Internal Rate of Return: What It Solves For and Where It Misleads

The internal rate of return is the discount rate at which a project's net present value is zero. Sankalp's automation cell costs Rs 50,00,00,000 and returns Rs 20,00,00,000 a year for five years. The stream solves at 28.65 per cent against a 12.00 per cent hurdle. The rate misleads in three ways: on scale, on timing, and where a cash flow stream changes sign more than once.

The trouble with this measure is not that it is difficult. The trouble is that the measure is easy, and easy in a way that hides what it threw away. Something small enough to hold in a single head is the right starting point. A woman runs a snack cart outside a college gate. She spends Rs 20,000 on a second gas burner and it brings her about Rs 9,000 a year of extra takings for the next five years. Somebody works it out and tells her the burner earns roughly thirty-four per cent a year. She is delighted, and she should be. A burner that pays back like that is a very good use of Rs 20,000.

Now her neighbour, who runs a tea stall, spends Rs 4,00,000 on a proper kitchen and a delivery scooter, and it brings in Rs 1,10,000 a year for five years. Somebody works that one out too: about eleven and a half per cent. If the only number either of them carries home is the percentage, the burner wins and the kitchen loses. But the burner adds a few thousand rupees a year and the kitchen adds more than a lakh, and no amount of staring at thirty-four against eleven will show that. The percentage is not wrong. The percentage is answering a question about how hard each rupee works, and it has been quietly promoted to answering a different question about how much money arrives.

The promotion of one question into another is where the trouble starts, and it starts on a far larger scale inside a company. Sankalp Industrial Systems Limited, an invented listed manufacturer of industrial valves, precision castings and aftermarket parts, has a list of five numbered projects in front of its board, and every one of them has been appraised at the company's own 12.00 per cent hurdle rateThe rate a project has to beat before it is worth doing. Sankalp's is 12.00 per cent, which is its weighted average cost of capital, and where that figure comes from is covered separately.. The worked project here is number 2, the automation cell.

What exactly does this measure solve for?

Take the question the value rule asks and turn it upside down. The value rule fixes the rate at 12.00 per cent and asks for the project's worth: an amount, in rupees, today. The internal rate of return fixes the amount at zero and asks what rate would produce it. That single swap is the whole of the difference between the two measures, and every property this one has, good and bad, falls straight out of it.

So the answer to the question is a rate rather than an amount. On Sankalp's automation cell it is 28.65 per cent. The figure repays care. Sliding past what it says is easy. The rate does not say the project earns 28.65 per cent in the way a deposit earns interest. The rate says that discounting this project's five receipts at 28.65 per cent instead of at 12.00 per cent would make the discounted receipts add to exactly the Rs 50,00,00,000 the project costs, leaving the project worth precisely nothing.

One picture holds the whole of the measure. Drawn against every discount rate that could be applied to it, the project's value traces one falling line. At a rate of nothing at all the project is worth the plain arithmetic difference between what goes out and what comes in: Rs 1,00,00,00,000 of receipts less a Rs 50,00,00,000 outlay, or Rs 50,00,00,000. As the rate rises the line falls. Each future receipt is being pushed down harder. At 12.00 per cent the line has fallen to Rs 22,09,55,240. Far enough along, it touches zero. The rate at which it touches zero is the internal rate of return, and it is a point on a curve already in hand rather than a separate calculation.

Two things follow immediately from that shape, and both matter later. The first is that the curve is falling everywhere, without a single wiggle, so it can only cross zero once. The second is that the crossing point is a property of the shape of the cash flows and of nothing else. Multiply every number in the project by ten and the curve gets ten times taller, but it still crosses at exactly the same place. Hold on to that. Indifference to size is the source of the measure's greatest convenience and of its worst failure, and the two are the same fact wearing two hats.

Why does the answer come out of a search rather than a formula?

