Net Present Value: The Rule and Its Assumptions

Net present value discounts every cash flow a decision changes at the cost of capital and then subtracts the outlay. Sankalp Industrial Systems Limited's third valve line costs Rs 2,00,00,00,000 and returns Rs 65,00,00,000 a year for five years, which is worth Rs 34,31,04,532 today at 12.00 per cent. Accept a positive figure. The rule carries three assumptions: reinvestment at that rate, the rate itself, and flows that are incremental and after tax.

The shape of this decision is older than any of the words used for it, so start with a kitchen table. A household has saved Rs 6,00,000. The mother wants to buy a small delivery van for the tiffin business she runs: it will cost the whole Rs 6,00,000 and it should bring in about Rs 1,80,000 a year for the next five years, after everything it costs to run and after the tax on the extra income. There is another use for the money. The household could put it against the outstanding balance of a loan it is paying 12.00 per cent on. So the question at that table is not whether the van makes money. The van obviously makes money. The question is whether it makes more money than the loan repayment saves.

The kitchen table question is the whole of it. Every capital decision a company makes is that kitchen table question at a larger size: the money has an alternative, and a project has to beat the alternative rather than merely be positive. A company writes it down more formally because it has more projects, longer lives and a board that has to be able to check the working, but nothing conceptual is added on the way from Rs 6,00,000 to Rs 2,00,00,00,000. The net present value rule is simply the arithmetic of that comparison, done properly, with the timing of the money taken seriously.

The invented company in this guide is Sankalp Industrial Systems Limited, a listed manufacturer of industrial valves, precision castings and the aftermarket parts and service that go with them. Sankalp is looking at a numbered list of five projects. The working here takes one of them, project 1, the third valve line, and borrows a second, project 4, the regional warehouse, at the one point where a second project makes something visible that the first cannot.

What does the rule actually instruct?

Four things, in order, and only the last two are arithmetic. Take every amount of cash the decision changes. Discount each one back to today at the company's cost of capitalThe blended return that the people who provided a company's money, its shareholders and its lenders, require on it.. Add those discounted amounts up. Subtract what the project costs at the start. If what is left is positive, the project passes; if it is negative, it does not.

Notice which instructions the rule leaves out. The rule does not say to accept the project with the biggest answer. Ranking is a separate question, and this rule is an acceptance test. It does not say the answer is profit. It does not say anything at all about how long the money takes to come back. The rule tests one thing and one thing only: whether the money the project produces, valued today, is worth more than the money it consumes today. Everything else a person might want to know about a project is a separate question with its own separate answer.

Sankalp discounts at 12.00 per cent. The 12.00 per cent is its weighted average cost of capital. How a company arrives at its own rate is settled separately, so the rate is restated here rather than rebuilt. Shareholders and lenders together require that 12.00 per cent on the money they have put in, so it is the return the company has to beat before a project has added anything at all. When a company uses that same figure as the bar a project must clear, it is called the hurdle rateThe return 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..

What is the arithmetic on the third valve line?

Project 1 is the third valve line. The line costs Rs 2,00,00,00,000, paid at the start, and it is expected to bring in Rs 65,00,00,000 a year for five years. The annual figures are already measured after tax at the company's own assumed effective rate of 25.0 per cent. Discount them at 12.00 per cent, one year at a time.

The working prints to the rupee and a careful reader will start checking, so two notes on precision come before anything is read off that column. The discount factorThe multiplier that brings one year's cash back to today. At 12.00 per cent, money arriving at the end of Year 5 is multiplied by 0.567426856. column is printed to nine decimals, but every rupee figure beside it was computed at full precision and rounded once, at the end. Rebuilding the column from the printed factors lands a few rupees away from these figures, and neither version is wrong; they are the same number rounded at different points. The last two digits rest on a forecast of Rs 65,00,00,000 a year that nobody can produce to the rupee, so on an answer this size they carry no information whatever. The figure of record is Rs 34,31,04,532; the figure a person says out loud is Rs 34.31 crore, the same amount rounded to the nearest lakh.

The whole of the work here is turning Rs 3,25,00,00,000 of future cash into Rs 2,34,31,04,532 of today's rupees and then asking whether that beats Rs 2,00,00,00,000. The discounted receipts do beat it, by Rs 34,31,04,532. The project passes.

There is a second reading of the same two amounts, and it is worth having because it is the reading a person reaches for when the two projects being compared are different sizes. Dividing the Rs 2,34,31,04,532 by the Rs 2,00,00,00,000 instead of subtracting gives 1.171552, printed as 1.1716. The ratio is project 1's profitability indexThe present value of a project's inflows divided by its outlay. Above one is the same test as a positive net present value.. The two figures are the same fact stated two ways: one as a difference and one as a ratio. Anything above 1.0000 is exactly the same set of projects as anything above nothing, so the two measures can never disagree about whether a project passes. The two measures can disagree about how a project ranks against one of a different size, and ranking is a separate subject covered separately.

