AtlasCalcsUtility Calculation Suite
2026-07-31 8 min

Internal Rate of Return Calculator

Calculate the internal rate of return (IRR) of a project with constant annual cash flows using bisection. Compare IRR against a reference rate to get an accept or reject decision.

Calculator

Unit System
Initial Investment Help
$

Upfront cost of the project or asset

Annual Cash Flow(/yr) Help
$

Net cash flow received each year of the project

Project Life(yr) Help
yr

Number of years the cash flows continue

Reference Rate (hurdle)(%) Help
%

Minimum acceptable return; IRR is compared against this rate

Results update live as you type

Results

Optimal

Internal Rate of Return

12.38%

Discount rate at which NPV equals zero

NPV at Reference Rate

6142.03$

Net present value evaluated at the reference rate

Total Cash Flow

140000.00$

Annual cash flow times project life

Net Profit (undiscounted)

40000.00$

Total cash flow minus the initial investment

Decision (1 = Accept, 0 = Reject)

1.00

Accept when IRR exceeds the reference rate

Visualizations

NPV at Reference Rate

6,142$
-100,000100,000

Formula

The internal rate of return is the discount rate that makes the net present value exactly zero. The calculator finds it by bisection: it repeatedly narrows the rate interval where NPV changes sign until the crossing point is found.

0 = −Investment + Σ CFt / (1 + IRR)^t

Alternative Form

Accept when IRR > ReferenceRate

Variables

SymbolNameUnitDescription
InvestmentInitial investment$Upfront cost of the project
CFtAnnual cash flow$/yrNet cash flow received each year
tYearyrYear index from 1 to the project life
ReferenceRateReference rate%Minimum acceptable return for the accept or reject decision

Explanation

IRR is the break-even earning rate of a project. If the project earns exactly the IRR, it pays back the investment and no more. Comparing the IRR to the reference rate tells you whether the project beats the minimum acceptable return.

Derivation

NPV is a decreasing function of the discount rate for conventional cash flows. The bisection method starts with a low rate where NPV is positive and a high rate where it is negative, then halves the interval 60 times until the rate where NPV crosses zero is pinned down to high precision.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Evaluate NPV at the low end of the rate range (0%).

NPV(0%) = −Investment + Σ CFt
Result: NPV at 0%: $40000.00
Step 2

Evaluate NPV at the high end of the rate range (50%).

NPV(50%) = −Investment + Σ CFt / 1.5^t
Result: NPV at 50%: $-51374.49
Step 3

Bisect the rate interval 60 times to find where NPV crosses zero.

IRR = rate where NPV = 0
Result: IRR: 12.38%
Step 4

Evaluate NPV at the reference rate.

NPVref = NPV(ReferenceRate)
Result: NPV at Reference Rate: $6142.03
Step 5

Compare IRR with the reference rate for the decision.

Accept if IRR > ReferenceRate
Result: Decision: Accept
Final Answer
12.38

Worked Examples

See how the calculator works with different scenarios.

Scenario

A $100,000 machine yields $28,000/yr for 5 years. The company reference rate is 10%.

Inputs

Investment:$100,000
Cash Flow:$28,000/yr
Life:5 yr
Reference Rate:10%

Calculation

At 12%: PV = 28,000 × 3.60478 = $100,933.8 → NPV = +$933.8 At 12.5%: PV = 28,000 × 3.56061 = $99,697.1 → NPV = −$302.9 IRR ≈ 12.4% (NPV crosses zero here) NPV at 10% = $6,142.03 IRR 12.4% > 10% → Accept

Result

IRR: ≈ 12.4%, NPV at 10%: $6,142.03, Decision: Accept

Interpretation

The machine earns about 12.4% per year, above the 10% reference rate. The project is accepted and the NPV at the reference rate is comfortably positive.

Understanding Your Result

Your Result: Internal Rate of Return: 12.38 %

Typical range: Equipment projects typically show IRRs of 10-30%. Energy savings projects often land at 8-20%. An IRR above the cost of capital means value creation; below it, the project should be rejected unless required for strategic reasons.

What This Means

IRR is the discount rate at which a project NPV equals zero — the break-even earning rate of the investment. Comparing it with the reference rate tells you whether the project beats the minimum acceptable return.