Because there is no formula. The absence of a formula sounds like an evasion and is not. The absence is a fact about the arithmetic. Written out, the question on project 2 asks for a number that satisfies an equation with the unknown rate appearing five separate times, once for each year, each time raised to a different power. The equation is a polynomial of degree five, and a polynomial of degree five cannot in general be rearranged into a tidy expression that spits out its own roots. Every tool that reports an internal rate of return, from a spreadsheet function to a Rs 500 calculator, is doing the same thing: trying rates and narrowing.

The narrowing is not mysterious either. The curve already supplies the direction of travel: a higher rate always means a lower value. So if a trial rate leaves the value positive, that rate was too low and the answer is higher. If the value comes out negative, the rate was too high. The gap is halved, the next rate tried, and the process repeated. The search is iterationTrying a value, seeing how wrong it is, and trying again in the direction that reduces the error. There is no formula, so the answer is found by repetition., and eight rounds of it take a bracket twenty four points wide down to one less than a tenth of a point wide.

The search runs like this on the automation cell. The opening bracket runs between 12.00 per cent, where the value is known to be positive, and 36.00 per cent, where it is clearly negative. The midpoint is 24.00 per cent, and there the project is worth Rs 4,90,76,883. A positive value means the answer is above 24.00. The midpoint of what is left, 30.00 per cent, gives minus Rs 1,28,86,050, so the answer is below 30.00. Six more rounds of exactly that and the bracket has closed to between 28.5938 and 28.6875 per cent. The bracket contains 28.6493 and rounds to the 28.65 the record carries.

Notice what the search never needed. The search never needed a formula, it never needed anything about the project beyond its cash flows, and it never needed to be right first time. It only needed the direction of travel, and the falling curve supplies that for free. An internal rate of return is a search result, and knowing that removes the impression that a spreadsheet is doing something a pencil and some patience could not.

Two practical consequences come with that. The first is that a tool stops when it is close enough rather than when it is exact, so a reported figure can be a little off in the last decimal place, and the record's 28.65 per cent is the full 28.6493 rounded for printing. The second is more serious and gets its own section below: a search can find more than one answer, and it will report whichever one it happened to land on without announcing that the others exist.

Where does 28.65 per cent actually come from on this project?

Out of one division, and this is why project 2 is the right project to learn the measure on rather than a messier one. The automation cell costs Rs 50,00,00,000 and pays Rs 20,00,00,000 a year, the same amount every year, for five years. The outlay divided by the annual receipt is 50 over 20, or exactly 2.50. The ratio of 2.50 is the entire input to the answer.

Here is why. When the receipts are identical, discounting all five of them is the same job as multiplying one of them by an annuity factorThe multiplier that turns a level annual amount into a present value. At 12.00 per cent over five years it is 3.6048, so five receipts of one rupee are worth 3.6048 rupees today.. The annuity factor is just the sum of the five discount factors. Asking at what rate this project is worth nothing therefore becomes asking at what rate the five year annuity factor equals 2.50, and at that rate Rs 20,00,00,000 times the factor comes to exactly the Rs 50,00,00,000 that went out. The answer is 28.65 per cent because the five year annuity factor at 28.65 per cent is 2.4999654, and at the unrounded 28.6493 per cent it is 2.5000000 to seven decimal places.

Doing the division once by hand makes the measure concrete rather than magical. A reader with a table of annuity factors and no spreadsheet at all can confirm the record's 28.65 per cent in about a minute. The hand version also makes something visible that a spreadsheet hides: the only two facts that went into the answer were the size of the outlay relative to the receipt, and the number of years. The absolute size of either never appeared.

When does the accept rule agree with the value rule?

The decision rule itself is one line: accept a project whose internal rate of return exceeds the hurdle rate. Project 2 returns 28.65 per cent against 12.00, so it clears by 16.65 points and the board would take it. Its net present value at that same 12.00 per cent is Rs 22,09,55,240. The value is positive, so the value rule says take it too. The two agree, and the interesting question is whether that agreement is luck.