Why are five identical receipts not five identical contributions?

Look down the right-hand column of that table again. The cash column is flat: Rs 65,00,00,000 five times, the same amount every year. The worth-today column is not flat at all. It falls from Rs 58,03,57,143 to Rs 36,88,27,456, and the last receipt contributes about 64 paise for every rupee the first one contributes.

The taper is not a subtlety and it is not a correction. It is the entire reason the rule exists. A person who adds up Rs 65,00,00,000 five times, gets Rs 3,25,00,00,000, compares it with Rs 2,00,00,00,000 and concludes the project is worth Rs 1,25,00,00,000 has not made a small error. The error is Rs 90,68,95,468, more than two and a half times the true answer. The taper down that column is the whole of what discounting does, and the size of the taper is the size of the mistake a person makes by ignoring it.

The household version is the tiffin van again. Rs 1,80,000 arriving next year can go against the loan a year earlier and stop four extra years of interest, so it is not the same thing to that household as Rs 1,80,000 arriving five years from now. The gap between them is not a matter of opinion or of inflation; it is the arithmetic of what the earlier rupee can be doing in the meantime.

What does the answer actually mean once it is computed?

The answer means that Sankalp Industrial Systems Limited is Rs 34,31,04,532 richer today, on the figures assumed, if it builds the third valve line than if it does the next best thing with Rs 2,00,00,00,000. A comparison between two uses of the same money, stated in today's rupees, is the entire content of the number.

Now the things the answer is not. Every one of them gets written into a real paper somewhere. The answer is not Rs 34,31,04,532 of profit in any year. It is not Rs 34,31,04,532 of cash the company will ever receive as a single amount. It is not an amount that will ever appear on any line of any statement the company publishes. The cash the project actually produces is Rs 65,00,00,000 a year for five years, being Rs 3,25,00,00,000 in total, and none of those figures is Rs 34,31,04,532.

The Rs 34,31,04,532 is not money; it is the distance between two ways of using money, measured today. Koller, Goedhart and Wessels put the same point as the reason value and accounting profit are different objects that happen to be denominated in the same currency, and a great deal of confusion in board papers comes from treating them as though the currency made them the same kind of thing.

Why is the test the sign rather than the size?

Because the rule is answering a yes or no question, and a yes is a yes. Project 4 on the same list, the regional warehouse, costs Rs 90,00,00,000 and brings in Rs 16,00,00,000 a year for ten years. Discounted at the same 12.00 per cent the warehouse is worth Rs 40,35,685, or Rs 0.40 crore. Project 1 is worth Rs 34,31,04,532. Both are positive. Both pass.

Rs 40,35,685 on a Rs 90,00,00,000 commitment looks like nothing at all, so the verdict feels wrong the first time a person meets it. But look at what a passing answer means. It means the project beats what the company would otherwise do with the money. A project that beats the alternative by Rs 40,35,685 still beats the alternative. If the company has the money and nothing better competing for it, turning that project down means choosing the thing worth Rs 40,35,685 less. The acceptance test and the ranking question are two different questions, and collapsing them is how a company ends up rejecting good projects because they are not exciting.

Size changes how much of the answer is at risk, not whether the project passes. A project worth Rs 34,31,04,532 has room. A project worth Rs 40,35,685 has none, and how to report that honestly is the subject of the assumption block below. The size matters enormously; it just does not matter to the acceptance test.

What timing convention is sitting underneath every one of these figures?

An assumption so quiet that most people who use the rule have never said it out loud. Every figure in this guide uses year-end discountingThe convention that a year's cash is treated as arriving in one lump on the last day of that year, rather than spread through it.: each year's Rs 65,00,00,000 is treated as though it arrives in one lump on the last day of that year.

The cash does not, of course. A valve line sells valves every month. The cash comes in through the year. A company that assumed each rupee arrived on 31 March when in fact half of it arrived by September has discounted every rupee by a few months more than it should have, and so understated the project's value. The alternative convention treats the cash as arriving in the middle of each year instead, and it produces a slightly higher answer for exactly that reason.

Neither convention is right; what is wrong is not saying which one was used. Two analysts who work the same project on the two conventions will produce two different answers and will spend an afternoon looking for an arithmetic error that is not there. Year-end discounting sits behind every figure above, and it is named wherever one of them is printed. The whole of the discipline is that: state the convention, then keep it.

What does the rule assume happens to the cash the project pays out?

