Net present value (NPV) and internal rate of return (IRR) are the two workhorse decision rules in corporate finance. Both ask the same question – is this project worth doing – but they answer it differently. NPV gives you a dollar number at your cost of capital. IRR gives you a single rate of return that summarizes the cash-flow pattern. Most of the time they agree. When they disagree, the textbook answer is unambiguous: trust NPV.
TL;DR: NPV is the present value of every future cash flow discounted at your cost of capital, minus the upfront cost. Accept any project with NPV greater than zero. IRR is the discount rate that makes NPV exactly zero – accept if IRR exceeds your hurdle rate. They disagree when projects are mutually exclusive with different scales or timing, or when cash flows switch sign more than once. In every conflict, NPV wins, because IRR implicitly assumes you can reinvest cash at the IRR itself – which is rarely true.
The formulas, in one place
NPV = Σ CFt / (1 + r)t
Where CFt is the cash flow in period t (negative for outflows, positive for inflows), r is your discount rate (usually the project’s weighted average cost of capital), and the sum runs from period 0 through the project’s life. Corporate Finance Institute presents the same formula.
IRR is the rate r* that makes NPV = 0
Solve Σ CFt / (1 + r*)t = 0 for r*. There is no closed-form solution for more than two periods – spreadsheets do it iteratively with Newton’s method. CFI’s IRR page states the same definition: “the discount rate that makes the net present value (NPV) of a project zero.”
Decision rules
- Single project, take-it-or-leave-it: NPV > 0 means accept. IRR > hurdle rate means accept. They agree.
- Two or more mutually exclusive projects (you can only pick one): rank by NPV. Higher NPV wins. IRR ranking can mislead.
- Cash flows switch sign more than once: IRR may have multiple solutions or no real solution. Use NPV.
A worked example: two real projects that disagree
You run a small industrial firm. Two factory upgrades cost $1,000 each up front. You can only do one. Cost of capital is 10%. The cash-flow streams look like this:
| Year | Project A (front-loaded) | Project B (back-loaded) |
|---|---|---|
| 0 | −$1,000 | −$1,000 |
| 1 | +$600 | +$100 |
| 2 | +$500 | +$400 |
| 3 | +$300 | +$1,100 |
| Total cash | $1,400 | $1,600 |
| NPV at 10% | $184 | $248 |
| IRR | 21.5% | 19.9% |
This is the classic conflict. NPV picks B. IRR picks A. Which is right? B – by $64. The fact that B has a lower IRR just means its average rate of return is slightly lower, but it does so over a much larger base of dollar value created. You cannot deposit a percentage in the bank. You deposit dollars. NPV measures dollars.
Why does IRR favor A here? Because IRR penalizes back-loaded cash. The IRR formula implicitly assumes that every dollar of intermediate cash flow gets reinvested at the IRR itself. Project A throws off $600 in Year 1 – the IRR calculation assumes you can plow that $600 back into something that earns 21.5%. In real life, you probably cannot. Project B’s Year 1 cash is smaller, so the unrealistic reinvestment assumption matters less.
Where they cross over
If you plot NPV as a function of the discount rate for each project, you get an “NPV profile.” The lines cross at the discount rate where both projects have equal NPV – the crossover rate. For our two projects it is about 16.9%. Below that, B wins NPV. Above it, A wins NPV (and IRR). At our 10% cost of capital we are well below the crossover, so B wins decisively.
When IRR breaks: multiple roots and no real solution
The polynomial equation that defines IRR can have more than one real root when cash flows switch sign more than once. Descartes’ rule of signs formalizes this: a cash-flow stream can have as many positive IRRs as it has sign changes.
The textbook example is a mining or remediation project. You invest $1,000 today, the mine produces $3,000 of cash in Year 1, then you owe $2,200 in cleanup costs in Year 2:
CF: −$1,000, +$3,000, −$2,200 — two sign changes
Solve for IRR and you get two positive roots: about 27.6% and 72.4%. Which is “the” IRR? Neither, really – the equation has two valid solutions, and the IRR decision rule (“accept if IRR > hurdle rate”) becomes ambiguous. If your hurdle rate is 10%, the project’s NPV is −$91 – reject it – but a naive IRR check against either root would say accept.