It is not. The agreement is a consequence of the shape in the first picture. If a project's cash flows are one outlay followed by nothing but inflows, its value curve falls continuously and crosses zero exactly once. Everywhere to the left of the crossing the value is positive and everywhere to the right it is negative, so the rate beating the hurdle and the value being positive at the hurdle are two ways of saying the identical thing. The rate test and the value test are not two tests that happen to agree. Both are one test read off two different axes.

Checked against all five of Sankalp's projects, there is not a single exception. Projects 1, 2, 3 and 4 return 18.72, 28.65, 21.86 and 12.11 per cent against a 12.00 per cent bar and all four have positive values. Project 5 returns 1.96 per cent and its value is negative. On an accept or reject decision taken one project at a time, the two measures cannot disagree, and any account suggesting otherwise has confused this decision with a ranking.

Project 4, the regional warehouse, is the row worth staring at. The warehouse returns 12.1094 per cent against a 12.00 per cent bar, a margin of eleven basis pointsHundredths of a percentage point. A hundred basis points make one point, so eleven of them is 0.11 of a point., and its value is Rs 40,35,685 on an outlay of Rs 90,00,00,000. Both figures say the same thing in different units: this project barely clears. Eleven basis points looks far more alarming than Rs 40,35,685, and Rs 40,35,685 looks far more alarming set beside the Rs 90,00,00,000 that had to be committed to earn it. Neither measure is lying. The two are just differently good at making a reader flinch.

What does the measure assume happens to the cash a project pays out?

One assumption almost nobody states is the heart of the difference between the two rules. Look again at what the arithmetic did. The arithmetic set the five discounted receipts against the outlay at the project's own rate of 28.65 per cent. Discounting at a rate and compounding at a rate are the same operation run in opposite directions, so saying that the receipts discounted at 28.65 per cent come to the outlay is the same as saying that the receipts, compounded forward at 28.65 per cent, come to what the outlay would have become at 28.65 per cent. The measure therefore carries a reinvestment assumptionWhat a measure silently assumes happens to the cash a project pays out once it has been paid out. It is never written down, and it is different for each rule.: that every Rs 20,00,00,000 goes on earning 28.65 per cent from the day it arrives until the project ends.

Say that out loud and it stops sounding modest. The assumption says Sankalp has somewhere to put a spare Rs 20,00,00,000, every year, at 28.65 per cent. If it did, that somewhere would already be a project on the list in front of the board, and the board would be discussing it rather than the automation cell. The value rule makes the other assumption: that the cash earns the company's 12.00 per cent cost of capital. The cost of capital is what the company actually pays for money, and therefore the rate it can always earn by simply paying money back. Of the two, the second is the more modest claim by a wide margin.

The picture below shows what the two assumptions do to a single rupee. The receipt at the end of Year 1 has four years left to run before the project closes. Compounded at 28.65 per cent it becomes 2.7393 rupees; compounded at 12.00 per cent it becomes 1.5735. The gap between the two is not a rounding difference. The earliest receipt is being assumed to nearly triple rather than to grow by about half, and the same gap, shrinking each year, applies to the receipts of Years 2, 3 and 4. The Year 5 receipt arrives on the last day and is never reinvested at all, so on that one receipt the two assumptions agree exactly and the whole argument disappears.

One boundary matters at exactly this point, where a reader reaches for a verdict. Naming the two assumptions is not the same as ruling between them, and when the two measures actually rank a pair of projects in opposite orders there is a determinate way to settle it. The resolution, and the rate at which the two answers cross over, are covered separately. The difference between the two measures lies in the assumptions, not in the arithmetic.

Why does a high percentage on a small project not beat a lower one on a large project?

Here is the first of the three failure modes, and it is the one that costs real money. Go back to what the answer was built from: a ratio, 2.50, and a number of years. Neither of those contains the size of the project. A percentage does not say what it is a percentage of, and no amount of care in computing it will put that information back.

