The Internal Rate of Return, Explained

After reading this you will be able to compute the internal rate of return of any cash-flow series by hand, read the NPV curve that produces it, and know when the number lies to you.

What the IRR actually measures

The internal rate of return (IRR) is the single annual growth rate that makes a project break even in present-value terms. You pay money out at the start, you receive money back over several periods, and somewhere there is a discount rate at which those inflows, discounted, exactly equal what you paid. That rate is the IRR.

Here is the hook. You invest 10000 today and receive 3000, 4000, 5000, and 3000 over the next four years. The raw sum of inflows is 15000, so you got back 1.5 times your money. But money next year is worth less than money today, so the true annual rate is lower than the 50 percent that ratio suggests. The IRR of this series is about 21.86 percent. This article shows exactly where that number comes from.

When to use IRR, and when not to

IRR is useful when you want one comparable percentage for a stream of uneven cash flows. A rental property, a business project, and a bond with irregular coupons can all be reduced to a single annual rate, which makes them easy to line up against a hurdle rate such as your cost of capital.

It fails, or misleads, in three situations. First, when cash flows change sign more than once (out, in, out again), a series can have several IRRs or none. Second, when you compare two projects of very different size or length: a small project with a high IRR can create less total wealth than a big project with a lower IRR. Third, when interim cash flows cannot realistically be reinvested at the IRR itself. For ranking projects, net present value at a rate you actually believe is the safer yardstick. Compute it with the NPV Calculator.

If your whole return arrives as a single lump at the end (one buy, one sell, nothing in between), you do not need IRR. Use the CAGR Calculator for the annual growth rate, or the ROI Calculator for the simple and annualized return.

The formula and the intuition

Start with net present value. If you invest C_0 today and receive cash flow C_t at the end of period t, then at discount rate r the net present value is:

\text{NPV}(r) = -C_0 + \sum_{t=1}^{n} \frac{C_t}{(1+r)^t}

Here C_0 is the initial outlay (a positive number that enters with a minus sign because it leaves your pocket), C_t is the inflow at the end of period t, n is the number of periods, and r is the annual discount rate. Each future flow is divided by (1+r)^t because a sum t years away, growing at r per year, would have needed to start smaller today.

The IRR is the value of r that makes this expression zero:

-C_0 + \sum_{t=1}^{n} \frac{C_t}{(1+\text{IRR})^t} = 0

There is no algebra that solves this for r in the general case, because it is a polynomial of degree n in \frac{1}{1+r}. So the calculator searches. It tries rates, watches whether NPV is positive or negative, and narrows the range until NPV is close enough to zero. The intuition: higher discount rates shrink future inflows, so NPV falls as r rises. The IRR is the crossing point where NPV passes through zero.

A worked example with the demo numbers

Investing 10000 for four uneven inflows

Use the demo data: an initial investment of 10000 and inflows of 3000, 4000, 5000, 3000. First check that a rate near 21.86 percent really zeroes the NPV.

Discount each flow at r = 0.2186. Year 1: 3000 / 1.2186 = 2461.8.
Year 2: 4000 / 1.2186^2 = 4000 / 1.485 = 2693.7.
Year 3: 5000 / 1.2186^3 = 5000 / 1.810 = 2762.9.
Year 4: 3000 / 1.2186^4 = 3000 / 2.206 = 1359.9.
Sum the discounted inflows: 2461.8 + 2693.7 + 2762.9 + 1359.9 = 9278.3. Subtract the outlay: 9278.3 - 10000 = -721.7.

That is close to zero but slightly negative, which tells you the true IRR is a touch below 21.86 percent. Refining the search gives an IRR near 21.86 percent once you carry more decimal places (the small residual above comes from rounding each step to four significant figures). To see the sensitivity, compare NPV at two rates:

NPV of the demo series at three discount rates
Rate r	Discounted inflows	NPV
10%	11569	1569
21.86%	10000	0
30%	9081	-919

At 10 percent the project looks good (positive NPV of 1569). At 30 percent it destroys value (negative 919). The break-even rate sits between them, and that is the IRR.

Reading the NPV curve

The calculator charts NPV against the discount rate. For a normal series (one outflow, then inflows) the curve slopes downward and crosses zero once. The crossing is the IRR. The chart below plots the demo series.

The curve falls as the rate rises and crosses zero near 0.2186. Left of the crossing NPV is positive, right of it NPV is negative.

Note that the points in that chart use exact arithmetic, so the zero crossing sits at 0.2186 even though the marker line is drawn there for reference. The steeper the curve near the crossing, the more precisely the IRR is pinned down; a flat curve means small data changes move the IRR a lot.

Common mistakes

The most damaging error is trusting IRR when cash flows change sign more than once. A series like -10000, +30000, -22000 can produce two IRRs (both mathematically valid), and the calculator can only report one. If your project has later outflows (a mine that must be restored, equipment replaced mid-life), read NPV instead of IRR.

A second error is mixing period lengths. IRR assumes each cash flow sits one full period after the previous one. If your flows are quarterly, the IRR you compute is a quarterly rate; annualize it with (1+r_q)^4 - 1, not by multiplying by four.

IRR assumes every interim cash flow is reinvested at the IRR itself. A project with a 40 percent IRR only delivers 40 percent overall if you can keep redeploying the cash at 40 percent, which is rare. When you cannot, your realized return is lower, and NPV at a realistic rate is the honest number.

How IRR relates to other measures

IRR, NPV, CAGR, and ROI answer overlapping questions. NPV gives a currency amount at a rate you choose. IRR gives the rate at which NPV is zero. For a single-in, single-out investment IRR equals the CAGR. For total return without timing, ROI is simpler. Reach for the tool that matches your question.

If you are projecting a portfolio with regular contributions rather than analyzing a fixed set of flows, use the Investment Growth Calculator. To see how inflation eats into a nominal rate, the Inflation-Adjusted Return Calculator applies the Fisher equation. To measure the drag from fund costs, try the Investment Fee Impact Calculator. For income-focused holdings there are the Dividend Yield Calculator and the Dividend Reinvestment (DRIP) Calculator. For trade-level work, the Stock Average Price Calculator, the Position Size Calculator, and the Crypto Profit Calculator cover cost basis, sizing, and fee-inclusive profit.

Frequently asked questions

Why does my project have no IRR?

If every cash flow has the same sign, or if the discounted inflows never reach the outlay at any rate, NPV never crosses zero, so there is no real IRR. A series of pure outflows or a project whose inflows are too small to recover the investment both produce this. Check the numbers: if total inflows are below the outlay, the IRR is negative or undefined.

Can IRR be negative?

Yes. If you invest 10000 and receive back only 8000 in total, you lost money and the IRR is negative. For the demo series, inflows total 15000 against a 10000 outlay, so the IRR is comfortably positive at about 21.86 percent.

Is a higher IRR always better?

No. A project with a 30 percent IRR on a 1000 investment adds less wealth than a 15 percent IRR on a 100000 investment. IRR ignores scale. Rank competing projects by NPV at your cost of capital when the sizes differ.

What period does the IRR represent?

Whatever period your cash flows are spaced by. If each flow is one year apart, the IRR is annual. If they are monthly, it is monthly, and you convert with (1+r_m)^{12} - 1 to get the annual equivalent.

Why does IRR differ from the average return I calculated?

A simple average ignores timing and compounding. Averaging the yearly cash flows of the demo series as a fraction of 10000 gives 37.5 percent, but that number double-counts money you never had at once and ignores discounting. The IRR of 21.86 percent respects when each amount actually arrives.

Try it yourself in the tool