Value vs Price: Why the Two Numbers Are Rarely the Same

A value is constructed from assumptions somebody chose. A price is observed. For Sankalp Industrial Systems Limited, invented, the discounted cash flow says Rs 21,28,13,79,094 and the market says Rs 22,40,00,00,000, a difference of Rs 1,11,86,20,906, being 5.26 per cent of the computed figure. Reverse arithmetic turns that distance into 80 basis points of growth or 33 of rate.

The whole of this subject already sits inside an ordinary household conversation, so a scooter is the place to start. A household decides to sell a five year old scooter. Before anybody comes to look at it, they work out what it is worth. The scooter has maybe four good years left in it. A new one of the same model costs a certain amount. There is a service due, and the tyres will need replacing this year. The household puts those judgements together and reaches a figure. The figure they reach is built. Every piece of it came from a judgement somebody made at the kitchen table, and every piece of it could have been made differently.

Then the neighbour two floors down sells an identical scooter, same year, same condition, and the money actually changes hands. The neighbour's figure is not built. Nobody assumed anything to produce it. Two people agreed, and it happened.

The built figure and the observed figure are not two attempts at the same thing, and the mistake worth preventing is treating them as though they were. When the built figure and the observed figure differ, the useful question is not which of the two people got it wrong. The question worth asking is what the observed figure would need the built one to assume before the two agreed. On a scooter that can be done in the head. On a company it takes arithmetic, and the arithmetic is the subject of everything below.

What is the difference between a value and a price?

One is an output and the other is a record. The whole distinction is carried in that one line, and almost every confused conversation in this subject is one in which somebody has quietly stopped believing it.

A valueA number constructed from assumptions somebody chose, all of which can be inspected. is what comes out of a construction. Somebody chose how many years to forecast, chose what would happen inside those years, chose a rate to bring the cash back to today, chose a method for everything after the forecast ended, and chose a set of conventions about timing and about what counts as part of the operating business. Then arithmetic ran, and a figure appeared. The figure is downstream of every one of those choices. Change one and it moves.

A priceA record of what somebody paid, with no inspectable assumptions inside it. is what comes out of a transaction. Somebody wanted to sell at a number, somebody else was willing to buy at that number, and the trade happened. There is nothing upstream of a price that a reader can open up. A price is not the answer to a question anybody wrote down. A price is a fact about an event.

So the two numbers are different kinds of object, and the difference is not that one is more accurate than the other but that one has its inputs attached and the other has none at all. A value comes with a bill of materials. A price comes with a receipt. A bill of materials can be argued with line by line. A receipt cannot. A receipt makes no claim; it records an event.

Where does each of the two numbers come from?

For Sankalp Industrial Systems Limited the two figures come from two completely separate processes that happen to produce figures in the same unit on the same day. A shared unit and a shared date are exactly why they get compared and exactly why the comparison is so easy to misread.

The computed figure comes from a discounted cash flow. The model rests on a five year cash flow forecast, a cost of capital of 12.00 per cent, a terminal growthThe rate cash flows are assumed to grow at forever after the forecast stops. rate of 5.00 per cent a year forever after Year 5, a return of 18.00 per cent on every rupee of new invested capital, and year-end discounting rather than mid-year. The model produces an enterprise value of Rs 21,28,13,79,094. How that model is built is covered separately; the figure is taken as given here and the work is on what sits beside it. The shape of the construction is what matters: five choices, one arithmetic engine, one answer.

The traded figure comes from a share price. The shares of Sankalp Industrial Systems Limited change hands at Rs 90.00, and there are 20,00,00,000 of them, so the market capitalisationThe traded equity value, being the share price times the number of shares. is Rs 18,00,00,00,000. Walk that up the same balance sheet lines the other side walks down, adding gross debt of Rs 6,00,00,00,000 and minority interest of Rs 60,00,00,000 and taking out cash of Rs 1,20,00,00,000 and non-operating assets of Rs 1,00,00,00,000, and the traded enterprise value is Rs 22,40,00,00,000 exactly. The bridge from a share price to an enterprise value is covered separately too.

Behind the first figure sit five statements somebody can be made to defend; behind the second sits one event that already happened. The asymmetry between them is not a weakness on either side. Each object is doing the job it exists to do.

Can a value be wrong in a way a price cannot?

Yes, and the word wrong has to be used carefully in both directions or the answer sounds like a verdict when it is a definition.

