Discounted Payback: Adding Time Value to a Simple Rule

Discounted payback counts the years a project needs to repay what it cost, once every year of cash has been discounted at the rate the company has to beat. Plain payback counted a Year 4 rupee as the equal of a Year 1 rupee, and discounting cures that flaw. The rule that survives is a screening measure. Discounted payback stops reading at the count and never asks how much the project is worth.

Underneath the rule there is one column of figures and nothing else. A project's cash flows are each brought back to today at the rate the company uses, and a running total is kept that starts in the hole by whatever the project costs. The single running column carries two separate answers. Asked when the running total first reaches zero, it gives a discounted payback. Asked where the running total finally stops, it gives something else entirely. Almost everything about the rule is a consequence of those two questions being asked of the same object.

The worked case throughout is project 3 of Sankalp Industrial Systems Limited, an invented company: a tooling upgrade. Five things about it are fixed before any arithmetic starts.

Everything from here on is arithmetic performed on those five lines. The cash is after taxMeasured once the charge on profit has already come out, so nothing further is waiting to be deducted downstream. before it reaches the column, so no further deduction is coming, and the 12.00 per cent is quoted where it is used rather than derived again.

What is the rule actually counting?

Think about a juice stall on a busy street corner. The owner buys a second refrigerator for Rs 60,000 because the first one runs out of chilled bottles by four in the afternoon. The obvious question is how many months of extra sales it takes before that Rs 60,000 has come back. The stall owner is asking a payback question, the most natural question anybody asks about money that has been spent. A company asks the same question about a new machine, a new warehouse or a new production line, only with more zeroes on it.

Discounted payback is that same question with one correction applied: before the money coming back is counted, each year of it is restated at what it is worth today. The outlayCash the company hands over on day one. Nothing has come back yet, so this is the hole the later years have to fill. is a figure in today's rupees already, because it is being paid today. The receipts are not. A receipt promised at the end of Year 4 is a promise about a rupee that has not arrived and cannot be put to work in the meantime, so counting it as though it were cash in hand quietly overstates how fast the hole is filling.

Both halves of that definition matter, and only one of them usually survives. The answer is a duration, so it comes out in years and never in rupees: a project does not have a discounted payback of Rs 6,00,00,000. And the count is built on discounted cash, so a rupee arriving later fills less of the hole than a rupee arriving sooner. Dropping the first half quotes an amount where a period was asked for. Dropping the second half is simply plain payback with extra arithmetic in front of it.

How is the number actually built?

Nothing about this rule is estimated. Every discounted payback figure is the output of four steps done in order, and no judgement about how long a project will take enters any of them.

The four steps are: discount, accumulate, locate the crossing, and split the crossing year. Step one applies the discount factorA multiplier smaller than one, fixed by the rate and by how many years away the money sits. Where it comes from is covered separately. for each year. Step two builds the running total, starting at whatever the project cost, expressed as a hole. Step three walks down that running total and finds the first year at which it is no longer negative. Step four looks inside that one year and works out how far through it the hole was actually filled.

What does the column look like, line by line?

Project 3 costs Rs 30,00,00,000 on day one and pays Rs 12,00,00,000 at the end of each of four years. Each of those four identical receipts sits a different distance away, so each is worth a different amount today. Here is the whole thing, with the running total beside it.

Each line above is rounded to the rupee for display while the running total is carried unrounded, so adding the printed lines by hand lands one rupee below the printed total in the last row. The one rupee gap is the rounding showing, not an error in the column.

Read the last column and the crossing is obvious: at the end of Year 3 the project is still Rs 1,17,80,248 short, and Year 4 brings Rs 7,62,62,169 of value in today's rupees, far more than enough to close it. So the hole closes somewhere inside Year 4. Divide the shortfall by the year that closes it and the answer is 0.1544704 of that year. Add the three full years already gone and the discounted payback (DPB) of project 3 is 3.15 years.

How real is the fraction after the decimal point?

The 0.15 of a year is the one part of the calculation that describes nothing real. InterpolationSplitting a whole period in proportion, so an answer landing part way through it can be quoted as a fraction rather than rounded to the nearest whole period. assumes the year's cash arrives in a smooth, even trickle from the first day to the last. Cash does not behave that way, and nothing in the forecast says it does. A tooling upgrade might bill quarterly, or seasonally, or in one lump when an annual contract settles.