Sankalp's own list contains the pair that shows it. Project 1 is the third valve line: Rs 2,00,00,00,000 out at the start and Rs 65,00,00,000 a year for five years, returning 18.72 per cent and worth Rs 34,31,04,532 today. Project 2 is the automation cell: Rs 50,00,00,000 out and Rs 20,00,00,000 a year for five years, returning 28.65 per cent and worth Rs 22,09,55,240. Project 2 earns 9.93 points more and banks Rs 12,21,49,291 fewer rupees. Projects 1 and 2 are mutually exclusiveTwo projects where taking one makes the other impossible. Sankalp's projects 1 and 2 need the same floor of the same building, so only one of them can be built.. The two need the same floor of the same building, so only one can be built and a ranking is forced.

The snack cart and the tea stall are back, in industrial clothing. A rate is a measure of intensity and a value is a measure of quantity, and when only one thing can be built it is the quantity that lands in the company's accounts. The conflict is real, and its cause is scale. Which of the two a board should take, and the rate at which the two answers cross over, are covered separately.

How do two projects of different lengths get compared unfairly?

The second failure mode is quieter than the first. Nothing looks wrong. An internal rate of return is quoted as a rate a year, and a rate a year invites the reader to assume that the number of years is somebody else's problem. It is not. Project 3, the tooling upgrade, costs Rs 30,00,00,000 and returns Rs 12,00,00,000 a year for four years at 21.86 per cent. Project 4, the regional warehouse, costs Rs 90,00,00,000 and returns Rs 16,00,00,000 a year for ten years at 12.11 per cent.

Put those side by side and 21.86 looks like it wins easily. But the two projects are not doing the same job. Project 3 has finished by the end of Year 4 and the company then has to find something else to do with the money. Project 4 is still running for another six years. Without a statement of what happens in Years 5 to 10, the two have not been compared at all: four years of one thing has been set against ten years of another, with the missing six years quietly assumed not to matter. The record is silent on those six years.

The same problem turns up inside a single length whenever two projects of the same duration have differently shaped receipts. A project that returns most of its cash in the first two years will score a higher rate than one that returns the same total later. Part of the difference is earned honestly: early money genuinely is worth more. Part of it is not: the measure is assuming that early cash goes on earning the project's own high rate for the remaining years. The front-loaded project is being rewarded twice, once correctly for arriving early and once questionably for what it is assumed to do after it arrives.

When can a project have more than one internal rate of return, or none at all?

The third failure mode is different in kind from the first two. Scale and timing are limits on what a rate can convey. The third is a limit on whether the rate exists or is unique at all, and the shape of the cash flow stream decides it before any arithmetic is done. The rule is about sign changesA point in a stream of cash flows where the amounts switch between negative and positive, or back again. Counting them takes a few seconds and decides how many answers are possible.: count the number of times the stream switches between money going out and money coming in, and that count governs how many answers the equation can have.

A conventional cash flowOne outlay at the start followed by inflows and nothing else, so the stream changes sign exactly once. Every project on Sankalp's list is of this kind. changes sign exactly once: money out at the start, money in thereafter, nothing else. Its value curve falls all the way and crosses zero once, so it has exactly one internal rate of return and the accept rule is safe. A project with a large end of life payment changes sign twice, and a stream that changes sign twice can produce two rates that both set the value to zero. And a stream that never turns negative at all has no internal rate of return. No rate is high enough to reduce a pile of pure inflows to nothing.

Which projects change sign twice in practice? A quarry that must be filled in and replanted when it is worked out. A plant that has to be decommissioned. A mine that carries a restoration obligation. In each case money goes out at the start, comes in for years, and then a large amount goes out again at the end. The equation can then have two roots, and when it does neither of them is a return on anything a reader can point at. A tool will report one of them, usually the first it stumbles across, and it will not mention the other.

