How to Resolve NPV and IRR Conflicts: The Crossover Rate

When two projects cannot both be built and the two appraisal rules rank them in opposite orders, find the crossover rate: the internal rate of return of the difference between them. For Sankalp Industrial Systems Limited's first two projects it is 15.2382 per cent, written 15.24. The crossing sits 3.24 points above the 12.00 per cent hurdle, so the larger project wins on rupees.

The shape of this problem is much older than any spreadsheet, so it is worth starting with something concrete. Suppose a household has saved a little and there is exactly one shop front available on the main road. The son wants to open a small tea and snacks counter: it will cost Rs 1,00,000 to fit out and it should bring in about Rs 40,000 a year. The daughter wants a printing and stationery shop in the same space: Rs 4,00,000 to fit out, and about Rs 1,10,000 a year. Only one of them can have the shop front.

Somebody at the table will say the tea counter is obviously the better use of the money, because Rs 40,000 a year on Rs 1,00,000 is a return of forty per cent while Rs 1,10,000 on Rs 4,00,000 is only twenty-seven and a half. The printing shop brings in Rs 70,000 a year more, so somebody else will say the printing shop is obviously better. Both of them are right about their own arithmetic, and they are answering two different questions. One is asking how hard each rupee works. The other is asking how many rupees come in. When only one thing can be built, those two questions can point in opposite directions, and somebody has to decide which question the household is actually asking.

Sankalp Industrial Systems Limited, an invented listed manufacturer of industrial valves and precision castings, is in exactly that position on two of the projects in front of its board. Both appraisal rules are covered separately and are assumed here. A conflict between the two has a determinate resolution rather than a preference between traditions.

When do the two rules actually disagree?

Far less often than the amount written about this would suggest, and never at all in the most common case. If two projects are independent, meaning the company can take both, there is no ranking to argue about. Each project is measured against the 12.00 per cent bar on its own, each either clears it or it does not, and the two rules give the same verdict every time. A project with a positive value at the hurdle rate is exactly a project whose internal rate of return exceeds that hurdle rate. Those are two ways of saying one thing.

The disagreement needs a second condition: the projects must be mutually exclusiveTwo projects where taking one makes the other impossible, usually because they need the same space, the same team or the same permission.. Taking one makes the other impossible. Now the company is not asking whether each project is good enough. It is asking which one to take, and a ranking is required. One measure ranks in rupees and the other ranks in percentages, and those two orderings are not the same ordering. A ranking is where the two measures can part company. Sankalp's projects 1 and 2 need the same floor of the same building, so only one can be built. Sharing the floor is a structural fact, established separately, and taken here as a precondition.

What is the conflict on Sankalp's own two projects?

Here it is, stated on the invented company's own figures and discounted at its own 12.00 per cent hurdle rateThe rate a company discounts a project at before deciding whether it clears the bar. Sankalp's is 12.00 per cent, which is its weighted average cost of capital., which is restated here rather than rebuilt. Project 1 is the third valve line: an outlay of Rs 2,00,00,00,000 at the start, then Rs 65,00,00,000 a year for five years. Project 2 is the automation cell: an outlay of Rs 50,00,00,000, then Rs 20,00,00,000 a year for five years. Every one of those flows is after tax at the company's own assumed effective rate of 25.0 per cent.

Read the last three rows again. The whole problem is in those three lines. The value rule prefers project 1, the rate rule prefers project 2, both are computed correctly, and the company can build only one of them. Neither figure is a mistake waiting to be found. Both measures are doing exactly what they were built to do, on the same cash flows, and arriving at opposite orders.

Two notes on precision before going further. The figures above are carried to the rupee. The record they are drawn from rounds the same arithmetic to the nearest lakh. Project 1's value of Rs 34,31,04,532 appears in the record as Rs 34,31,00,000 and project 2's Rs 22,09,55,240 as Rs 22,10,00,000, so the difference is carried there as Rs 12,21,00,000 against the Rs 12,21,49,291 above. The two records hold the same arithmetic to the nearest lakh. And the two internal rates of return are 18.7189 and 28.6493 per cent, printed as 18.72 and 28.65. Two decimal places are as much precision as a five year forecast can honestly carry.