So why bother with the fraction at all? Because the alternative is worse. Without it every project answers in whole years, and a proposal that closes its hole on the second day of Year 4 scores the same as one that closes it on the last day. The fraction restores some ordering between projects that would otherwise be tied, and it costs nothing as long as everybody understands it as a convention. The rule to carry is simple: treat the whole number of years as a fact about the column and the digits after the point as a convention for breaking ties. Quoting a discounted payback to three decimal places is claiming a precision the cash timing cannot support.

What does discounting actually repair?

Plain payback has one arithmetic property that ruins it, and it is easy to miss because the arithmetic itself is correct. Plain payback adds a rupee arriving in Year 4 to a rupee arriving in Year 1 as though the two were the same object. The two rupees are not the same object. Time valueWaiting has a price. The same amount is worth less the further into the future somebody promises to hand it over, because until then it cannot be put to work. says the later one is worth less today, and no amount of care taken over the forecast repairs an addition that was wrong before the forecast was written.

All four of project 3's receipts are the same amount, so the size of the correction shows cleanly. Rs 12,00,00,000 arriving at the end of Year 1 is worth Rs 10,71,42,857 today. The identical Rs 12,00,00,000 arriving at the end of Year 4 is worth Rs 7,62,62,169. The same cash buys less repayment every year it is delayed, and by the fourth year it is filling barely more than three quarters of the hole its first-year twin would have filled.

In household terms: a neighbour who owes Rs 50,000 and offers to settle next month is offering something close to Rs 50,000. The same neighbour offering to settle in four years is offering considerably less, and the difference is felt instinctively even without a discount factor to hand. Plain payback is the rule that fails to feel it. Discounted payback is the rule that does.

Why does one column answer two questions?

The running total answers a second question as well as the first, and the second question is the one that decides whether a project should be done. Ask when the running total first reaches zero and the answer is 3.15 years. Ask where the running total finally stops and the answer is Rs 6,44,81,922, project 3's net present valueAdd every discounted receipt, take away what was spent, and whatever is left over is this. Net present value has a rule of its own..

The shape of the running total is why the two measures can never contradict each other about whether a project passes at all. If the running total reaches zero at any point inside the project's life, it must end above zero, so the project has a positive value. If it never reaches zero, it must end below zero. Agreement on the pass or fail question is structural, and it costs the two rules nothing. The two rules can disagree, violently, about which of two passing projects is the better one.

What does the rule still throw away?

The first question does something worth noticing with the rest of the column. Discounted payback stops reading at 3.15 years. Whatever the running total does after that instant is outside the rule's field of view, a property of the rule rather than an accident. And on project 3, everything between 3.15 years and the end of Year 4 adds up to exactly Rs 6,44,81,922, the whole of the value the project creates.

Neither of those things is an accident peculiar to this one project. The running total is at zero at the moment the rule stops reading, so whatever it climbs to afterwards is the entire net present value, always. The rule discards precisely the part of the column that answers whether the project is worth doing. Everything it does read is the part where the project is still working off what it cost.

Can two proposals score the same and be worth wildly different amounts?

An argument made in the abstract is easy to nod at and easy to forget. Sankalp Industrial Systems Limited's project list contains a pair that settles the argument permanently. The pair falls straight out of the numbers, with no construction needed.

Project 2 carries a value of Rs 22,09,55,240. Project 3 carries Rs 6,44,81,922. Yet the two of them score identically on both payback measures. Not approximately, not to the second decimal place as a matter of luck, but identically, and for a reason visible in the inputs.

The reason sits in one ratio. Project 2 costs Rs 50,00,00,000 and returns Rs 20,00,00,000 a year, so its cost is 2.50 times its annual receipt. Project 3 costs Rs 30,00,00,000 and returns Rs 12,00,00,000 a year, so its cost is also 2.50 times its annual receipt. Once two projects share that ratio and share a discount rate, both the plain count and the discounted count depend on the ratio alone and not on the size of anything. The two proposals cross zero at the same instant because they are the same shape, and the sizes cancel out of the arithmetic entirely.

Then project 2 does something project 3 cannot. Project 2 carries on for a fifth year and collects another Rs 20,00,00,000. Both measures stopped counting at 3.15 years, a point inside the fourth year, so neither of them ever looks at the fifth.

Where does the missing money actually sit?

The gap between the two proposals is Rs 15,64,73,318, being the difference between the two locked values. The fifth year explains most of that gap, though not the whole of it.

So most of the difference is one year of cash that arrives after both rules have closed their books, and every rupee of the rest is invisible to them too. This is the bias the rule builds in, stated as plainly as it can be put: discounted payback prefers short projects. That preference is sometimes exactly what a reader wants, which is why the rule survives, and it is never a statement about which project is worth more.

Where does the maximum itself come from?

A payback figure on its own decides nothing. The figure becomes a rule the moment somebody names a maximum, and the maximum is where the trouble starts. Suppose the board of Sankalp Industrial Systems Limited writes a policy saying no project may take longer than four years to repay on a discounted basis. Here is what that policy does to the project list.

A four year maximum passes projects 2 and 3, at 3.15 years each. The same maximum rejects project 4, whose 9.92 years is nowhere close. And it rejects project 1, at 4.07 years. Project 1 misses by 0.07 of a year, and project 1 is the most valuable proposal on the whole list at Rs 34,31,04,532. Move the maximum to five years and project 1 comes back in with room to spare, while project 4 still fails. A single change of policy wording, four rather than five, decides the fate of the largest source of value in the set.

Now hold that policy next to the 12.00 per cent it sits beside in the same document. The 12.00 per cent is a derived figure: shareholders name a return they require, lenders charge a rate, and the two get combined in the proportions the company actually funds itself in. Somebody can argue with it on evidence, and somebody regularly does. The four years is derived from nothing whatever. The four years came out of a meeting. Nothing in a 12.00 per cent cost of capital says four years rather than five, or three, or seven, and no arithmetic anywhere can be asked to produce that figure.

Both numbers arrive in the same policy document, printed in the same font, and a reader who does not know the difference between them will treat them as equally solid. The two numbers are not remotely equally solid. One is an estimate of something real. The other is a preference, and a legitimate one, but a preference all the same.

When is this rule the right one to reach for?

Discounted payback fails as a way of deciding anything. As a cheap way of computing something it survives in real approval processes, for two honest reasons.

The first is triage. A group with forty proposals on the desk and enough analyst time to appraise twelve of them properly has a real problem, and it is not a valuation problem. Somebody has to cut the list. A discounted payback figure can be produced from a forecast and a rate in a couple of minutes, it uses a defensible rate rather than none at all, and it will reliably remove the proposals that take a decade to come back. Used that way it is doing a job nothing else does as cheaply. Opportunity costThe return given up by putting money here rather than into the closest alternative use of it. A required return is a stand-in for it. applies to analyst time as much as to capital.

The second is exposure. A lender reading a project proposal is asking a different question from an owner. An owner wants to know how much the project is worth. A lender wants to know how long the borrower's money is committed and unrecovered. The window before recovery is where something can go wrong and there is nothing to show for the spending. A discounted payback figure is a crude but genuine reading of that window, and it improves on the plain version because it prices the waiting. A household thinks the same way about rooftop solar panels: the question is not only whether the panels pay for themselves, but how many years the household is out of pocket before they do. A roof repair or a job change inside that window changes everything.

What can this rule never be asked to do?

There is one instruction this rule can never carry out, and it is the instruction it gets given most often: rank two proposals that both pass. Projects 2 and 3 both clear a four year maximum comfortably. The rule has now said everything it has to say about them. Asking it which of the two is better is asking a stopwatch which of two runners is richer.

The five numbered proposals are a list under consideration by Sankalp Industrial Systems Limited. The list sits outside the company's five year forecast, whose capital spending line is the run rateThe pace of spending a business is running at before anybody puts a new proposal on the table. already approved. Both objects get discounted at 12.00 per cent, and that shared rate is the whole of the connection between them. Whether approving one of these proposals would push that spending line upward, or come out of what already sits inside it, is not something the record says.

Moving the discount rate moves the crossing, and how far it moves across all five proposals at once is set out under the discounted payback rate tool.

Sankalp Industrial Systems Limited is invented.
Educational material. Not advice on any investment, tax, budget or market position.