A value can be wrong in the ordinary sense. A model can carry an arithmetic error. A model can discount a cash flow that belongs to shareholders at a rate built for the whole business, and that mismatch is simply incorrect. A model can build a terminal value on a growth rate that its own reinvestment assumptions cannot fund. A model can double count cash by adding a balance and deducting net debt in the same walk. Every one of those is a defect a reader can find by opening the model and checking a line, and every one of them produces a figure nobody should use.

A price cannot be wrong in that sense at all. A price is a record of an event, and it is accurate about the one thing it records: somebody paid this, on this day, for this. A price has no internal lines, so there is no internal line inside a price to be inconsistent with any other. Asking whether a price is wrong is asking whether a receipt is wrong, and a receipt can only be wrong about whether the transaction happened.

Whether a price is wrong about the future is a completely different question. How prices are formed, what a market does or does not know, how quickly anything is reflected in a figure that people trade at: none of that is the subject here, and none of the arithmetic below depends on any position about it. Only one thing is needed from the traded figure, and a traded figure can always supply it: a number somebody actually agreed to.

How far apart are the two numbers for this company?

Rs 1,11,86,20,906 apart on the enterprise side. There are two enterprise values available and dividing by the wrong one quietly changes the answer, so the very next thing that sentence needs is the base it should be read against.

The subtraction is Rs 22,40,00,00,000 less Rs 21,28,13,79,094. The answer is Rs 1,11,86,20,906. As a share of the computed figure that is 5.26 per cent. As a share of the traded figure it is 4.99 per cent. Both are correct and they are not the same statement, so the base is named every single time one of them is printed. The computed figure is the one being measured from, so the computed figure is the default base.

Now, is Rs 1,11,86,20,906 a lot? Every reader arrives with that question, and it has no answer in that form. A hundred and eleven crore sounds enormous said out loud, and it is 5.26 per cent of a figure that was built out of five assumptions, any one of which could reasonably have been set a little differently. The rupee figure is genuinely uninformative on its own, and the rest of this guide is about converting it into a form that is not.

Before that, one thing about the drawing of it. A difference this size is where a picture can do the arguing that the words have refused to do. Two figures that are 5.26 per cent apart look nearly identical on an axis that starts at zero and look like a chasm on an axis that starts just below them. Both drawings are accurate. Neither is neutral.

Why is the difference on the equity side the same rupees?

Because the lines that sit between an enterprise value and an equity value are the same four lines on both sides, so anything that happens in the operating value arrives at the shareholders untouched.

Worked through on the computed side: the enterprise value of Rs 21,28,13,79,094, plus cash of Rs 1,20,00,00,000, plus non-operating assets of Rs 1,00,00,00,000, less gross debt of Rs 6,00,00,00,000 and less minority interest of Rs 60,00,00,000. The computed equity value is Rs 16,88,13,79,094. On the traded side, the same four lines applied to Rs 22,40,00,00,000 land on Rs 18,00,00,00,000, the market capitalisation the walk began from. The check is that the walk is reversible and uses the same balance sheet in both directions.

Subtract the two equity figures. Rs 18,00,00,00,000 less Rs 16,88,13,79,094 is Rs 1,11,86,20,906. The equity difference matches the enterprise difference to the rupee. The four bridge lines are constants in this comparison, so the whole of the difference in the operating value lands on the shareholders and none of it is absorbed on the way.

The percentage does change, and the percentage is where a careless reader gets caught. The same Rs 1,11,86,20,906 is 5.26 per cent of the computed enterprise value and 6.63 per cent of the computed equity value. The denominator shrank by Rs 4,40,00,00,000 on the way down the bridge and the numerator did not move at all. Both are true. Quoting one of them without saying which figure it was divided by is how two people end up arguing about a disagreement neither of them actually has.

What does the same difference look like on one share?

Rs 5.59, and putting it in that unit is worth doing once because it is the unit most readers actually think in, even when the work above it was done in crore.

The computed equity value of Rs 16,88,13,79,094 divided by 20,00,00,000 shares is Rs 84.41 a share, rounding once from the full figure rather than from any printed intermediate. The shares trade at Rs 90.00. The difference is Rs 5.59 a share. Multiply Rs 5.59 by 20,00,00,000 shares and Rs 1,11,86,20,906 arrives for the third time, in a third unit.