Steps one and two: are the preconditions even met?

Before computing anything, two checks, and they take about ten seconds each. The first is the one just made. Are the projects genuinely mutually exclusive? If the answer is no, there is nothing to resolve and the work stops. Sankalp's projects 1 and 2 share a factory floor, so the answer is yes.

The second is whether each project has a conventional cash flowOne outlay followed by inflows, with exactly one change of sign in the whole stream.: one outlay followed by inflows, with exactly one change of sign. The sign change matters. A stream that turns negative again later can produce more than one internal rate of return, and then there is no single answer to which rate is being compared. Both of Sankalp's streams are one outlay and five inflows, one sign change each, so each has exactly one internal rate of return and the comparison in the table above is well defined. If that second check fails, the disagreement on the table is not the one resolved here; it belongs to the rate measure's own failure modes, which are covered separately.

Step three: what is causing this particular disagreement?

There are exactly two things that can cause it, and naming which one applies is part of the answer rather than a decoration on it. The first is a scale conflictA disagreement caused by one project being much larger than the other, so a big percentage on a small base loses to a smaller percentage on a large one.: one project is much bigger than the other, so a high percentage on a small base can be worth fewer rupees than a lower percentage on a large one. The second is a timing conflictA disagreement caused by two projects of similar size returning their cash at different speeds, one early and one late.: the projects are similar in size but return their cash at different speeds, one loading its receipts early and the other late, so a change in the discount rate hurts one far more than the other.

On Sankalp's pair the cause is scale, and it is not subtle: project 1 costs four times what project 2 costs, Rs 2,00,00,00,000 against Rs 50,00,00,000. Timing is not involved at all here, and the shapes show it. Both projects run for exactly five years and both pay a level annual amount, so their profiles are the same profile at two different sizes. Nothing about when the cash arrives differs between them. Naming the cause matters for how the result is explained to a board. A scale conflict is explained by saying a percentage cannot see how big the base is. A timing conflict is explained by saying a discount rate punishes late cash more than early cash. Same technique, different sentence.

Here is the household version again, and it is the same fact. Forty per cent on Rs 1,00,000 is Rs 40,000 and twenty-seven and a half per cent on Rs 4,00,000 is Rs 1,10,000. The percentage is a ratio and it has thrown away the size of the thing it is a ratio of. A company banks rupees, not ratios, and that observation is the seed of the answer reached at the end. But it is only a seed, because saying rupees matter more than ratios is an assertion, and what follows replaces it with a computation.

Step four: what does the extra money actually buy?

This is the restatement that dissolves the problem, and everything after it is arithmetic. Choosing project 1 over project 2 is exactly the same decision as choosing project 2 and then also making one further investment: the difference between them. The company does not have to think of it as a contest between two projects at all. It can think of it as taking the small project and then being asked, separately, whether to spend a further Rs 1,50,00,00,000 to turn it into the big one.

The difference between the two projects is a real proposal with its own cash flows. Analysts call it the incremental projectThe difference between two projects, treated as a project in its own right and appraised like any other., and it is built by subtraction, one date at a time. Its incremental cash flowThe larger project's cash flow less the smaller one's, year by year, including at time zero. at time zero is project 1's outlay less project 2's outlay: Rs 2,00,00,00,000 less Rs 50,00,00,000, an outlay of Rs 1,50,00,00,000. In each of years 1 to 5 it is Rs 65,00,00,000 less Rs 20,00,00,000, being Rs 45,00,00,000 a year.

Look at what that bottom row does to the argument. The decision is no longer a beauty contest between a valve line and an automation cell. The bottom row is one ordinary project with one outlay and five receipts, of the sort the company appraises every week, and both measures agree about how to treat an ordinary project. The restatement has turned a stand-off between two rules into a single appraisal that neither rule disputes.

Step five: what rate does that extra project earn?