Other pathological cases:
- No real IRR. If the polynomial has no real positive root, IRR is undefined. Happens when total cash never recovers the upfront cost at any positive rate.
- Scale mismatch. A $1,000 project with 50% IRR creates $500 of value at a 10% discount rate. A $1,000,000 project with 15% IRR creates $45,455. IRR rewards the small project; NPV rewards the big one. If they are mutually exclusive and you have $1M to deploy, the small project wastes $999,000 of capacity.
- Different project lives. Comparing a 2-year and a 10-year project on IRR alone ignores that you may need a replacement project in Year 3 for the short one.
The reinvestment problem and MIRR
IRR’s hidden assumption – that intermediate cash flows can be reinvested at the IRR itself – is the deepest reason finance textbooks prefer NPV. NPV uses a single, externally determined discount rate (your cost of capital) for every cash flow. IRR implicitly uses the project’s own rate. For a project with a 35% IRR, that is fantasy.
The fix is MIRR (modified internal rate of return), which lets you specify a separate reinvestment rate – usually the firm’s cost of capital. MIRR almost always sits between IRR and the reinvestment rate and removes the multiple-root problem. It is what disciplined practitioners quote when they need a single “return” number for a project. But MIRR is still a derived metric – the primary decision rule remains NPV.
Why IRR refuses to die
If NPV is unambiguously better, why does every banker, private-equity associate, and corporate development team still lead with IRR? Three reasons:
- It is unit-free. “28% IRR” is comparable across deals of vastly different size; “$184 NPV” is not, without context.
- It does not require a discount-rate assumption. NPV forces you to commit to a cost of capital. IRR avoids that conversation – the IRR itself is the answer, and the user can compare it to whatever hurdle they want.
- LP fund returns are quoted in IRR. Private equity, venture capital, and infrastructure funds report performance as IRR (with multiple of invested capital as a supplement). So practitioners build the same habit for individual deals.
The honest practitioner reports both, uses NPV to decide, and is suspicious of any high IRR that comes from front-loaded cash flows or short project lives.
Common mistakes that show up in real models
- Forgetting the sign of Year 0. The upfront investment must be negative. A spreadsheet that mis-signs the initial outlay will return nonsense.
- Using IRR on cash flows that switch sign. Mining, decommissioning, large mid-project capex – any of these can produce multiple IRRs. Switch to NPV or MIRR.
- Comparing IRRs across projects with different scales. A higher IRR on a smaller project does not beat a lower IRR on a larger one if the larger NPV is meaningfully bigger.
- Discounting at the wrong rate. NPV’s answer depends entirely on the discount rate. Use the project’s risk-adjusted cost of capital, not the firm’s blended WACC for a project that is much riskier or safer than the firm average.
- Ignoring inflation consistency. Discount nominal cash flows at a nominal rate; real cash flows at a real rate. Mixing the two embeds a hidden adjustment.
Where this connects
NPV is the engine behind every DCF stock valuation – a DCF is just an NPV calculation with a longer cash-flow horizon and a terminal value. The discount rate it uses is the weighted average cost of capital, which leans on the equity-risk concepts in our beta, alpha, and CAPM explainer. On the corporate side, both NPV and IRR are inputs to the LBO model that private-equity firms use to evaluate buyouts. If you can read an NPV/IRR analysis cleanly, the rest of corporate finance starts to fall into place.
Sources & further reading
- Corporate Finance Institute – NPV formula and decision rule.
- Corporate Finance Institute – IRR definition and intuition.
- Investopedia – MIRR, multiple IRRs, and Descartes’ rule of signs.
- IRS Publication 542 – Corporations – source for the 21% US federal corporate tax rate used in standard after-tax cost-of-capital examples.
- Federal Reserve H.15 Selected Interest Rates – source for risk-free rate inputs to the discount rate (10-year Treasury at 4.49% as of June 17, 2026).
- Brealey, Myers & Allen, Principles of Corporate Finance – the standard academic treatment of NPV, IRR, and the reinvestment assumption.
- Related on ECMSource: DCF valuation, WACC explained, ROIC explained, LBO 101.
Disclosure: This article was produced with AI assistance and reviewed before publication. It is for informational purposes only and is not investment advice.