Valuation Range: Four Methods, Four Numbers, One Span

Run several defensible methods on one company and they do not agree. Sankalp Industrial Systems Limited, invented, produced four enterprise values on one day: Rs 21,28,13,79,094, Rs 22,46,40,00,000, Rs 24,48,00,00,000 and Rs 27,36,00,00,000. The lowest to the highest is a span of Rs 6,07,86,20,906, being 28.56 per cent of the low end. The span is the output, and the width of it gets explained rather than closed.

Before any of that, a flat in a familiar building. A household has lived in a two room flat for twenty two years and has decided to sell it. Over one afternoon four people give them a price, and not one of the four is being careless.

The broker says a similar flat two floors up changed hands last month, and after knocking a little off for the kitchen this one should fetch about the same. The bank valuer, who is there because the buyer will need a loan, works out what it would cost to build the structure today and adds a share of the land, and lands lower. The cousin who has rented out three flats says forget both of those: what it is worth is whatever the monthly rent will support after the maintenance bill, and he lands lower still. And the neighbour on the same landing has wanted to knock through into the next flat for six years. To him this is not a flat, it is a second bedroom for his son, so he quietly says he will pay more than any of them.

Four prices for one flat in one afternoon, and the argument is not about which person can add up. The four numbers differ because the four people asked four different questions. The broker asked what similar flats have fetched. The valuer asked what it would cost to rebuild. The cousin asked what the cash it throws off will support. The neighbour asked what it is worth to him specifically. Nobody else in the room was even asking that question.

Four answers to four differently worded questions: that is a range. Every valuation of a company has the same shape, and the interesting part is never the four numbers. The interesting part is which of the four sits at each end, and why.

What is a valuation range actually claiming?

Less than people think, and something more useful than they expect. A valuation rangeThe span between the lowest and the highest of several defensible valuations of the same thing. claims exactly this: several methods were run properly, none of them was thrown away, and here is where each landed. The claim ends there.

Notice what is not in it. There is no statement that the truth is somewhere in the middle. There is no statement that a figure near the centre is more likely than one near an edge. No probability was ever computed, so no probability is attached to any part of it. A range is a set of separate answers to separate questions, and it is not a distribution.

The distinction matters because of how ranges get drawn. Draw one as a smooth bar and the eye immediately reads a middle, a centre of gravity, a most likely value. Nobody wrote that. The middle arrived free with the shape. Draw the same four numbers as four upright spikes with flat nothing between them and the drawing tells the truth about what is known. Four points, and a lot of empty space.

The alternative to a range is a point estimateA single number offered as the answer, with the assumptions behind it no longer visible., and a point estimate makes a much bigger claim than most people realise. A point estimate says the assumptions behind it are settled. The same single number says one method was the right method and the others were noise. Nobody working on a live company can honestly say either of those. The range survives for that reason, and the single number keeps being asked for anyway.

What has to be true before four values can share one axis?

Two conditions, and both get checked before a single mark is made on the chart. Skipping either leaves the drawing wrong in a way that looks completely reasonable.

The first condition is that every figure is the same kind of value. All four here are enterprise valuesThe value of the operating business itself, before any question of who funded it and how., meaning each one is a value for the operating business before any question of who funded it. An enterprise value is not the only kind of value in circulation. The same company also has an equity value of Rs 16,88,13,79,094. An equity value measures what is left for the shareholders after the lenders, the cash, the minority interest and the assets that sit outside the operating business have all been dealt with. The equity value is perfectly correct and it belongs to a different axis. Put it on this one and it would look like the lowest valuation in the set, when in fact it is not a valuation of the same thing at all.

The second condition is that they all sit on the same denominator. Each of the four here is a multiple of the same Year 0 earnings before interest, tax, depreciation and amortisation (EBITDA) of Rs 2,88,00,00,000 for the same company. Because of that, the four values can be restated as 7.39 times, 7.80 times, 8.50 times and 9.50 times, and those four figures can be compared with nothing further done to them. One unit of a multiple is called a turnOne whole unit of a multiple, as in one turn of EBITDA. Two turns is two whole units., so the distance from the bottom of that list to the top is 2.11 turns.

Both conditions are conditions about comparability, and comparability is a property of the pair of numbers rather than of either one alone. A figure is never wrong for being an equity value or for being built on a different denominator. A figure is wrong only once somebody sets it beside a figure it cannot be compared with and draws a line between the two.

Where do the four values land when they are drawn on one scale?

In an order, and the order is the first thing worth reading. How each one was built is settled elsewhere, so the four are restated below in a single line each.

The discounted cash flow gives Rs 21,28,13,79,094, being 7.39 times. Trading comparablesA valuation built from the multiples at which similar listed companies are currently trading. give Rs 22,46,40,00,000, being 7.80 times. The leveraged buyout entry value is Rs 24,48,00,00,000, being 8.50 times. And precedent transactionsA valuation built from the multiples paid in completed acquisitions of similar businesses. give Rs 27,36,00,00,000, being 9.50 times.

The range runs from Rs 21,28,13,79,094 to Rs 27,36,00,00,000, and only two of the four methods have anything to do with that. The discounted cash flow sets the low end and the precedent set sets the high end. The trading comparables and the buyout entry sit inside, and if either of them moved by a crore the width of the range would not change by a rupee. Sitting inside is not a criticism of those two methods. The position of a marker is a fact about this particular set of four, on this particular day. Knowing which methods carry the ends is worth more than knowing how many were run.

How wide is the span, and which end was it divided by?

The spreadThe distance between the two ends of a range, given in rupees or as a percentage of a named end. is Rs 6,07,86,20,906. In rupees that figure is unambiguous and there is nothing to argue about. The moment it becomes a percentage, an argument appears, and it is an argument nobody announces in advance.

Divide that spread by the low end of Rs 21,28,13,79,094 and it is 28.56 per cent. Divide the identical spread by the high end of Rs 27,36,00,00,000 and it is 22.22 per cent. Both divisions are correct. Both describe the same gap between the same two numbers. The two answers differ by more than six percentage points because the two denominators differ by more than six hundred crore.

Shopping teaches this already. A shirt marked at Rs 1,000 and sold at Rs 800 is twenty per cent off, dividing by what it was marked at. The same two hundred rupees is twenty five per cent of the price actually paid. Nobody is lying in either sentence; the base moved. The rule that falls out of it is short and it holds everywhere in this subject.

A percentage without its base named is not a comparable figure, so every spread in this guide says which end it was divided by in the same sentence as the number. The habit costs four words and it removes an entire class of argument, including the one where two people quote the same range at each other and think they disagree.

One more note on the arithmetic. Every figure below is printed to the rupee. The spread is Rs 6,07,86,20,906 exactly. