What Is Internal Rate of Return (IRR)? Formula and Examples

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The internal rate of return (IRR) is the discount rate at which the net present value (NPV) of all of an investment’s cash flows equals zero. Put another way, it is the single annual rate of return that makes the present value of everything an investment pays out equal to the present value of everything put into it. Analysts use IRR to estimate how profitable a project, acquisition, property or fund could be, and to compare that estimate with the minimum return the money needs to earn.

What IRR Means

The “internal” in internal rate of return means the rate depends only on the investment’s own cash flows and their timing. Outside rates such as the company’s cost of capital, inflation or market rates come in afterward, when you decide whether the IRR is high enough.

A useful way to picture IRR is as the interest rate on an imaginary savings account. Suppose you deposited the investment’s outflows into that account and withdrew its inflows on the same dates. The IRR is the account interest rate that would leave a balance of exactly zero after the last cash flow. If a project has an IRR of 12%, its cash flows behave as if your money earned 12% a year, compounded, for as long as it stayed invested.

Because the same formula applies to any set of cash flows, IRR works across very different investments: a new factory, a rental building, a software startup, a bond held to maturity or an equipment lease. That is a large part of why it is used to rank opportunities. Two projects with the same IRR can still carry very different risk, size and duration.

In most applications IRR is a forecast built on estimated cash flows. The return actually earned will usually differ, sometimes widely, and will rarely arrive as the same rate every year.

The IRR Formula

IRR is defined by setting the net present value equation equal to zero:

0 = CF0 + CF1/(1 + IRR)^1 + CF2/(1 + IRR)^2 + ... + CFn/(1 + IRR)^n

Where:

• CFt = net cash flow in period t (inflows positive, outflows negative)
• CF0 = cash flow at the start, usually the initial investment, entered as a negative number
• t = the period number, from 0 to the final period n
• IRR = the internal rate of return, the unknown you solve for

The NPV formula is identical except that it uses a chosen discount rate r and returns a dollar amount. The IRR formula fixes the result at zero and asks which rate produces it. That makes IRR the break-even discount rate for the project: discount the cash flows at a lower rate and NPV is positive (for a conventional project), discount them at a higher rate and NPV turns negative.

The periods do not have to be years. If cash flows are monthly, the formula gives a monthly IRR, which must be annualized before it is compared with annual rates. A monthly rate of 1% compounds to (1.01)^12 - 1, or about 12.68% a year, not 12%.

How to Calculate IRR

The one-period and two-point case

When an investment has only one outflow and one inflow, IRR can be solved directly:

IRR = (Ending cash flow / Initial investment)^(1/n) - 1

Hypothetical example: you invest $1,000 today and receive $1,210 in two years. IRR = (1,210 / 1,000)^(1/2) - 1 = 1.10 - 1 = 10%. Check: $1,210 / (1.10)^2 = $1,000, so NPV at 10% is zero.

Once there are several cash flows at different dates, the equation becomes a polynomial in (1 + IRR) with no practical algebraic solution, so the rate is found by trial and error.

Trial and error with a multi-year project

Hypothetical example: a company considers a machine that costs $10,000 today and is expected to produce after-tax cash flows of $3,000, $4,000, $4,000 and $3,000 over the next four years.

Step 1: Pick a starting rate and compute NPV. At 10%:

• $3,000 / 1.10 = $2,727.27
• $4,000 / 1.21 = $3,305.79
• $4,000 / 1.331 = $3,005.26
• $3,000 / 1.4641 = $2,049.04

Present value of inflows = $11,087.36, so NPV = $11,087.36 - $10,000 = +$1,087.36. NPV is positive, so the IRR is higher than 10%.

Step 2: Raise the rate and recompute. The results at a few rates:

Step 3: NPV changes sign between 14% and 15%, so the IRR lies between them. Linear interpolation gives an approximation:

IRR ≈ 14% + 185.56 / (185.56 + 21.41) × 1% ≈ 14.9%

Recomputing at 14.9% gives an NPV of roughly -$1, close enough to zero for a decision. The machine’s IRR is about 14.9%.

