Net Present Value (NPV)

In one sentence: 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.

$\text{NPV} = \sum_{t=0}^{n} \frac{C_t}{(1 + r)^{t}}$

Read it out loud: “N P V equals the sum, from t equals zero up to t equals n, of C t divided by one plus r, all raised to the power t. NPV means net present value. C t is the cash flow at time t, r is the discount rate per year, and the sigma sign means add up all these terms.”

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.

Exponents, like $(1.1)^{3}$: A small raised number, the exponent, says how many times to multiply a number by itself. $(1.1)^{3} = 1.1 \times 1.1 \times 1.1 = 1.331$ . An exponent of 0 always gives 1, so $(1.1)^{0} = 1$ , which is why a payment at time zero is not divided at all.
The sigma sign $\sum$: The Greek capital letter $\sum$ , called sigma, means add up a list of terms. The letter under it is the counter, which starts at the value below, here $t = 0$ , and goes up by one each step until it reaches the value on top, $n$ . So $\sum_{t=0}^{n} C_t$ means $C_0 + C_1 + \cdots + C_n$ .
Cash flows and their signs: A cash flow is money that moves in or out at one moment in time, measured in dollars. Money paid out is negative and money received is positive. So a 1000 dollar purchase at the start is written as $C_0 = -1000$ dollars.
Time periods and the index $t$: Time is counted in periods, usually years. The index $t$ is the number of the period: $t = 0$ means now, $t = 1$ means one year from now, and $t = 3$ means three years from now. Each cash flow gets its own time period, so it can be discounted by the right amount.
Discount rate and present value: Money later is worth less than money now, because money in hand can earn a return. The discount rate $r$ is the yearly return you could earn elsewhere, written as a decimal, so 10% per year is $r = 0.10$ . The present value of a future amount is what it is worth in today’s money at that rate.
Net present value (NPV): NPV is the total of all the cash flows of a plan, each one converted into today’s money. A plan with a positive NPV is worth more than it costs at the chosen rate, and a plan with a negative NPV is worth less.

What each symbol means

Symbol	Meaning
$\text{NPV}$	Net present value, the total value of the plan in today’s money, measured in dollars. A positive NPV means the plan is worth more than it costs at the chosen rate, and a negative NPV means it is worth less.
$\sum$	The Greek capital letter sigma, which means add up a list of terms. The letter below it, $t = 0$ , is where counting starts, and the letter above it, $n$ , is where counting stops.
$C_t$	The cash flow at time $t$ , the money that moves in or out at that moment, measured in dollars. Money paid out is negative, such as the price of an oven at time zero, and money received is positive.
$t$	The time period, counted in years from now. $t = 0$ means today, $t = 1$ means one year from now, and so on up to the last period $n$ .
$n$	The last time period, a whole number of years. It is the year of the final cash flow in the plan.
$r$	The discount rate, the yearly return you could earn on your money elsewhere, written as a decimal per year. For example, 10% per year is written 0.10.

The formula piece by piece

$C_t$: Start with one cash flow, the money that moves at time $t$ , measured in dollars. A negative value means money paid out, such as the price of an oven, and a positive value means money received.
$(1 + r)^{t}$: The growth factor for $t$ years. Adding 1 to the rate and raising the result to the power $t$ says how much one dollar invested today grows to after $t$ years at rate $r$ . Dividing by this factor shrinks a future amount back to today’s money.
$\frac{C_t}{(1 + r)^{t}}$: One cash flow in today’s money. The amount is divided by its growth factor, which gives the present value of that single payment.
$\sum_{t=0}^{n} \frac{C_t}{(1 + r)^{t}}$: Add up these present values for every time period, from time zero up to the last period $n$ . The sum is the total value of all the cash flows in today’s money.
$\text{NPV} = \sum_{t=0}^{n} \frac{C_t}{(1 + r)^{t}}$: The whole formula in one sentence: net present value is the sum, over all time periods, of each cash flow divided by one plus the rate raised to the power of its time period.

What the formula is doing, and why

Net present value, or NPV, answers one question: how much is this plan worth today, after counting every payment it makes or costs? A plan usually has money going out at the start, such as the price of a machine, and money coming in later, such as the extra sales the machine brings. The formula puts every cash flow on the same footing, which is today’s money, and then adds them up. The result is one number in dollars.

Why divide by $(1 + r)^{t}$ ? Money received later is worth less than money received now, because money in hand can earn the rate $r$ . A payment of 1 dollar received $t$ years from now is worth $\frac{1}{(1 + r)^{t}}$ dollars today. Dividing each cash flow by $(1 + r)^{t}$ does exactly this. The higher the rate or the longer the wait, the bigger the division and the smaller the value today.

Why add? Each cash flow is converted into today’s money on its own, and only then are they added together. Adding is fair only when every number is in the same kind of money, which is why the division comes first and the sum comes second. The total includes both the cost at time zero and every later gain, so the cost and the gains are compared in one step.

Why the first term is usually negative. The cost of buying something happens at the start, so $C_0$ is usually a negative number, and adding it reduces the total. Its discount factor is $(1 + r)^{0} = 1$ , so the cost is not shrunk at all, because it is already in today’s money.

