NPV & IRR

Two numbers that decide whether money moves: what future cash is worth today, and what return a stream of cash flows actually earns.

The one idea: money has a time price

A euro next year is worth less than a euro today — not philosophy, but arithmetic: today's euro can be invested at the going rate and become more than a euro by next year. Discounting runs that logic backwards, translating every future cash flow into today's money:

$$ \mathrm{PV} = \frac{CF_t}{(1+r)^t} \qquad\qquad \mathrm{NPV} = \sum_{t=0}^{T} \frac{CF_t}{(1+r)^t} $$
  • Net present value (NPV): discount every cash flow — the negative ones too — and add them up. Positive NPV: the project creates value at that discount rate. Negative: it destroys it.
  • The discount rate r is the decision's whole personality: it encodes what the money could earn elsewhere at similar risk (the "opportunity cost of capital", or hurdle rate).
  • Every bond price in this atlas is an NPV; every yield curve point is a discount rate with a maturity attached.

IRR: the break-even discount rate

Ask the reverse question: at what discount rate would this project's NPV be exactly zero? That rate is the internal rate of return — the return the cash-flow stream itself earns:

NPV falls as the discount rate rises; the crossing point is the IRR. A project "clears its hurdle" when the IRR sits right of the hurdle rate.
IRRNPV of the projectDiscount rateNet present value
$$ \sum_{t=0}^{T} \frac{CF_t}{(1+\mathrm{IRR})^t} = 0 $$

The decision rule follows from the picture: invest when IRR > hurdle rate (equivalently, when NPV at the hurdle is positive). The two rules agree for ordinary projects — one outflow, then inflows — and that covers most of life.

Interactive: cash-flow NPV & IRR

Enter up to six annual cash flows (year 0 is usually the negative investment) and a hurdle rate. The solver finds the IRR numerically — exactly what a spreadsheet's IRR() does.

NPV at hurdle
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IRR
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Verdict
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Try flipping year 5 to −600 (a cleanup cost): with two sign changes there can be two IRRs or none — the solver says so, and the honest answer becomes "use NPV, not IRR".

Where IRR misleads — the professional's checklist

  • The reinvestment assumption: IRR implicitly assumes interim cash flows are reinvested at the IRR. A 40% IRR fund did not compound your money at 40% unless you re-deployed every distribution at 40% — you didn't.
  • IRR vs. MOIC: a quick 1.5× flip in one year is a 50% IRR; a patient 3× over ten years is ~12%. Which was better? MOIC measures wealth created, IRR measures speed — private equity quotes both because either alone can flatter (the MOIC/IRR and waterfall calculators show the interplay).
  • Timing games: subscription credit lines delay capital calls, mechanically boosting reported fund IRRs without creating a cent — a now-standard practice that makes early-life IRRs largely marketing.
  • Multiple sign changes: mining projects with cleanup costs, structured deals with capital returns — multiple IRRs or none. Descartes' rule of signs, applied to your term sheet.
  • Scale blindness: a 100% IRR on €1,000 is worth less than a 15% IRR on €1m. NPV in euros ranks projects; IRR in percent ranks percentages.

The same machine everywhere

  • Bonds: yield to maturity is the IRR of the bond's cash flows at its price — the YTM solver is this page's solver wearing a bond costume.
  • Equities: the dividend discount model is an NPV with an infinite horizon.
  • Private markets: fund returns are IRRs on called-and-distributed cash — with every caveat above, at scale.
  • Corporate life: every capex committee meeting is an argument about which hurdle rate, applied to whose cash-flow forecast.