Later cash flows can also be negative, for example a major overhaul in year three or a cleanup cost at the end. They stay in the equation with their sign and, as covered under limitations, can change how many IRRs the equation has.

How to calculate IRR in Excel or Google Sheets

Spreadsheets run the same search automatically. Excel and Google Sheets both provide three related functions.

IRR for evenly spaced cash flows:

1. List the cash flows in chronological order in one column or row, starting with the initial investment as a negative value.
2. Include every period, entering 0 for any period with no cash flow so the spacing stays correct.
3. In an empty cell, type =IRR(values), where values is the range, for example =IRR(B2:B6) for the machine above. The result is about 14.9%.
4. An optional second argument, =IRR(values, guess), gives the function a starting point. Excel starts at 10% if no guess is supplied. If the function cannot converge, it returns a #NUM! error, and a different guess may solve it.

The function assumes the cash flows occur at equal intervals and that the first value falls at the start of the first period. The interval is whatever your data uses, so monthly flows produce a monthly IRR.

XIRR for irregular dates: =XIRR(values, dates) takes a second range with the actual date of each cash flow and returns an annualized rate. This is the better choice for real investment histories, such as a brokerage account with deposits on random days, a real estate deal that closes mid-month, or a private fund with irregular capital calls and distributions.

MIRR for a stated financing and reinvestment rate: =MIRR(values, finance_rate, reinvest_rate) calculates the modified internal rate of return, explained below.

Irregular timing and a missing initial cash flow

IRR needs at least one negative and one positive cash flow. If every value in the range is positive (for example, you list future receipts but leave out what you paid), no discount rate can bring NPV to zero and the function returns an error. The fix is to add the purchase price or initial investment as a negative value at its correct date.

If the investment did not start at the beginning of a regular period, or the cash flows are not evenly spaced, IRR will misstate the result because it treats every value as one period apart. Use XIRR with actual dates instead. When the first cash flow is a positive amount (a loan received, for instance) followed by repayments, IRR still works, but it then measures the cost of the financing rather than the return on an investment.

How to Interpret IRR

The IRR rule

The basic decision rule compares IRR with a required rate of return, often called the hurdle rate:

• IRR above the hurdle rate: the project earns more than the minimum required, and it can be accepted on this test.
• IRR below the hurdle rate: the project does not earn enough to justify the capital, and rejecting it may be the better choice.
• IRR equal to the hurdle rate: NPV is zero at that rate, so the project just covers the required return.

For conventional, independent projects, where outflows come first and all later cash flows are inflows, this rule gives the same answer as the NPV rule.

Comparing IRR with WACC and the required rate of return

Companies usually judge IRR against their weighted average cost of capital (WACC), which blends the cost of each source of capital: common stock, preferred stock, bonds and other long-term debt, weighted by their share of the financing. A project whose IRR exceeds WACC earns more than the capital costs, which is the condition for adding value.

Many firms set a required rate of return (RRR) above WACC to leave a margin for estimation error and for risk that varies by project. A riskier venture may face a higher hurdle than a routine equipment upgrade. Clearing the RRR does not guarantee approval: management also weighs how much IRR exceeds the hurdle, how much capital the project needs, how long it takes, and its strategic fit.

IRR can also be compared with returns available in financial markets. If no internal project offers an IRR above what the company could earn from securities with similar risk, investing in the market or returning cash to shareholders may be the better use of funds.

Analysts typically pair IRR with NPV at several discount rates and with scenario analysis. Showing NPV under pessimistic, base and optimistic cash flow assumptions indicates how fragile the result is.

What IRR Is Used For

Capital budgeting. The classic use is choosing among projects: building a new plant versus expanding an existing one, replacing equipment, launching a product line or entering a new market. IRR sits alongside NPV and payback period in most discounted cash flow evaluations of these decisions.

Share buybacks and acquisitions. A company that spends heavily on repurchasing its own shares is implicitly betting that its stock offers a better return than the alternatives, such as new outlets or acquisitions. Estimating the IRR of each use of cash gives a way to compare them on the same footing.

