Internal rate of return

The internal rate of return is the discount rate at which NPV = 0. It's the project's implied yield, the r that makes the PV of what you get equal what you paid. For the project in the net present value example, IRR ≈ 14.3%, so at any hurdle below that, NPV > 0.

People love IRR. It's one number, it needs no pre-agreed discount rate, and "this deal yields 24%" sounds better in a meeting than "NPV is ₹15.4 crore at an 11% hurdle." Private equity runs on it. So you have to speak fluent IRR, and you have to know where it lies. There are six traps. Learn each one through its example.

Why can IRR give two answers, or none?

Trap 1: multiple IRRs when signs change more than once. NPV as a function of r is a polynomial in 1/(1+r). By Descartes' rule of signs, it can have up to as many positive roots as the cash-flow stream has sign changes. A normal project (−, +, +, …) has one sign change, so it has one IRR. Now take a project that pays to open, earns, then pays again to close for reclamation. Mining is the classic shape. Its CFs are −100, +230, −132, with two sign changes. Solve NPV = 0 by multiplying through by (1+r)²:

−100(1+r)² + 230(1+r) − 132 = 0 → with x = (1+r): 100x² − 230x + 132 = 0 → x = (230 ± √(230² − 4·100·132))/200 = (230 ± 10)/200 → x = 1.10 or 1.20.

Both 10% and 20% are IRRs. The NPV profile confirms it: negative below 10%, positive between, negative above:

So is the project "yielding 10%" or "yielding 20%"? The question has no answer. IRR is undefined as a return here. Excel's IRR() will return whichever root is nearest its guess argument, which defaults to 10%, and it won't warn you. Two analysts with two guesses get two different "IRRs." NPV at your actual hurdle stays perfectly well behaved. At r = 15%, NPV = +0.19, so you accept, if barely. You'll find this shape in mines and nuclear plants with decommissioning costs, in projects with mid-life capex or working-capital reversals, and in LBOs with dividend recaps followed by a final equity injection.

Trap 2: no IRR at all. Take the stream +1,000, −3,000, +2,500. You're paid upfront. Then you owe midway, then collect at the end, much like insurance. Its NPV is +500 at r = 0, +236 at 20% and +100 at its minimum near r ≈ 67%, and it never crosses zero. The quadratic's discriminant is negative, so no real root exists. What's the IRR? There isn't one. NPV is positive at every rate, so you accept. The rule that never breaks keeps working.

Why does IRR ignore the size of a project?

Trap 3: scale blindness. IRR is a percentage, so it can't see size. Project A turns −100 into +150 in a year. Project B turns −10,000 into +12,000. IRR says A, at 50% against 20%. NPV at a 10% hurdle says A creates ₹36.4 while B creates ₹909.1, twenty-five times more value. If you can do only one, B makes you richer. IRR maximizes the rate on whatever happens to be invested, but you eat amounts. When sizes differ, the proper fix is the incremental IRR on the difference stream B−A. Here that's −9,900 turning into +11,850, an incremental IRR of 19.7%, comfortably above the 10% hurdle, so the extra 9,900 is well spent. Notice, though, that you needed a hurdle rate anyway, and at that point you've re-derived NPV.

What is MIRR, and why does reinvestment matter?

Trap 4: the reinvestment assumption, and MIRR. Compounding to an IRR of 20% assumes every interim cash flow is itself reinvested at 20% until the end. For a high-IRR deal that's fantasy. Your other opportunities earn your opportunity cost. They don't earn this deal's exceptional rate. So quoted IRRs overstate the realized compound return you'll actually experience whenever IRR > reinvestment rate. The repair is the Modified IRR:

MIRR = [FV(inflows, compounded at the reinvestment rate) / |PV(outflows, at the finance rate)|]^(1/n) − 1

Take a synthetic stream: −1,000, then +500, +400, +300, +200 over four years. Plain IRR = 17.8%. Now compound the inflows forward at a realistic 10% reinvestment rate: FV = 500·1.1³ + 400·1.1² + 300·1.1 + 200 = 665.5 + 484.0 + 330.0 + 200 = 1,679.5. MIRR = (1,679.5/1,000)^(1/4) − 1 = 13.8%. Four points of that "17.8%" were reinvestment fantasy. MIRR also has a single value whatever the sign changes, so it cures Trap 1 too. When a PE deck says "our fund IRR is 28%," your reflex should be two questions. Early distributions reinvested at what? That's the MIRR question. And on how much capital, for how long? That's the MOIC question, the money-on-money multiple that is IRR's essential companion in private equity.

When do IRR and NPV disagree?

Trap 5: ranking conflicts and the crossover rate. Even with single sign changes and equal outlays, IRR and NPV can rank two projects differently when their cash-flow timing differs. Project L is late-loaded: −1,000, then 100, 300, 400, 675. Project S is early-loaded: −1,000, then 500, 400, 300, 100. Their IRRs are 13.5% for L and 14.5% for S, so S "wins." At an 8% hurdle, though, L's NPV is +163.5 against S's +117.6, and L wins by a lot. The two NPV profiles cross at the crossover rate, the IRR of the difference stream L−S, which here is 12.0%. Below 12%, L is worth more, because its big late cash flows are valuable when the opportunity cost of waiting is low. Above 12%, S wins, because early money can be reinvested at attractive rates. IRR is blind to your actual opportunity cost, so it always votes for the early-loaded project. NPV votes correctly at your rate, the only rate that describes your world.

Trap 6: the borrowing orientation. The stream +100, −130 has "IRR = 30%." But you received money first. It's a loan to you, and when you borrow you want the lowest rate. The mechanical rule "accept if IRR > hurdle" flips for financing-shaped streams without telling you. NPV at 10% is 100 − 130/1.1 = −18.2. Reject it, and you never had to think about which way the rule points today.

What are XNPV and XIRR, and when should you use IRR?

XNPV and XIRR: the real world runs on dates. Actual deals pay on 17 March and 4 November, which "t = 1, 2, 3" can't represent. Date-aware versions measure each gap in days:

XNPV = Σ CFᵢ / (1+r)^(dᵢ/365) where dᵢ = days from t = 0 to cash flow i; XIRR is the r making XNPV = 0.

You can do a single-pair version by hand. Invest ₹1,00,000 and receive ₹1,12,000 after 300 days. Over those 300 days the return is 12%, and annualized it's 1.12^(365/300) − 1 = 14.78%. That's XIRR. PE and VC reporting, SIP return statements ("your SIP's XIRR is 13.2%") and any irregular stream all run on it. It inherits every one of IRR's conceptual traps, because it's IRR with better calendar manners.

The verdict to internalize. NPV is the decision rule. IRR is a communication device, and a useful summary when the stream has one sign change and you're comparing like scale and like life. Quote IRR, decide on NPV, and when they disagree, NPV is right. The disagreement itself tells you something about timing or scale that IRR is hiding.