Here is the assumption almost nobody states, and it is built into the arithmetic rather than added to it. When the rule discounts Rs 65,00,00,000 arriving at the end of Year 3 by 0.711780248, it is doing exactly the same arithmetic, backwards, as growing an amount at 12.00 per cent for three years. Discounting at a rate and compounding at that rate are the same operation read in two directions. So the rule has already assumed that a rupee is worth 12.00 per cent a year, in both directions, over the whole period.

Turn that round and it says something about the cash the project pays out. The reinvestment assumptionWhat an appraisal measure silently assumes happens to the cash a project pays out, once the company has it. in the net present value rule is that each Rs 65,00,00,000, from the moment the company receives it, earns 12.00 per cent. Not more, not less.

Is that a strong assumption? The reinvestment assumption here is the most modest one available, and Aswath Damodaran presses that point hardest in his material on capital budgeting: a measure's reinvestment assumption has to be one the company can actually meet. 12.00 per cent is what Sankalp's own capital costs, so Sankalp can always meet it. If a rupee comes in and there is nothing to do with it, the company can pay down borrowing and save what that borrowing costs. If there is something to do with it, the next project on the list that clears the bar returns at least 12.00 per cent by definition. Clearing the bar means nothing else. The rule assumes the company can do with a spare rupee exactly what the company can, in fact, always do with a spare rupee.

The reinvestment assumption is not the only one available, and the alternative is not obviously silly. Another appraisal rule in this sequence solves for the rate that makes a project worth nothing, rather than valuing the project at a rate handed to it, and that rule carries a different assumption about the same cash. The two can therefore rank a pair of projects that cannot both be built in opposite orders, and when they do, the difference in the reinvestment assumption is the reason. The rate that other rule solves for, where it misleads, and how a disagreement between the two is resolved are all covered separately.

What does the rule assume about the rate itself?

Two things, and the second is the quieter and the more expensive of them.

The first is that 12.00 per cent is right. The 12.00 per cent is not a fact about the world; it is an estimate the company built, and it was built out of judgements. The rate rests on an assumed risk-free rate of 7.75 per cent, an assumed total equity risk premium of 5.00 per cent, a levered beta of 1.25 and market weights of 75 per cent equity and 25 per cent debt. Every one of those four inputs is a judgement made elsewhere, restated here and not rebuilt. Change any of them and the 12.00 per cent moves, and every project's answer moves with it.

The second is that project 1 carries the same risk as Sankalp Industrial Systems Limited as a whole. Applying the company's own rate to a project means exactly that, and it is almost never said. A third valve line in an existing plant, selling to the customers the company already has, is probably about as risky as the rest of the business, so the assumption is a fair one here. A project to enter a business the company has never been in is not, and discounting it at 12.00 per cent prices it wrongly in a direction nobody will notice for years. The error sits in the rate rather than in any figure a person can point at.

Getting the rate wrong does not produce an answer that is a bit off; on a project with no room it produces an answer with the wrong sign.

What does being wrong about the rate actually cost?

Read it off the one project on the list where there is nothing to spare. Project 4, the regional warehouse, is worth Rs 40,35,685 at 12.00 per cent, and its own return is 12.11 per cent. The warehouse clears the hurdle by eleven basis points. Eleven basis points is what its entire case is made of.

So suppose the 12.00 per cent is a little too low and the true figure is 12.50 per cent. Project 4 is then worth minus Rs 1,41,71,069 and it fails. Not smaller: negative. Half a point of an estimated rate did not shave the answer, it reversed it. Do the same to project 1 and it falls from Rs 34,31,04,532 to Rs 31,43,69,422, a real reduction that changes nothing about the decision.

Project 4 is a marginal projectA project whose value sits barely above nothing, so that a small change in the rate or the forecast reverses the answer., and the honest way to report one is not to print Rs 40,35,685 and move on. It is to say that the project clears the bar by eleven basis points, that the bar itself is an estimate, and that the answer is therefore inside the width of the estimate. A sentence like that is more useful to a board than the figure is. The figure invites a decision; the sentence describes what is actually known.

Before the control below is touched, a guess is worth making. Project 1 is worth Rs 34,31,04,532 at 12.00 per cent, and that sounds like a comfortable cushion on a Rs 2,00,00,00,000 outlay. At what rate does that cushion run out completely?

Move the control and one thing becomes hard to unsee. The Rs 34,31,04,532 that looked like a comfortable cushion on a Rs 2,00,00,00,000 outlay turns out to be worth 6.72 points of discount rate, and the curve falls the whole way. A project's whole cushion, expressed in the units the rate is measured in, is almost always smaller than it looks when it is expressed in rupees. None of that is a criticism of the rule. The falling curve is the rule speaking plainly, and it is why the second assumption set out here is the one worth arguing about.