One distance has now been stated in three ways and none of the three has yet given a reader anything to form a view about. Rs 1,11,86,20,906, or 5.26 per cent of the computed enterprise value, or Rs 5.59 on a Rs 90.00 share. Each is arithmetically exact and each leaves a reader in the same place: two numbers that differ, no way to think about the distance, and a strong pull towards deciding which one is right. The pull towards picking a winner is what the arithmetic below defuses.

What would the model have to assume to produce the traded figure?

Asking what the model would have to assume turns a rupee difference into something a person can actually think about, and asking it is the whole technique. The move has a name. Running a valuation backwards from an answer to the input that would produce it is reverse arithmeticRunning a valuation backwards from an answer to the input that would produce it., and what it recovers is an implied assumptionWhat a model would have to assume in order to produce an observed figure.: not what the market believes — nobody can know that — but what a model of this particular shape would need to assume before it landed on the observed figure.

Notice what has changed in that sentence. Nobody is being asked which number is right. The model is being held fixed in every respect except one, that one input is solved for, and the answer is a statement about arithmetic rather than about anybody's opinion. The answer is checkable. Two people running it independently get the same figure or one of them has made a mistake.

Two inputs are the ones a reader would most naturally reach for, so there are two ways to run reverse arithmetic on Sankalp Industrial Systems Limited. Both are worth having: each is a view of the same distance. Both rebuild the terminal value using the growing perpetuity Gordon set out, with the reinvestment made consistent with the growth being assumed. Making reinvestment consistent with growth is the argument Damodaran is named for. In each route exactly one input moves and everything else is held.

Route one, holding the cost of capital at 12.00 per cent

Hold the rate, hold the five year forecast, hold the 18.00 per cent return on new capital, hold the year-end convention, and ask what terminal growth rate produces an enterprise value of Rs 22,40,00,00,000. The answer is 5.80 per cent rather than the 5.00 per cent the model assumes.

The mechanism is worth seeing rather than taking on trust. Raising terminal growth does two things at once and they pull in opposite directions. Year 6 net operating profit after tax grows faster, so faster growth raises the cash flow the perpetuity starts from. Faster growth also raises the share of that profit which has to be reinvested to fund it. At 18.00 per cent on new capital a company growing at 5.00 per cent forever must reinvest 5 divided by 18 of its profit, and one growing at 5.80 per cent must reinvest 5.80 divided by 18. And faster growth shrinks the gap between the rate and the growth, and that gap is the denominator of the perpetuity. The third effect dominates, so the curve rises, and it rises faster the closer growth gets to the rate.

Run the growth rate up from 4.00 per cent and the value the model produces sweeps upward through the traded figure at one place, and that crossing is the entire content of the comparison. Below it the model sits under the traded figure, above it the model sits over it, and at 5.80 per cent the two meet.

Route two, holding terminal growth at 5.00 per cent

Now do it the other way. Hold the growth rate at 5.00 per cent, hold the forecast and the return on new capital, and solve for the cost of capital that produces an enterprise value of Rs 22,40,00,00,000. The answer is 11.67 per cent rather than 12.00.

The mechanism here is simpler, and the reason is that the rate does only one thing and does it everywhere. The rate discounts the five explicit years harder or more gently, and the rate sets the denominator of the perpetuity. Lower the rate and both effects push the value up. There is no second effect pulling the other way. A much smaller move in the rate therefore does the same work as a larger move in the growth rate.

Two routes, one distance, and the fact that they disagree about how large the move is says something useful about which input the answer is actually sensitive to. A tenth of a point on the rate is worth roughly two and a half tenths of a point on the terminal growth here, which is a property of this particular model at this particular pair of settings and not a general law. How a full two-way table over the rate and the growth rate is built and read is covered separately. One row of that relationship is used here to find a crossing rather than to fill in a grid.

Why are basis points the right unit for stating this?

Because basis points make the two routes comparable, and because they are small enough that a person can hold both answers in their head at once. None of the three rupee statements above manages that.

A basis pointOne hundredth of a percentage point. is one hundredth of a percentage point. Going from 5.00 per cent to 5.80 per cent is 80 basis points. Going from 12.00 per cent to 11.67 per cent is 33 basis points. Eighty basis points of growth and 33 of rate are the honest statements of the distance between the computed value and the traded price for this company, and there is nothing else to add to them.