Real estate. Property investors calculate IRR from the purchase price and closing costs (outflows), net operating cash flows during the holding period, and the expected sale proceeds at exit. Two versions are common. Unlevered IRR uses the property’s cash flows before debt, measuring the asset itself. Levered IRR uses cash flows after mortgage payments, with only the investor’s equity as the initial outflow, so it reflects the effect of borrowing. Leverage can raise levered IRR above unlevered IRR when the property earns more than the cost of the debt, and it pushes levered IRR lower when the property underperforms.

Private equity and venture funds. Funds report IRR on capital called from investors and distributions paid back. Because IRR is highly sensitive to timing, these funds are usually also judged on multiples of invested capital, which show how much money was made in total.

Portfolio returns. IRR is the calculation behind the money-weighted rate of return (MWRR). It accounts for the size and timing of every deposit and withdrawal, so it measures the return an investor actually experienced. A time-weighted return, by contrast, strips out the effect of cash flows and is used to judge a manager who does not control when clients add or remove money.

Insurance, annuities and personal decisions. IRR can compare products with complicated cash flows, such as annuities or permanent life insurance, by treating premiums as outflows and benefits or surrender values as inflows. For a life insurance policy, the IRR to the beneficiary is extremely high if the insured dies after only a few premium payments and falls as the policy ages, which shows how strongly the result depends on the assumed timeline. Individuals can also use IRR to judge a rental unit, a solar installation or a paid-off lease.

IRR vs. NPV

NPV and IRR use the same cash flows and the same discounting, but they answer different questions. NPV says how many dollars of value a project creates at a given discount rate. IRR says what rate of return the project earns, without stating how big that return is in dollars.

For a single conventional project, they point the same way. For mutually exclusive projects, where you can pick only one, they can disagree.

Hypothetical example, different sizes:

• Project Small: invest $1,000, receive $1,500 in one year. IRR = 50%. NPV at 10% = $1,500 / 1.10 - $1,000 = $363.64.
• Project Large: the $10,000 machine above. IRR ≈ 14.9%. NPV at 10% = $1,087.36.

Ranked by IRR, Small wins easily. Ranked by NPV, Large creates about three times as much value. If the firm has access to capital at 10% and can only do one, NPV gives the better answer: a 50% return on $1,000 adds less wealth than a 14.9% return on $10,000. This is why corporate finance texts treat NPV as the primary criterion and IRR as a supporting measure, especially when projects compete for the same slot.

IRR vs. ROI

Return on investment (ROI) measures total gain or loss relative to the amount invested:

ROI = (Ending value - Beginning value) / Beginning value × 100

ROI is not annualized and ignores when the gains arrive. In the two-year example above, $1,000 growing to $1,210 has an ROI of 21% but an IRR of 10% a year. Over a single one-year period with one outflow and one inflow, the two figures are the same. Over longer periods they diverge, and ROI gives no credit for getting money back sooner.

ROI’s strength is simplicity: it applies to almost any outlay with a measurable outcome, from a marketing campaign to a kitchen renovation. Its weakness is that the term is used loosely and can mean different things to different people. IRR has a precise mathematical definition, which is one reason investment professionals prefer it when cash flows are spread over time.

IRR vs. CAGR

The compound annual growth rate (CAGR) is the constant annual rate that takes a starting value to an ending value:

CAGR = (Ending value / Beginning value)^(1/n) - 1

When an investment has only a starting amount and an ending amount, CAGR and IRR give the same answer. The $1,000 that becomes $1,210 in two years has a CAGR of 10% and an IRR of 10%.

The difference appears with interim cash flows. CAGR ignores dividends paid out, rent collected, additional contributions and partial withdrawals unless they are folded into the ending value. IRR includes each cash flow at its own date. For a rental property that pays income every year and is sold at the end, CAGR on the price alone understates the return, while IRR captures both the income stream and the sale.

IRR vs. MIRR

Standard IRR carries an implicit assumption: interim cash flows are treated as if they can be reinvested at the IRR itself. For a project with a 40% IRR, that means assuming every cash inflow along the way also earns 40% until the end, which is rarely realistic.

