CB1 · explained

NPV against IRR, and the three times IRR gives the wrong answer

A percentage return feels more informative than an amount of money, which is why internal rate of return is the measure boards ask for. It is also why boards pick the smaller project. IRR fails in three specific, identifiable ways, and each one can be shown with arithmetic a reader can repeat.

CB1 · Business Finance · 9 min read

After this page you can

Calculate net present value and internal rate of return, recognise the three situations in which IRR ranks projects wrongly, and use an incremental appraisal to show which project a shareholder should prefer

What each measure is

Net present value is an amount of money: the value of everything a project brings in, discounted at the cost of capital, less what it costs. A positive NPV is the amount by which the owners of the business are better off.

Internal rate of return is a percentage: the discount rate at which the NPV would be exactly zero. It is the rate the project "earns" on the money while it is tied up.

Both use the same cash flows and the same discounting. They disagree about which project to choose, and when they do, NPV is right. The rest of this page is why.

Why NPV is the one with authority

The objective of the business is to make its owners better off, and NPV is denominated in exactly that. It also has a property IRR lacks, and the property is the whole argument.

NPV adds up. The NPV of doing two projects is the NPV of one plus the NPV of the other. So a company with a list of projects can add their NPVs and get the value of the programme.

IRR does not add up. Two projects with IRRs of 30% and 22.5% do not combine into a project with an IRR of 52.5%, and they do not generally combine into one with an IRR of 26.25% either. A percentage with no amount attached to it cannot be added to another percentage, and a measure you cannot add cannot rank a portfolio.

That is enough to settle the matter in principle. In practice you also need to recognise the situations where the two diverge, because an exam question will hand you one.

Failure one: scale

IRR has no idea how big the project is. Earning 30% on a small sum can look better than earning 24% on a large one, while adding far less money.

The resolution is the incremental project, and it is the technique worth learning from this page. Take the larger project, subtract the smaller one cash flow by cash flow, and appraise the difference as a project in its own right. It is the extra money and the extra return, and the question "is the extra outlay worth making?" is answered by its NPV and its IRR like any other project.

The worked example does this and finds exact agreement: the NPV of the smaller project plus the NPV of the increment equals the NPV of the larger. That is NPV's additivity being used to repair IRR's ranking.

Failure two: more than one answer, or none

IRR is the root of a polynomial. A project whose cash flows change sign once — money out, then money in — has exactly one root, and IRR behaves. A project whose cash flows change sign twice can have two roots, and then the phrase "the IRR" has no meaning.

This is not an exotic case. Any project with a large cost at the end has it: a mine that must be restored, a plant that must be decommissioned, equipment that must be removed.

The worked example constructs one with roots at exactly 10% and 20%. At a cost of capital of 8% the project has a negative NPV, while the rule "accept if the IRR exceeds the cost of capital" sees 10% against 8% and accepts. The rule is not merely ambiguous there; it is wrong.

And some projects have no real IRR at all, so there is nothing to compare. NPV is defined in every one of these cases, because it is a sum rather than a root.

Failure three: what the measure assumes about the cash you get back

NPV discounts each receipt at the cost of capital, which is a claim about what the money is worth once you have it: it is worth what the business's capital is worth. IRR, by its construction, treats every intermediate receipt as though it continued to earn the IRR itself.

For a project with a 30% IRR, that is an assumption that the cash coming out of it can be put back to work at 30%. If the company had a supply of 30% opportunities it would be doing them already. The higher the IRR, the less credible its own implicit assumption — which is an uncomfortable property for a measure whose whole purpose is to flag the best projects.

Why the IRR survives anyway

It should not be dismissed, and a one-sided answer loses marks. A percentage travels: it can be compared to a borrowing rate, to a hurdle rate, to last year's projects, without anyone needing to know the size of the outlay. It needs no cost of capital to compute, so it can be quoted before that argument has been had. And it gives an immediate feel for margin of safety — a project with an IRR of 30% against a cost of capital of 11% survives a lot of disappointment, and one at 12% does not.

The sound position is that IRR is a good description and a bad decision rule. Report it; decide on NPV.

Questions students ask

What if the company cannot afford every positive-NPV project?

Then the projects are competing for a scarce resource, and the ranking is by NPV per unit of the scarce resource — usually NPV per rupee of outlay — rather than by NPV alone. Note that this is still not IRR: it is NPV with a denominator chosen to match the constraint that actually binds.

Does the payback period have any use?

As a secondary test, yes. It ignores everything after the payback date and ignores the time value of money, so it cannot rank projects. But it is a crude measure of exposure, and in a business where forecasts beyond three years are guesses, a short payback is genuine information. Use it to describe risk, never to choose.

How is the modified IRR different?

It replaces IRR's implicit assumption with an explicit one: intermediate receipts are assumed to be reinvested at the cost of capital, and the rate is recomputed. That repairs failure three and the multiple-root problem, and it leaves the scale problem exactly as it was, because it is still a percentage. Worth knowing, not worth deciding on.