Why It Matters

IRR is the most quoted capital budgeting metric because it summarizes a project as a single percentage. Management can compare it directly with the cost of capital, interest rates, and other opportunities.

Professional Interpretation

Professionals treat IRR as a screening metric and NPV as the decision metric. For mutually exclusive projects, NPV ranks correctly while IRR can mislead because of reinvestment assumptions and project size differences.

Recommendations

  • •Compare IRR against the company cost of capital, not zero
  • •Use NPV as the final arbiter between competing projects
  • •Watch for projects where no IRR exists or multiple crossovers occur
  • •Verify IRR with the NPV at the reference rate
  • •Re-run IRR when cash flow estimates change materially

Educational Content

Introduction

The internal rate of return answers a natural question: what rate of return does this project earn? Unlike ROI, it accounts for the timing of every cash flow by finding the discount rate that exactly recovers the investment.

Concept Explanation

NPV falls as the discount rate rises. The IRR is the rate where the NPV curve crosses zero. The bisection algorithm brackets the zero crossing between a low and high rate and narrows the bracket by halves until the crossing is found.

Applications

▸Capital project approval
▸Machine purchase justification
▸Energy savings verification
▸Lease versus buy analysis
▸R&D project ranking

Benefits

  • •Summarizes a project as one comparable percentage
  • •Directly comparable to borrowing and equity costs
  • •Accounts for the timing of cash flows
  • •Widely understood by executives and lenders
  • •Easy to communicate in capital requests

Limitations

  • •Can be undefined when cash flows never recover the investment
  • •May produce multiple rates for non-conventional cash flows
  • •Can rank mutually exclusive projects incorrectly

Best Practices

  • •Pair IRR with NPV at the reference rate
  • •Use the cost of capital as the reference rate
  • •Apply IRR only to conventional cash flow patterns
  • •Prefer NPV for ranking competing projects
  • •Document the reinvestment assumption in reports

Common Misconceptions

Myth: IRR is the profit percentage of the project.
Truth: IRR is the discount rate that zeroes NPV, not the accounting profit rate; the two differ for most projects.
Myth: A higher IRR always means a better project.
Truth: Between competing projects of different sizes, the higher IRR project can create less dollar value; NPV decides.
Myth: IRR assumes cash is reinvested at the IRR.
Truth: The IRR formula implicitly assumes reinvestment at the IRR; the modified IRR (MIRR) uses the cost of capital instead.

Practical Tips

  • •Set the reference rate from your cost of capital
  • •Check the NPV at the reference rate alongside the IRR
  • •Watch the NPV curve in the diagram for the zero crossing
  • •Use NPV instead of IRR when comparing projects of different size
  • •Treat a "not computable" IRR as a strong rejection signal

Common Mistakes

Comparing IRR to the wrong hurdle rate
Use the weighted cost of capital, not inflation or the bank loan rate.
Ranking projects by IRR instead of NPV
Use NPV for ranking; IRR can favor small projects with high rates.
Trusting an IRR when cash flows change sign
Non-conventional cash flows can have multiple IRRs; use NPV instead.
Ignoring the reinvestment assumption
Use MIRR or NPV when reinvestment at the IRR is unrealistic.

Assumptions

This calculator makes the following assumptions to provide estimates:

Constant Annual Cash Flow: The same net cash flow arrives every year of the project.
Conventional Cash Flow Pattern: One investment outflow followed by positive inflows; other patterns can break IRR.
Reinvestment at the IRR: Intermediate cash flows are implicitly reinvested at the IRR.
Single IRR in 0-100% Range: The bisection searches only between 0% and 100%.

Limitations

Be aware of these limitations when using this calculator:

Multiple or Missing IRRs: Sign-changing cash flow patterns can produce several IRRs or none at all.
Size Blindness: IRR ignores project scale; a small high-return project can beat a large value-creating one.
Reinvestment Assumption: Implicit reinvestment at the IRR overstates returns for high-yield projects.

Frequently Asked Questions

Answers to common questions about this calculator.