The modified internal rate of return (MIRR) replaces that assumption with rates you choose. Positive cash flows are compounded forward to the final period at a reinvestment rate (often the cost of capital), negative cash flows are discounted back to the start at a finance rate, and MIRR is the single rate connecting the two:

MIRR = (FV of positive cash flows at reinvestment rate / PV of negative cash flows at finance rate)^(1/n) - 1

Hypothetical example: take the $10,000 machine and assume inflows are reinvested at 10%. The future value of the inflows at the end of year four:

• $3,000 × 1.10^3 = $3,993
• $4,000 × 1.10^2 = $4,840
• $4,000 × 1.10 = $4,400
• $3,000 × 1.00 = $3,000

Total = $16,233. MIRR = (16,233 / 10,000)^(1/4) - 1 ≈ 12.9%, compared with an IRR of about 14.9%. The gap is the effect of assuming 10% reinvestment rather than 14.9%. MIRR also produces one answer even when cash flows change sign several times, which removes one of IRR’s technical problems.

Limitations of IRR

Multiple IRRs. A conventional project has one sign change in its cash flows (negative, then positive) and therefore one IRR. When cash flows change sign more than once, the equation can have several solutions. Hypothetical example: a project costs $100, returns $230 in year one, then requires a $132 cleanup payment in year two. NPV equals zero at both 10% and 20%, and it is positive only for discount rates between them. Neither IRR alone tells you whether to accept the project at a 15% or 25% cost of capital. NPV at the actual hurdle rate, or MIRR, resolves the ambiguity.

Scale. IRR is a percentage and ignores dollar size. As the Small versus Large example shows, a high IRR on a small amount can create less value than a moderate IRR on a large one.

Duration. IRR favors projects that return money quickly. Hypothetical example: Project Short turns $1,000 into $1,200 after one year (IRR 20%). Project Long turns $1,000 into $2,000 after five years (IRR about 14.9%). At a 10% cost of capital, Short’s NPV is about $90.91 and Long’s is about $241.84. Short looks better by IRR, yet Long creates more value, assuming the firm can wait and the year-five cash flow is as reliable as the year-one cash flow. Comparing NPV, or ROI alongside IRR, helps here.

Reinvestment assumption. Because IRR treats interim cash flows as reinvested at the IRR itself (see MIRR above), very high IRRs overstate what the capital will actually earn.

A single discount rate. IRR is compared with one hurdle rate, but for long projects the appropriate discount rate may change over time as interest rates, risk or financing change. NPV can use a different rate for each period. IRR cannot.

Sensitivity to estimates and timing. IRR is only as good as the cash flow projections behind it, and those are hard to forecast years ahead. It is also sensitive to timing: pulling a distribution forward or delaying a capital outlay can raise IRR noticeably even when the total profit barely changes. When comparing reported IRRs, check how the cash flow dates were set.

Nominal versus real. IRR on nominal cash flows includes inflation. Comparing it with a hurdle rate that excludes inflation, or the reverse, mixes two different measures. To convert a nominal IRR to a real one, use (1 + nominal IRR) / (1 + inflation rate) - 1, drawing the inflation assumption from a source such as U.S. inflation rate history.

Other considerations. IRR does not capture risk, strategic value, management effort or how easily the investment can be exited. A project with a slightly lower IRR, much less up-front capital and fewer complications can be the better choice.

What Is a Good IRR?

There is no universal good IRR. An IRR is good when it exceeds the return the investor needs for that level of risk, which in a company is the WACC or the project’s hurdle rate, and for an individual is the return on the next-best alternative of similar risk. A 9% IRR can be excellent for a low-risk, stable infrastructure asset and inadequate for an early-stage venture where many investments fail.

When comparing opportunities, a higher IRR is better only when risk, size, duration and confidence in the cash flow estimates are roughly similar. If a modestly lower IRR comes with far less risk or effort, many investors will reasonably choose it. A practical check uses three numbers together: IRR for the rate, NPV at your hurdle rate for the dollar value created, and a multiple or ROI for the total money returned.

FAQs

The Bottom Line

IRR matters to anyone weighing projects, property, funds or financial products with cash flows spread over time. It works best as one check among several: compare it with WACC or a hurdle rate, confirm the dollar value with NPV at that rate, and look at a multiple or ROI for total money returned. When projects differ in size or length, or cash flows change sign more than once, rely on NPV or MIRR.