Put the two beside each other on scales of the same width and something becomes visible that no rupee figure conveys. The growth move is a little over twice the rate move. Both are small. A difference that sounded like a serious disagreement when it was Rs 1,11,86,20,906 turns out to be eight tenths of a point on an assumption about the far future, or a third of a point on a rate. The reversal from a hundred crore to eight tenths of a point is the single most useful move in the whole comparison, and it is available to anybody who does the subtraction and then does one more step.

What does 5.80 per cent mean once inflation sits beside it?

Something quite different from what it means on its own, and stating an implied assumption without its inflation is stating half of it.

Every growth rate above is a nominal growthGrowth including inflation. rate, meaning it includes inflation. Expected inflation in this worked example is 5.00 per cent a year. The 5.00 per cent is an assumption of the example rather than a measured rate, and a different inflation assumption would move every real figure below. Put 5.00 per cent inflation beside the model's terminal growth rate of 5.00 per cent and the reading changes completely. The model is assuming this business grows at exactly the rate of the price level, giving real growthGrowth after inflation is taken out. of nothing at all, forever.

Now the implied 5.80 per cent. Take inflation out properly. Removing inflation is a division rather than a subtraction: 1.058 divided by 1.05 is 1.0076, so the implied real growth is about 76 basis points a year forever, not 80. The traded figure implies that this business grows about three quarters of a point faster than the price level forever, where the model implies it grows exactly with it. That is the same distance again, stated in the form that most readers will find easiest to think about, and it is still a statement about what the arithmetic implies rather than a view about whether either is the better assumption.

Where does the arithmetic stop, and what does it not establish?

The arithmetic stops at the two implied assumptions, and it refuses to write the sentence that would come next.

Here is everything the arithmetic supports, in full. The model gives an enterprise value of Rs 21,28,13,79,094 and the traded enterprise value is Rs 22,40,00,00,000. The difference is Rs 1,11,86,20,906, being 5.26 per cent of the computed figure. At a 12.00 per cent cost of capital the traded figure implies terminal growth of 5.80 per cent rather than 5.00. Once 5.00 per cent inflation is taken out, 5.80 per cent is about 76 basis points of real growth. Holding growth at 5.00, it implies a cost of capital of 11.67 per cent rather than 12.00. The whole distance is 80 basis points of growth or 33 basis points of rate. The arithmetic stops there.

The arithmetic does not say that 5.80 per cent is high or that 5.00 per cent was low. Nor does it say that 11.67 per cent is a sensible rate for a business like this one, or that 12.00 per cent was. Nor does it say that the shares of Sankalp Industrial Systems Limited are cheap, expensive, undervalued, overvalued, fairly valued or attractive. Nor does it say the model is missing something, and nor does it say the market is missing something. The arithmetic states what the traded figure implies, and it stops there.

Why so absolute about it? Because every one of those sentences requires a fact nobody here has. Calling 5.80 per cent too high would require knowing what this business can sustain forever, and nothing in this worked example establishes that; the 5.00 per cent in the model is a choice somebody made, exactly as much as the 5.80 per cent the traded figure implies. The only difference between the two is which one belongs to the person doing the writing, and that is not a reason to prefer it.

Why is stopping there the analysis rather than a failure of nerve?

Because of what a reader is holding before the arithmetic and after it. Comparing the two is a better test than any rule about what may be written.

Before the reverse arithmetic, a reader has two numbers that differ by Rs 1,11,86,20,906 and no way to think about the distance between them. The only question available is which one is right, and nobody in the room can answer it. The future has not happened, and neither figure is a measurement of it. An answer supplied at that point would be supplied from nowhere.

After the reverse arithmetic, the reader has a different question. Should a business whose revenue growth is falling from 10.00 per cent to 7.14 per cent across the five forecast years be assumed to grow at 5.00 per cent forever, or at 5.80? The growth question is small, it is specific, and evidence can address it. Somebody who knows this industry, this product line and this customer base can bring something to it. The arithmetic here is what converts an unanswerable question into an answerable one, and the conversion is the deliverable.

Supplying the answer as well would undo the conversion. The value of handing a reader an 80 basis point question rather than a hundred and eleven crore one is that they can now exercise their own judgement on it, and an answer supplied for them takes back the only thing the arithmetic produced. Stopping is not a refusal to conclude. Stopping is the conclusion.