What does the IRR calculator compute?
It computes the internal rate of return, NPV at the reference rate, total cash flow, net profit, an NPV-versus-rate curve, and an accept or reject decision.
What is the internal rate of return?
The discount rate at which the net present value of a project equals zero.
How is the IRR found?
The calculator evaluates NPV across rates and uses bisection — repeatedly halving the rate interval where NPV changes sign — until the zero crossing is pinned down.
What is the reference rate?
The minimum acceptable return, usually the company cost of capital or hurdle rate.
What is the decision rule?
Accept when IRR exceeds the reference rate; reject when it falls below.
Why does the calculator show NPV at the reference rate too?
It verifies the decision: IRR above the hurdle and positive NPV at the hurdle should agree.
What does "not computable" mean?
NPV is negative at every rate from 0% to 50%, so the project cannot earn any positive return — a strong rejection signal.
How is IRR different from ROI?
ROI ignores timing; IRR discounts every cash flow to find the break-even earning rate.
How is IRR different from NPV?
NPV uses your discount rate and reports dollars; IRR reports the rate that zeroes NPV.
What is a typical IRR for equipment?
Most equipment projects fall between 10% and 30%, with energy savings projects often at 8-20%.
What inputs are needed?
Initial investment, annual net cash flow, project life, and the reference rate.
Can IRR be negative?
Yes, if the investment is never fully recovered the IRR can be negative or not computable.
What happens if cash flows change sign mid-project?
There can be multiple IRRs; the NPV curve and NPV at the reference rate are the trustworthy results.
Does IRR account for inflation?
Only implicitly. Use nominal cash flows with a nominal reference rate, or real values with a real rate.
Why does the NPV curve matter?
It shows how value changes with the discount rate and where the IRR crossing occurs — the diagram makes the result visual.
Is IRR the same as the accounting rate of return?
No. The accounting rate divides accounting profit by book value; IRR is a discounted cash flow measure.
What is MIRR?
Modified IRR, which reinvests intermediate cash flows at the cost of capital instead of the IRR.
How accurate is the result?
The bisection runs 60 iterations, giving precision far beyond display needs; accuracy depends on the inputs.
What does the decision value 1 or 0 mean?
1 means accept (IRR above the reference rate) and 0 means reject.
Can students use this calculator?
Yes, the NPV curve and bisection behavior make it an excellent teaching tool.
Is this calculator free?
Yes, it runs entirely in your browser, is free, and no data leaves your device.
Why is my IRR neutral but the decision bad?
When no IRR exists the rate shows neutral and the decision flags reject because the project cannot meet the hurdle.
Should I use IRR or payback?
Payback measures risk exposure; IRR measures earning rate. Use both, plus NPV, for capital decisions.
How does IRR handle salvage value?
Add salvage value to the final-year cash flow so the model includes it.
What is the difference between IRR and the discount rate?
The discount rate is chosen by the investor; the IRR is a property of the project cash flows themselves.
Why would IRR and NPV disagree on ranking?
Different project sizes and cash flow timing can reverse rankings; NPV is the correct decision metric.
Does the calculator work on mobile?
Yes, it is fully responsive and works on phones, tablets, and desktops.
What is the best way to verify the IRR?
Evaluate NPV at the computed IRR (it should be near zero) and confirm the NPV curve crosses zero there.
Why does the reference rate affect the decision but not the IRR?
The IRR is fixed by the cash flows; the reference rate only sets the comparison threshold for the decision.
What is a good IRR for an energy project?
8-20% is typical; anything above the cost of capital adds value and is worth pursuing.

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Glossary

Technical terms used in mortgage calculations explained in plain language.

IRR
Internal rate of return: the discount rate that makes NPV equal zero.
Reference Rate
The minimum acceptable return used to test an IRR decision.
Hurdle Rate
Another name for the minimum acceptable return.
NPV Curve
A plot of NPV against discount rate showing where it crosses zero.
Bisection
An iterative method that halves a search interval to find a function zero crossing.
Conventional Cash Flow
An investment outflow followed by positive inflows, with one sign change.
MIRR
Modified internal rate of return with reinvestment at the cost of capital.
Cost of Capital
The return required by investors to fund a project.
Zero Crossing
The point where the NPV curve passes through zero — the IRR.

Authoritative References

Trusted sources and standards referenced by this calculator.

Applicable Standards

AACE 17R-97ISO 15686-5ASMENIST GMP

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Page Information

Last Updated:

2026-07-31

Version:

2.0.0

License:

Free to use for educational and professional purposes.