What does the rule assume about the cash flows going in?

Three conditions, and all three hold for project 1 by construction. Holding by construction is exactly what makes it a worked example rather than a warning.

The first is that every amount is an incremental cash flowAn amount that differs between taking a decision and not taking it. Anything that happens either way is not incremental.: it differs between building the valve line and not building it. Anything that happens either way does not belong in the calculation, and money already spent never appears in one for that reason. The incremental test was settled earlier in this sequence and is used here rather than argued for.

The second is that every amount is measured after tax. Sankalp's 12.00 per cent is built with an after-tax cost of debt of 6.00 per cent, so it is an after-tax rate. Discounting a pre-tax cash flow at an after-tax rate compares two things measured on different bases, and it flatters every project it touches, in the same direction, every time. The Rs 65,00,00,000 used here is already net of tax at the company's own assumed effective rate of 25.0 per cent.

The third is that the list is complete. Every consequence of the decision belongs in it, including the ones that are not receipts: the working capital a new line ties up in stock and receivables while it is running, and the release of that working capital when it stops. A project appraisal that counts the sales and forgets the stock the sales require has not made a rounding error; it has left out a real outflow.

The three assumptions are not warnings printed on the side of the rule; they are the rule, and a person who cannot state all three does not know what their own answer means. Assumption three is concrete and somebody usually checks it, so it is the one most often got right. Assumption one is the one most often not noticed. Assumption two quietly does the most damage. A wrong rate produces a clean, checkable, professionally formatted answer to the wrong question.

Why can these answers be added together when returns cannot?

Because discounting is linear, and this small technical property is what turns the rule from something that runs on one project into something that runs on a list.

Projects 3 and 4 come from the same list. Project 3, the tooling upgrade, costs Rs 30,00,00,000 and brings in Rs 12,00,00,000 a year for four years, and at 12.00 per cent it is worth Rs 6,44,81,922. Project 4 is worth Rs 40,35,685. Added together they come to Rs 6,85,17,606. The other direction gives the same thing. Merging the two cash flow streams into one leaves the company spending Rs 1,20,00,00,000 at the start and receiving Rs 28,00,00,000 a year for four years and Rs 16,00,00,000 a year for the six years after that. Discounted at 12.00 per cent, that single merged stream is worth Rs 6,85,17,606. The same figure. The match is not a coincidence and not an approximation; it is what linear means.

One honest note on that column, of the kind that gets reported as a defect. Adding the two printed figures gives Rs 6,85,17,607, one rupee more than the unrounded sum of Rs 6,85,17,606. Nothing is wrong. Each project's figure was rounded once for printing, and the total was computed on the unrounded values and rounded once. Neither number is adjusted to make the column foot.

Now try the same thing with the returns. Project 3 returns 21.86 per cent and project 4 returns 12.11 per cent. So what does the pair return? Not 33.97 per cent, obviously. Not 16.99 per cent either. The average of two rates is only meaningful when the two things are the same size and run for the same length of time, and these two are neither. The pair does have a rate of its own, but it has to be solved out of the merged cash flow stream; there is no arithmetic that builds it from 21.86 and 12.11.

Values add and rates do not, and that single asymmetry is why one appraisal measure can be run across a whole list and the other has to be run one project at a time. The word additivityThe property that separate values can be added together. Rupee amounts have it; percentage returns do not. names that property, and it is doing more work than its dryness suggests. Additivity is the reason a company can look at the total value of a set of approvals, ask what a combination is worth, and get a meaningful answer.

How does this go wrong when the arithmetic is faultless?

Where does the rule stop being enough on its own?

At the exact point where the company has to choose rather than accept. The rule tests projects one at a time against a bar. The rule has nothing to say about what to do when two projects both pass and only one can be built, and nothing to say about what to do when six projects pass and there is only enough money for three.

Both of those situations are real on this very list. Projects 1 and 2 need the same floor of the same building, so only one of them can exist, and the acceptance test says yes to both. And a company with a fixed amount to spend has to rank rather than accept, a different exercise with a different answer. Both are covered separately.

There is a third limit worth naming, and it is not an arithmetic one. Project 5 on this list, the effluent treatment plant, costs Rs 45,00,00,000 and saves Rs 5,00,00,000 a year for ten years. At 12.00 per cent it is worth minus Rs 16,74,88,849. The rule says no, clearly and correctly. The plant is nonetheless mandatory under the site's own consent to operate, so the rule's answer is not the decision. The rule tells a company what a project is worth; it does not tell a company what it is obliged to do. Where a project has to happen, the useful question is not whether but which way of complying costs least, and that is the same rule applied to a different set of options.