Internal Rate of Return (IRR)

In one sentence: 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.

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

Read it out loud: “Zero 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. The number r that makes this true is the internal rate of return, IRR. It is the yearly rate at which the net present value of the plan is exactly zero.”

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 $1.05^{2}$: A small raised number, the exponent, says how many times to multiply a number by itself. So $1.05^{2} = 1.05 \times 1.05 = 1.1025$ . An exponent of 0 always gives 1, so a cash flow 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.
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 a yearly return 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 a plan’s cash flows, each converted into today’s money at a chosen rate. The internal rate of return is the rate at which this total comes out exactly zero.
Trial and error: Trial and error means guessing a value, testing it, and using the result to choose a better guess. Each test is called a trial. When the answer changes sign between two guesses, the true answer lies between them, so the search can get closer step by step.

What each symbol means

Symbol	Meaning
$r$	The internal rate of return, the unknown we are solving for. It is a yearly rate written as a decimal, so 10% per year is $r = 0.10$ . It is the rate that makes the sum come out to exactly zero.
$\sum$	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.

The formula piece by piece

$C_t$: Start with one cash flow, the money that moves at time $t$ , in dollars. In this plan $C_0$ is negative because the oven is bought, and $C_1$ and $C_2$ are positive because the oven brings in cash.
$(1 + r)^{t}$: The growth factor over $t$ years at the unknown rate $r$ . Adding 1 to the rate and raising it to the power $t$ says how much one dollar grows in $t$ years. Dividing by this factor brings a later amount back to today.
$\frac{C_t}{(1 + r)^{t}}$: One cash flow in today’s money. The amount is divided by its growth factor, giving the present value of that single payment at the rate $r$ .
$\sum_{t=0}^{n} \frac{C_t}{(1 + r)^{t}}$: Add up the present values of every cash flow, from time zero up to the last period $n$ . At a given rate, this total is the NPV of the plan.
$0 = \sum_{t=0}^{n} \frac{C_t}{(1 + r)^{t}}$: The whole formula in one sentence: the internal rate of return is the rate $r$ at which the present values of all the cash flows add up to exactly zero.

What the formula is doing, and why

The internal rate of return, or IRR, is the rate a plan earns on the money tied up in it. It comes from the net present value formula by setting the answer to zero. A plan with an IRR of 10% per year breaks even at a discount rate of 10%, which means its gains, measured in today’s money, exactly equal its cost. The formula is the same as the NPV formula, but now $r$ is the unknown that must be found.

Why set the NPV to zero? The NPV tells you how much a plan is worth at a chosen rate. If you lower the rate, future money is worth more today and the NPV rises. If you raise the rate, the NPV falls. Somewhere in between, the NPV crosses zero, and at that exact rate the plan has broken even. That rate is the IRR, and it measures the plan’s own return instead of a rate chosen by you.

Why trial and error? The unknown $r$ sits in the bottom of each fraction, raised to a different power for each year. For a plan with several years of cash flows there is no simple formula that isolates $r$ . So we try a rate, see whether the NPV is positive or negative, and move the guess toward zero. Each try narrows the range where the answer must lie. Spreadsheets and financial calculators have an IRR function that does these tries for you.

How to read the answer. Compare the IRR with the return you could earn elsewhere, which is sometimes called the hurdle rate. If the IRR is higher, the plan earns more than that alternative. If it is lower, the plan earns less. The IRR is a rate, so it does not show the size of the gain in dollars. A small project can have a high IRR and still add less money than a large one.

When the answer can mislead. If the sign of the cash flows switches more than once, for example money out, money in and then money out again for cleanup, the NPV can cross zero at more than one rate, and the search can find more than one IRR. In that case, check the NPV at the rate you are using, as the example does, before you trust the answer.

An everyday picture

Think of a puzzle with a dial. A small shop has paid 1000 dollars for a machine that pays back in two installments, but nobody has told you the return it earns. You find it by turning the dial. Each test rate tells you what the installments are worth today. If a test rate makes them worth more than the price, the dial is set too low, and if they are worth less, it is set too high. Turning the dial until the two sides balance reveals the hidden return, and that return is the IRR.

Worked example: What return does an oven earn?

The same oven from the NPV page costs 1000 dollars. This time the return is unknown. We only know the cash it brings in: 550 dollars at the end of year 1 and 605 dollars at the end of year 2. What internal rate of return does the purchase earn?