Reading the answer. If the NPV is positive, the plan earns more than the rate $r$ , so it is worth doing at that rate. If the NPV is negative, the plan earns less than $r$ and loses value in today’s money. If the NPV is exactly zero, the plan earns exactly $r$ . Changing $r$ changes the answer, so the rate is a choice that must always be stated.

An everyday picture

Think of a bag of future paychecks, each one a different amount and each arriving on a different day, plus a price you must pay now. To compare them fairly, you first ask how much each paycheck is worth today. A paycheck that arrives later is worth less today, because you could have put money aside and earned a return on it. The discount rate is like an exchange rate between today’s money and later money. NPV converts every paycheck into today’s money, adds them up, and subtracts the price you pay now.

Worked example: Is a 1000 dollar oven worth buying?

A small shop is thinking about buying an oven for 1000 dollars. The extra sales it brings in are expected to add 400 dollars of net cash at the end of each of the next three years. The shop could earn 10% per year on its money elsewhere, so it uses that as the discount rate. Is the oven worth buying at that rate?

Write each cash flow with its time. The shop pays 1000 dollars today, at time zero, so $C_0 = -1000$ dollars. The oven then brings in 400 dollars at the end of year 1, year 2 and year 3, so $C_1 = C_2 = C_3 = 400$ dollars. Each amount is tagged with its year, because the year decides how much it is shrunk.
Choose the yardstick. The discount rate $r = 0.10$ per year stands for the return the shop could get elsewhere. Each later dollar is compared with what the same money could earn if it were invested from today, which is why the rate must appear in the formula.
Discount the year 1 cash flow. Dividing by $1 + r = 1.1$ shrinks a payment that arrives one year later. So $\frac{400}{1.1} = 363.636$ dollars, which is what the year 1 payment is worth in today’s money.
Discount the year 2 and year 3 cash flows. For two years of waiting we divide by $(1.1)^{2} = 1.21$ , giving $\frac{400}{1.21} = 330.579$ dollars. For three years we divide by $(1.1)^{3} = 1.331$ , giving $\frac{400}{1.331} = 300.526$ dollars. The longer the wait, the bigger the division and the smaller the value.
Add the three present values. The sum over the years is $363.636 + 330.579 + 300.526 = 994.741$ dollars. Keeping three decimal places stops the rounding from blurring the total, and this is what the three future payments are worth together in today’s money.
Add the cost at time zero. The cost is already in today’s money, because its discount factor is $(1 + r)^{0} = 1$ . So $\text{NPV} = -1000 + 994.741 = -5.259 \approx -5.26$ dollars. The negative sign appears because the cost is larger than the value of the gains.
Read the sign and try a lower yardstick. A negative NPV means that at 10% per year the oven gives back slightly less than it costs in today’s money, so it does not quite pay its way. At 8% per year the same payments are worth more, because each one is shrunk less: $-1000 + \frac{400}{1.08} + \frac{400}{1.08^{2}} + \frac{400}{1.08^{3}} \approx 30.84$ dollars, which is positive.

Answer: At a 10% yearly yardstick the oven has an NPV of about minus 5.26 dollars, so it falls short by a small amount. At 8% per year the NPV would be about 30.84 dollars, so the same oven would pass. The answer depends on the rate, which is why the rate must always be stated.

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

Common mistakes

Adding the cash flows without discounting them. 400 dollars received next year is not the same as 400 dollars today, so each payment must be divided by its growth factor before the adding starts.
Entering the upfront cost as a positive number. If the 1000 dollar cost is left positive, the NPV looks far better than it is, because the cost has been added instead of subtracted.
Starting the count at $t = 1$ for money paid today. Money at time zero has the factor $(1 + r)^{0} = 1$ and must not be divided at all, so count today as $t = 0$ .
Writing the rate as a whole number. Use $r = 0.10$ for 10% per year. Writing $r = 10$ makes each divisor $11$ raised to the power $t$ , which gives values that are far too small.

Where you will meet it

Capital budgeting, where a company decides whether to buy a machine, build a factory or launch a new product.
Comparing business plans, where each plan’s cash flows are turned into one value in today’s money so that the plans can be ranked.
Valuing investments such as property or infrastructure, where the costs come first and the income arrives over many years.
Personal decisions, such as whether a solar panel system saves enough electricity money to be worth its price.

Questions people ask

Why is the first cash flow not divided?

Because it happens today. Its growth factor is $(1 + r)^{0} = 1$ , so a payment at time zero is already in today’s money and needs no shrinking.

What discount rate should I use?

Use the return you could earn on similar money with a similar level of risk. The example uses 10% per year as an illustration. Choosing the rate is a judgment, and a different rate can change the answer, so always state the rate you used.

Is a positive NPV always a good decision?

It means the plan is worth more than its cost at the rate you chose, which is a good sign. But the result depends on the cash flow estimates and the rate, so it is only as reliable as those inputs.

How is NPV different from the internal rate of return?

NPV gives a value in dollars at a chosen rate. The internal rate of return gives the rate at which the NPV is exactly zero. Both come from the same formula and answer related questions.

Internal Rate of Return (IRR): The internal rate of return is the yearly rate a plan earns on its money: the single rate at which all of its cash flows exactly balance in today’s money.
Gordon Growth Model: The Gordon growth model says a share is worth its next dividend divided by the gap between the return you want and the rate the dividend grows.
Discounted Return: The discounted return is the total reward an agent can expect from now on, where rewards that arrive later count for a little less.