The important part is what to do about it, and it is not to hunt for the real rate. There is no real one to find. The correct response to a stream with more than one sign change is to stop asking for a rate and work in rupees instead. The value at the company's own hurdle rate is a single unambiguous number no matter how many times the stream turns. There is also a modified version of this measureA variant that requires the rate at which interim cash is reinvested to be stated, instead of leaving the project's own rate buried inside the answer.. The modified version replaces the buried reinvestment assumption with one stated out loud, and is covered separately.

All five of Sankalp's projects are of the first kind. Every one of them is an outlay at time zero followed by inflows and nothing else. None of them can produce more than one answer, and the record contains no example of the awkward case at all. The absence is a fact about this list rather than about the world, and a reader who meets a project with a restoration cost attached should count the sign changes before quoting anything.

Can a project have a real, positive rate and still destroy value?

A project can, and Sankalp's list contains one. Project 5 is the effluent treatment plant: Rs 45,00,00,000 out at the start, and Rs 5,00,00,000 a year of cost savings for ten years. Its internal rate of return is 1.96 per cent. The 1.96 per cent is real, it is positive, it is correctly computed, and the project's net present value is minus Rs 16,74,88,849.

There is nothing paradoxical about it once what the rate has to be compared with is kept in view. A positive rate is not a pass; the test is whether the rate exceeds the cost of the capital the project consumes, and 1.96 per cent does not come close to 12.00 per cent. Money that earns 1.96 per cent while costing 12.00 per cent is losing about ten points a year on every rupee committed, and Rs 16,74,88,849 is what that loss is worth in today's rupees. The two statements are the same statement.

The plant is built anyway. Building it is mandatory under the site's own consent to operate, so the board is not choosing whether to do it. For a project like that the appraisal question changes shape entirely: not whether to spend the money, but which of the ways of complying costs least. The measure still has a job there, comparing the alternatives, and it has no job at all in deciding whether to comply.

So what is the measure genuinely good for?

Rather a lot, once its boundary is drawn. Nothing above says the rate is a bad measure. The rate is a measure of one thing, and people use it for a second thing it was never built to do. The rate is safe and useful on a single project with a conventional cash flow stream, tested against a hurdle rate, and it becomes unsafe the moment it is asked to rank two projects that cannot both be taken.

Within that boundary it does two jobs nothing else does as well. The first is that it states how much room a project has, in units everybody in the room already understands. Saying that project 1 clears the bar by 6.72 points while project 4 clears it by eleven basis points tells a board something about fragility that Rs 34,31,04,532 and Rs 40,35,685 do not say nearly as loudly. The second is that it needs no rate as an input. The rate can be computed before anybody has agreed what the hurdle rate should be. Real conversations often happen in exactly that order.

The discipline that goes with it is mechanical and takes one extra clause. Never quote a rate without the outlay it was earned on and the number of years it was earned over. The outlay and the life are precisely what the measure discarded on its way to producing a percentage. Twenty eight point six five per cent, said on its own, is a rumour. Twenty eight point six five per cent, on Rs 50,00,00,000, over five years, is a fact somebody else can check.

How does this go wrong when the arithmetic is faultless?

What is left to other subjects?

Two things, and both are worth naming. The first is the ranking conflict. At Sankalp's own 12.00 per cent the rate ordering and the rupee ordering come apart on projects 1 and 2, and scale is the cause. The rate at which the two answers would cross over, the two value curves meeting, and which project the board should take are all covered separately.

The second is the relationship between this list and the company's own forecast. The five numbered projects are a list under consideration and are not inside Sankalp's locked five year forecast, whose capital expenditure of Rs 1,34,80,00,000 in Year 1 rising to Rs 1,54,00,00,000 in Year 5 is the existing approved run rate. The two are never added together. The project list is assessed against the same 12.00 per cent the forecast is discounted at, and whether it is additional to that run rate or part of it is a question this record does not settle.