A project with two rates of return, so really with none

−2−100%10%20%30%Rate the company must beatValue today (₹ lakh)10%20%
  • the only stretch where the quarry is worth opening at all
The quarry pays out, then takes in, then pays out again, and a value that starts below zero, climbs above it and sinks back crosses zero twice. Both 10% and 20% are internal rates of return, so asking what rate this project earns has no single answer. Asking what it is worth today still does.

A worked example

Two invented examples, built for this page. All figures are made up and chosen so that every division is exact and every result can be reproduced on a calculator.

  1. Part 1: the higher IRR is the worse project

    A coaching business in Indore can open one new centre, not both. The cost of capital is 10%. Each option costs money now and returns a single sum in one year.

    OptionOutlay nowReceived in one year
    Small centre (S)₹110 lakh₹143 lakh
    Large centre (L)₹550 lakh₹682 lakh

    The IRRs. With one receipt a year out, the IRR is just the growth in the money.

    S: 143 ÷ 110 − 1 = 1.30 − 1 = 30.0%
    L: 682 ÷ 550 − 1 = 1.24 − 1 = 24.0%

    The NPVs, discounting the receipt at 10%.

    S: 143 ÷ 1.10 − 110 = 130 − 110 = ₹20 lakh
    L: 682 ÷ 1.10 − 550 = 620 − 550 = ₹70 lakh

    S has the higher percentage and L makes the owners ₹50 lakh better off. The board that asked for IRRs would open the small centre and leave ₹50 lakh on the table.

  2. Part 1 continued: the incremental test, which settles it

    Opening L instead of S means spending 550 − 110 = ₹440 lakh more now to receive 682 − 143 = ₹539 lakh more in a year. Appraise that as a project.

    IRR of the increment: 539 ÷ 440 − 1 = 1.225 − 1 = 22.5%
    NPV of the increment: 539 ÷ 1.10 − 440 = 490 − 440 = ₹50 lakh

    The extra ₹440 lakh earns 22.5%, comfortably above the 10% cost of capital, so the extra spending is worth making — and the test is now a straight comparison of a rate with the cost of capital, which is the one job IRR does reliably.

    Check it against the NPVs above. Using the printed figures, ₹20 lakh + ₹50 lakh = ₹70 lakh, which is exactly L's NPV. That is NPV's additivity, and it is the reason the incremental argument is not a trick: L simply is S plus the increment.

    And watch IRR fail to add. L's 24.0% is not 30.0% + 22.5%, nor is it the average of the two. It happens here to be the outlay-weighted average — (110 × 30.0% + 440 × 22.5%) ÷ 550 = (33 + 99) ÷ 550 = 132 ÷ 550 = 24.0% — and that works only because every cash flow in this example falls at the same date. Move one receipt to year two and the weighting breaks. Do not carry that shortcut away; carry away that NPV adds and IRR does not.

  3. Part 2: one project, two IRRs

    A quarry, invented for this page. It costs ₹100 lakh to open, yields ₹230 lakh at the end of year one, and must then be restored at a cost of ₹132 lakh at the end of year two.

    TimeCash flow
    Now(₹100 lakh)
    End of year 1₹230 lakh
    End of year 2(₹132 lakh)

    The signs run minus, plus, minus — two changes of sign, so there can be two roots. There are. Verify each by substitution rather than taking it on trust.

    At 10%: 230 ÷ 1.10 = 209.0909 and 132 ÷ 1.21 = 109.0909, so

    NPV = −100 + 209.0909 − 109.0909 = 0.0000

    At 20%: 230 ÷ 1.20 = 191.6667 and 132 ÷ 1.44 = 91.6667, so

    NPV = −100 + 191.6667 − 91.6667 = 0.0000

    Both are IRRs of the same project. Asking for "the IRR" of this quarry is asking a question with two answers.

  4. Part 2 continued: the shape of the NPV, and why the rule breaks

    Four more discount rates, each computed the same way:

    Discount ratePV of year 1PV of year 2NPV
    0%230.0000132.0000(₹2.0000 lakh)
    8%212.9630113.1687(₹0.2057 lakh)
    15%200.000099.8110₹0.1890 lakh
    30%176.923178.1065(₹1.1834 lakh)

    Each NPV is −100 + (PV of year 1) − (PV of year 2) from the two printed columns; for the 8% row, −100 + 212.9630 − 113.1687 = −0.2057.

    So the NPV is negative below 10%, positive between 10% and 20%, and negative again above 20%. Now apply the familiar rule at a cost of capital of 8%: the quarry's IRR of 10% exceeds 8%, so accept. But its NPV at 8% is negative ₹0.21 lakh. The rule does not merely give an ambiguous answer here; it gives the wrong one, and a company following it would destroy value.

    NPV has no such difficulty. It is a single number at the company's actual cost of capital, it exists for every set of cash flows, and at 8% it says no.

Where this fits in CB1

This page explains one idea from CB1 Business Finance. The full CB1 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

Other CB1 topics explained

Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.