Write each cash flow with its time. The shop pays 1000 dollars at time zero, so $C_0 = -1000$ dollars. The oven brings in 550 dollars at the end of year 1, so $C_1 = 550$ dollars, and 605 dollars at the end of year 2, so $C_2 = 605$ dollars. Every amount is tagged with its year before the formula is used.
Set up the equation. The IRR is the rate $r$ that makes the sum of the present values equal zero, so $0 = -1000 + \frac{550}{1 + r} + \frac{605}{(1 + r)^{2}}$ . We do not know $r$ yet, so we will try values and watch whether the answer is positive or negative.
Try a low rate of 5% per year. Set $r = 0.05$ . The year 1 cash is divided by 1.05, and the year 2 cash is divided by $1.05^{2} = 1.1025$ . So the sum is $-1000 + \frac{550}{1.05} + \frac{605}{1.1025} \approx -1000 + 523.81 + 548.75 = 72.56$ dollars. The result is positive, so 5% is too low: at that rate the cash is worth more than the price.
Try a high rate of 15% per year. Set $r = 0.15$ . Now the divisors are 1.15 and $1.15^{2} = 1.3225$ , so the sum is $-1000 + \frac{550}{1.15} + \frac{605}{1.3225} \approx -1000 + 478.26 + 457.47 = -64.27$ dollars. The result is negative, so 15% is too high.
Narrow the range. The NPV is positive at 5% and negative at 15%, so the IRR must lie between those two rates. The midpoint of that range is 10%, so 10% is the next guess to test. Each new guess shrinks the gap in which the answer can sit.
Test 10% per year. Set $r = 0.10$ . Then $1 + r = 1.1$ and $(1 + r)^{2} = 1.21$ , so $\frac{550}{1.1} = 500$ and $\frac{605}{1.21} = 500$ . The sum is $-1000 + 500 + 500 = 0$ dollars, exactly zero, so the IRR is 10% per year.
Check the answer by putting it back. Each year’s cash is worth exactly 500 dollars in today’s money at 10%, and the two together make back the 1000 dollar cost. Because the sum is zero, the test passes, and the NPV at the IRR leaves nothing over.

Answer: The oven purchase has an internal rate of return of 10% per year. The money tied up in the oven earns 10% a year, so the purchase is worth doing if the shop cannot earn more than 10% elsewhere. The cash comes to 155 dollars more than the price, but the IRR measures the rate, not that total.

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

Common mistakes

Trying to solve for $r$ by rearranging like an ordinary equation. The rate sits in the bottom of the fractions, so simple algebra does not isolate it. Use trial values, as in the example, or a spreadsheet IRR function.
Stopping at a rate whose NPV is small without checking its sign. The IRR is the rate where the NPV is exactly zero, so a rate with a small positive NPV is not yet the answer, and you should test a higher rate.
Reading the IRR as the dollar profit. The IRR is a yearly rate, not an amount of money. The dollar profit in the example is 155 dollars, which is the sum of the cash flows minus the cost.
Comparing IRRs of plans of very different size without checking the NPV. A small plan can show a high IRR and still add less total value than a large plan with a lower IRR.

Where you will meet it

Ranking investment projects inside a company, alongside net present value, so that managers can see the yearly return each project offers.
Bond analysis, where the yield to maturity of a bond is the IRR of its promised payments.
Private equity and venture capital reports, which quote an IRR as the annual return on money that goes in and comes out at different times.
Comparing energy-saving upgrades, such as a new boiler, by the yearly return on the money spent on them.

Questions people ask

Is the IRR the same as the interest rate on a loan?

For a standard loan, yes. Seen from the lender’s side, the money out is the amount lent and the money in is the repayments, and the only rate that balances those two is the interest rate charged.

Why can an IRR be misleading?

When the cash flows switch between money in and money out more than once, the NPV can cross zero at more than one rate, so there can be several IRRs. Then no single rate describes the plan well, and the NPV at a chosen rate is the safer test.

How do I find an IRR without trial and error?

Use a spreadsheet function called IRR, or a financial calculator. The computer does the trial and error for you by testing rates until the NPV is close to zero. Knowing the idea helps you read the result.

Should I always pick the plan with the highest IRR?

No. A plan with a high IRR may be small and add little money in total. Check the NPV at your rate as well, because it shows how much value each plan adds in today’s money.

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.
Compound Annual Growth Rate (CAGR): CAGR says how fast something grew each year, as if it grew by the same share every year, from its starting value to its ending value.
Return on Investment (ROI): Return on investment says how much profit you made for each dollar you spent, so you can compare a small investment with a large one.