The extra project is then appraised the way anything else would be. Its internal rate of return is 15.2382 per cent, written 15.24, and that number is the crossover rateThe discount rate at which two projects have exactly the same net present value, so neither is preferred on rupees.: the single discount rate at which projects 1 and 2 have exactly the same net present value. The two facts are the same fact, and that identity is why the technique works at all. If the difference between two projects is worth precisely nothing at some rate, then at that rate the two projects are worth the same. Setting the difference to zero and setting the two values equal are one equation written two ways.

The shape of it can be checked in a single line without any solver. The extra project pays a level amount, so its value is the annual amount times the five year 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., less the outlay. Setting that to zero means the annuity factor must equal Rs 1,50,00,00,000 divided by Rs 45,00,00,000, which is 3.3333. So the crossover rate is simply the rate at which the five year annuity factor equals 3.3333, and that rate is 15.2382 per cent. At the company's own 12.00 per cent the same factor is 3.6048. A larger factor means the extra project is worth more than nothing at 12.00 per cent.

How much more? Rs 45,00,00,000 times 3.6048 is Rs 1,62,21,49,291, less the Rs 1,50,00,00,000 outlay, giving Rs 12,21,49,291. And that is the same Rs 12,21,49,291 the table showed as the gap between the two projects, which is the check that the extra project was built correctly. The match is not a coincidence and not an approximation. Discounting is a linear operation, so the present value of a difference always equals the difference between the present values. If the extra project's value does not equal the gap between the two originals, something has been subtracted wrongly, and it needs finding before anything else proceeds.

Step six: what happens when the hurdle rate is put beside it?

Everything, because this is where the answer comes out. The company discounts at 12.00 per cent. The crossover rate is 15.2382 per cent. The hurdle sits 3.2382 points below the crossing, written 3.24, and below the crossing the larger project has the higher net present value. So project 1 wins, and it wins by an amount already computed twice above: Rs 12,21,49,291.

Say it the other way and it becomes something a board can act on. The company is being asked to invest a further Rs 1,50,00,00,000 in order to receive a further Rs 45,00,00,000 a year for five years. That proposal earns 15.24 per cent. The money to do it costs 12.00 per cent. The proposal clears its cost of capital by 3.24 points, so the company should make it. Making it is the same thing as building project 1 instead of project 2. There is nothing left in that sentence that two rules could argue about.

Notice how much the answer depends on where the hurdle happens to sit. At 14.00 per cent, still below the crossing, project 1 would still win, by a much thinner Rs 4,48,86,436. At 16.00 per cent, now above the crossing, project 1 would lose: it would be worth Rs 12,82,90,875 against project 2's Rs 15,48,58,731. The ranking is not a property of the two projects on their own; it is a property of the two projects and the rate together. The ranking's dependence on the rate is the claim most people never quite believe until they watch it happen, and the control below lets them watch it.

What does the whole thing look like as a picture?

Two lines, and once they have been seen the technique never needs to be memorised again. The discount rate runs along the bottom and each project's net present value up the side. A higher discount rate makes future cash worth less today, and both of these projects are nothing but future cash bought with money spent now, so every project drawn this way slopes downwards. The only question is how steeply each one falls.

The whole argument is legible in that one picture. Project 1 starts far higher at a low discount rate, falls much faster, meets project 2 at 15.2382 per cent, and is worth less than it at every rate above. To the left of the red line the value rule says project 1; to the right it says project 2. An internal rate of return is a property of a cash flow stream alone and does not move when the discount rate moves, so the rate rule says project 2 everywhere. That is the deepest reason the two measures can disagree: one of them responds to the company's cost of capital and the other one has never heard of it.

Here is the same picture as a ladder of readings, so the claims survive without the drawing. The record carries these rounded to the nearest lakh, as Rs 59,53,00,000 against Rs 29,85,00,000 at 8.00 per cent and so on down to a negative Rs 5,61,00,000 against Rs 9,81,00,000 at 20.00 per cent; the table below is the same arithmetic carried to the rupee.

