Internal Rate of Return (IRR)

What is IRR?

The internal rate of return is the discount rate that makes the NPV of a project's cash flows equal to zero. It's the "break-even rate" — below IRR the project is profitable, above it the project loses money (for a conventional project with an upfront cost and later inflows).

Formal definition

Solve for $r$ in: $NPV(r) = \sum_{t=0}^{T} \dfrac{CF_t}{(1 + r)^t} = 0$

For our problem: $-1{,}000 + \dfrac{400}{1 + r} + \dfrac{400}{(1 + r)^2} + \dfrac{400}{(1 + r)^3} = 0$

There's no closed-form solution for general cash flows — you solve numerically (bisection, Newton's method, or Excel's IRR).

Step-by-step solution

Step 1 — Try $r = 10\%$: $NPV(0.10) = -1000 + 400/1.10 + 400/1.21 + 400/1.331$ $\approx -1000 + 363.64 + 330.58 + 300.53 = -5.26$

Close to zero but slightly negative → IRR is a bit below 10%.

Step 2 — Try $r = 9\%$: $NPV(0.09) = -1000 + 400/1.09 + 400/1.1881 + 400/1.2950$ $\approx -1000 + 366.97 + 336.67 + 308.87 = 12.51$

Now positive. IRR is between 9% and 10%.

Step 3 — Interpolate or refine.

Linear interpolation: $r \approx 9\% + \dfrac{12.51}{12.51 + 5.26} \cdot 1\% \approx 9\% + 0.704\% = 9.704\%$

Step 4 — Verify at $r = 9.70\%$: $NPV(0.0970) \approx -1000 + 364.63 + 332.39 + 303.00 \approx 0.02$

Boxed answer: IRR ≈ 9.70%.

Decision rule

For a conventional project (outflow first, inflows later):

• Accept if IRR > required rate of return (hurdle rate).
• Reject if IRR < hurdle rate.

At a hurdle of 10%, our 9.70% IRR rejects. At a hurdle of 8%, it accepts.

NPV-vs-rate curve

Plot NPV against $r$:

• $r = 0\%$: NPV = $-1000 + 400 \cdot 3 = +200$
• $r = 5\%$: NPV $\approx +89.30$
• $r = 9.70\%$: NPV $\approx 0$ ← IRR
• $r = 15\%$: NPV $\approx -86.33$
• $r = 25\%$: NPV $\approx -232$

It's a downward-sloping curve crossing zero at the IRR.

Key properties

• Independent of scale. A $10M project and a$10 project can have the same IRR.
• Independent of hurdle rate. IRR is intrinsic to the cash-flow pattern, not the investor's required return.
• Assumes reinvestment at IRR — this is a big caveat. MIRR (modified IRR) relaxes that assumption.

Known pitfalls

1. Multiple IRRs. Cash flows that change sign more than once (outflow → inflow → outflow) can have several rates where NPV = 0. Example: $-100, +230, -132$ has two IRRs (10% and 20%).
2. No IRR. Some cash flows don't have a real solution at all (if the project always has positive NPV at any positive rate, or negative at all rates).
3. Project size blindness. A $100 project with 50$10M project with 15% IRR — NPV captures scale, IRR doesn't.

For these reasons, NPV is the preferred decision criterion in corporate finance; IRR is a supplementary metric valued for its intuitive "rate" interpretation.

Common mistakes

• Using IRR to compare projects of different sizes. Use NPV.
• Applying IRR when cash flows switch signs multiple times. Check for multiple IRRs.
• Assuming intermediate cash flows are reinvested at IRR. They're typically reinvested at the cost of capital instead — use MIRR if this matters.

Try it in the visualization

A curve of NPV vs. $r$ with a dot that slides left-right as you adjust a rate slider. The zero crossing is highlighted as the IRR. Side bars show each cash flow discounted at the current rate.