest rate per period, written as a decimal. Four percent per year is r = 0.04 when the periods are years. It has no unit of its own beyond the period it refers to. A bigger rate means a smaller amount is needed today.
$n$	Number of periods, the count of time steps between today and the later date, such as years or months. It must use the same period as the rate: a yearly rate needs the number of years, and a monthly rate needs the number of months.

The formula piece by piece

$FV$: $FV$ is the future amount in dollars, the known target that we want to reach at a later date. It is the starting point of the calculation, and it is always the larger of the two amounts when the rate is positive.
$r$: $r$ is the interest rate per period as a decimal. It is the share of the money that is added each period, so 0.04 means 4 percent of the amount is added each year.
$1 + r$: Add one to the rate. $1 + r$ is the amount that one dollar becomes after one period. If $r = 0.04$ , then $1 + r = 1.04$ , which means one dollar grows to 1.04 dollars after a year.
$(1 + r)^{n}$: Multiply $1 + r$ by itself $n$ times. This is the growth factor over all the periods, the number that turns an amount today into the amount at the later date. For five years at 4 percent it is about 1.216653.
$\frac{FV}{(1 + r)^{n}}$: Divide the future amount by the growth factor. Dividing undoes the growth, so this gives the amount that must be held today.
$PV = \frac{FV}{(1 + r)^{n}}$: The whole formula in one sentence: the present value equals the future amount divided by one plus the rate, raised to the power of the number of periods, which is the amount you need today to reach that future amount.

What the formula is doing, and why

Money you will receive later is worth less than the same amount in your hand today, because money you hold today can earn interest. The present value answers the reverse of the question that compound interest answers. Compound interest asks how much a starting amount grows to. Present value asks how much you must start with to reach a known amount later. The formula gives that starting amount as PV.

Why divide? Growing an amount forward over $n$ periods multiplies it by the growth factor $(1 + r)^{n}$ , which is one factor of $1 + r$ for each period. To go the other way, from the future amount back to today, you divide by that same factor. Dividing by a number bigger than 1 gives a smaller result, which is the mathematical way of saying that money promised later is worth less now.

Why the power $n$ ? Each period, the money earns interest on the interest that was added in earlier periods. That repeated growth is why $(1 + r)$ is multiplied by itself $n$ times, and why the exponent sits on the whole bracket rather than on $r$ alone. Five years at 4 percent is not 5 times 4 percent. It is $1.04$ multiplied by itself five times, which gives a bigger number than the simple total.

The rate is what turns time into a number. Doubling the wait does not halve the amount needed today, because each extra period compounds on the last. If the rate were zero, the present value would equal the future value, because nothing would grow and waiting would cost nothing.

The formula assumes a single payment and one rate that stays the same, with interest added once per period. Cases with several payments need a sum of discounted amounts, which is what the present value of an annuity page covers.

An everyday picture

Think of a seedling that grows by the same fraction every season. If you know how tall the tree must be at a later date, you can work backward: divide by the growth factor for each season to find how tall the seedling must be at the start. The present value is that starting height, and the interest rate plays the role of the growth fraction.

Worked example: How much to deposit today for a 2000 dollar goal in 5 years

A family wants 2000 dollars for a school trip in 5 years. They can put money into a savings account that pays 4 percent per year, with interest added once a year. How much must they deposit today? In the formula, $FV = 2000\ \text{dollars}$ , $r = 0.04$ per year and $n = 5$ years. These numbers are an illustration, not a real account.

Write down the inputs. We list the three numbers before touching the formula, so nothing is forgotten. The target is $FV = 2000\ \text{dollars}$ , the rate is 4 percent per year, written as $r = 0.04$ , and the time is $n = 5$ years. Keeping the rate and the time in the same unit, years, is what makes the formula give a correct answer.
Work out the growth factor. The bracket $1 + r = 1.04$ says that each year the money becomes 1.04 times as large. We multiply that factor by itself five times: $(1.04)^{5} = 1.04 \times 1.04 \times 1.04 \times 1.04 \times 1.04 \approx 1.216653$ . This means one dollar left alone for five years grows to about 1.216653 dollars.
Divide the target by the growth factor. Dividing undoes the growth, so we find how much must be in the account at the start: $PV = \frac{2000\ \text{dollars}}{1.216653} \approx 1643.85\ \text{dollars}$ . Dividing by a number above 1 gives a smaller amount, which fits the idea that the money now is worth less than the money later.
Check the answer by growing it forward. If the family deposits 1643.85 dollars today, the account should reach the target after five years. Multiplying by the growth factor gives $1643.85 \times 1.216653 \approx 2000\ \text{dollars}$ , which matches the goal, so the answer is consistent.
Find the interest the account adds. The gap between the target and the deposit is the interest earned over the five years: $2000 - 1643.85 = 356.15\ \text{dollars}$ . The family puts in about 1644 dollars, and the savings account adds about 356 dollars of interest along the way.

Answer: The family must deposit about 1643.85 dollars today. Over five years the account adds about 356.15 dollars of interest, so the deposit grows to the 2000 dollar goal. A real account may pay a different rate or add interest on a different schedule, and either change would change the answer.

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

Common mistakes

Multiplying the future amount by the growth factor instead of dividing by it. Multiplying gives a bigger number, but when the rate is above zero the present value must be smaller than the future value.
Mixing periods, such as a yearly rate with a count of months. Four percent per year with 60 months gives a wrong answer, so convert the rate or the count first.
Writing the rate as 4 instead of 0.04. The formula needs a decimal, so 4 percent means 0.04. Typing 4 would make the growth factor 5 to the power 5, which is 3125, and the answer would be far too small.
Thinking that $(1 + r)^{n}$ is the same as $1 + rn$ . Simple interest adds the same amount each period, but compound growth multiplies, so the exponent matters more the longer the wait.

Where you will meet it

Deciding how much to save today so that a fund reaches a known goal, such as a deposit, a trip or a school fund.
Comparing a payment offered now with a larger payment promised for later, by putting both on the same footing: value today.
Valuing a loan or a bond by discounting its future payments back to today.
Calculating the net present value of a project by discounting each year’s cash flow to today.

Questions people ask

Why is the present value smaller than the future value?

Money you hold today can earn interest, so a payment due later has to be discounted to show what it is worth now. The growth factor $(1 + r)^{n}$ is bigger than 1 when the rate is above zero, and dividing by it gives a smaller number. Only at a rate of zero are the two amounts equal.

Which rate should I use?

Use the rate that the money can actually earn over the same period as the count of periods. The formula does not choose the rate for you. The 4 percent in the example is only an illustration, and a real account or investment has its own rate, which may change over time. This formula assumes one rate that stays the same.

How is this different from the future value formula?

The future value formula grows an amount forward in time by multiplying by $(1 + r)^{n}$ . The present value formula runs the same idea backward. The two are one relation written in opposite directions: $FV = PV (1 + r)^{n}$ .

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.
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.
Net Present Value (NPV): Net present value says whether a plan is worth its cost: it adds every future cash flow, shrunk back to today’s money, and the total is the plan’s value in dollars today.
Effective Annual Rate: The effective annual rate says what a stated yearly rate really adds up to over a year, once interest is added more than once and earns interest of its own.