The crossing row deserves a word, because a careful reader will notice that the two figures are not identical. The two values differ by Rs 1,314 on about Rs 16,66,00,000, a difference in the seventh significant figure. The residue is the crossover rate being printed to four decimal places rather than carried in full: at 15.2382 per cent the two are Rs 1,314 apart, and at the exact rate they are equal. The rate has been rounded, not the arithmetic, and a difference in the last printed decimal is not a defect to be chased.

Why do the two lines cross at all, and why only once?

They cross because they start at different heights and fall at different speeds, and the faster one starts higher. At 8.00 per cent project 1 is worth Rs 59,52,61,524 and project 2 Rs 29,85,42,007, so project 1 begins almost exactly twice as high. Between 8.00 and 20.00 per cent project 1 falls by Rs 65,13,63,633 while project 2 falls by only Rs 20,04,19,579. Project 1 falls more than three times as far, so a line that started twice as high has to pass through the other one somewhere.

Why does it fall faster? Because it has far more money committed and it has it committed for the same five years. A discount rate is a charge on waiting, and project 1 is waiting on Rs 65,00,00,000 a year while project 2 is waiting on Rs 20,00,00,000 a year. Raise the charge and the bigger stream loses more in absolute rupees, every single time. The household version: if the printing shop and the tea counter both take five years to pay for themselves and the cost of borrowing rises, the printing shop is hurt more, because there is more money sitting out there waiting to come back.

Why only once? Because both curves are falling everywhere and neither one wiggles. A stream of one outlay and five positive receipts produces a value that decreases smoothly as the rate rises, without turning back up, which is the same property that gave each project exactly one internal rate of return in step two. Two such curves can meet once and then separate, and having separated they cannot meet again. That is why there is a single crossover rate rather than a region of them, and it is why step two was worth the ten seconds it took.

Which rule wins, and what is the reason rather than the convention?

Take the higher net present value. Take project 1. Project 1 is the answer, and the answer is not a house style or a preference for one tradition over another. The choice follows from a difference in what the two measures quietly assume about the cash a project pays out, and once that difference is on the table the choice makes itself.

Every appraisal measure carries a reinvestment assumptionWhat a measure silently assumes happens to the cash a project pays out before the end of the appraisal period., whether or not anybody states it. Project 2 hands the company Rs 20,00,00,000 at the end of each of five years. The Rs 20,00,00,000 does not sit in a drawer. The money goes back into the business, or it pays down debt, or it sits in a deposit. The measure being used has already made an assumption about what it earns while it does that, and the two measures make different assumptions.

The internal rate of return assumes those receipts go on earning the internal rate of return itself, which for project 2 is 28.65 per cent. Net present value assumes they earn the company's cost of capital, 12.00 per cent. The two assumptions are not close, and only one of them describes something Sankalp can actually do. If the company genuinely had somewhere to put spare cash at 28.65 per cent, that somewhere would be on the list of projects in front of the board, and the board would be discussing it instead.

Look at what the left panel is claiming. The left panel says that every rupee coming back was immediately put to work at 28.65 per cent, so by the end of Year 5 the company will be holding Rs 1,76,20,28,564 from project 2's receipts alone. The right panel says Rs 1,27,05,69,472, because every rupee that came back earned what capital costs the company. The gap of Rs 49,14,59,092 is not something project 2 produces; it is something the rate measure assumes on project 2's behalf. That is the entire reason net present value is the rule to follow here, and it is a reason rather than a convention.

One honest note on the left column. Compounding the five receipts at 28.65 per cent and growing the Rs 50,00,00,000 outlay at the same rate should give the same figure, because that is what an internal rate of return means. The two land about Rs 25,000 apart, and the whole of that residue is the rate being printed as 28.65 rather than carried as 28.6493. There is also a modified internal rate of returnA version of the rate measure that states its reinvestment rate explicitly instead of assuming it. Named here, taught separately., which states the reinvestment rate openly instead of assuming it. The modified measure is named here and not taught; it belongs to the rate measure's own subject.

What if the crossover rate had come out below the hurdle rate?

Then there would have been nothing to resolve, and this is the part that stops the answer from being memorised as a slogan. If the crossing had come out at 9.00 per cent while the company still discounted at 12.00 per cent, the hurdle would sit above the crossing, and above the crossing the smaller project has the higher net present value. The rate rule prefers project 2 at every discount rate, so project 2 would have the higher internal rate of return too. Both measures would point at project 2 and the meeting would be over in a minute.

The same thing shows on the ladder without changing anything. At 16.00 per cent project 2 is worth Rs 15,48,58,731 against project 1's Rs 12,82,90,875, and project 2 also has the higher rate at 28.65 against 18.72. At 16.00 per cent the two measures agree completely, so a company with a 16.00 per cent cost of capital would never discover that this pair of projects has a conflict in it at all. The conflict is not a property of the projects on their own. It is a property of the projects together with the rate the company happens to use.

Sankalp's rate is 12.00 per cent, which sits 3.2382 points below the crossing, and that is the width of the move it would take before the disagreement disappeared. Put plainly: the company's cost of capital would have to rise by more than three and a quarter points before the two measures started agreeing about these two projects. Computing the crossover rate is therefore a test as much as a technique. Quite often it shows that there was never a conflict to begin with.

How does this go wrong in a real meeting?

What is the honest qualifier on this answer?

Two things, and an answer given without them is half a finding. The first is the size of the win. Project 1 is worth Rs 12,21,49,291 more than project 2 at the company's own 12.00 per cent. Set against Sankalp's discounted cash flow enterprise value of Rs 21,28,13,79,094, that is 0.57 per cent; set against its traded enterprise value of Rs 22,40,00,00,000, the same amount is 0.55 per cent. Naming which of the two bases was divided by is not pedantry, because there are two enterprise values available for this company and they are not the same figure. Either way the win is real, it is worth having, and it is not the difference between one company and another.

The second is the capital it consumes. Project 1 costs Rs 1,50,00,00,000 more than project 2, and that money has to come from somewhere. Whether the board has it, and what else the same Rs 1,50,00,00,000 could have done, is a completely separate question with its own answer, covered separately in this sequence. The crossover rate settles the ranking between two mutually exclusive projects; affordability and the year's capital budget are covered separately.

One ruling travels with those figures and it is worth stating plainly. The five numbered projects in front of Sankalp's board are a list under consideration; they are not inside the company's locked five year forecast. The forecast's 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, and no project outlay is added to it anywhere. The list is assessed against the same 12.00 per cent hurdle the forecast is discounted at, and whether it is additional to the run rate or part of it is a question this record does not settle.

Where does this leave the internal rate of return?

Exactly where it was, and not discredited. The rate measure is correctly computed on both projects and it is not being dropped. The claim is narrower: when two mutually exclusive projects of different sizes are being ranked, the rate measure is answering a question about the intensity of a return and the decision needs a question about the quantity of one. A percentage is the right answer to how hard money works. It is the wrong answer to how much money the company ends up with, and only one of those two questions is on the table when a floor can hold one machine.

The rate measure keeps everything else it was useful for. The rate measure is the natural language for comparing a project against a borrowing cost, it needs no discount rate to compute, and it travels well between people who do not share a view about what the cost of capital should be. Aswath Damodaran's valuation material at pages.stern.nyu.edu is the standard reference for the argument that a measure's reinvestment convention has to be one the company can actually meet, and that argument is what settles the question here rather than any preference between the two. Koller, Goedhart and Wessels make the same point from the other direction, by putting growth, return on capital and value into a single expression in which rupees rather than ratios are the output.

The habit worth carrying away is smaller than any of this: when two appraisal measures disagree about two projects, do not argue about the measures, subtract the projects. The difference is a project, the project has a rate, and that rate against the hurdle is the answer. It works on Sankalp's valve line and automation cell, and it works on a tea counter and a printing shop with one shop front between them.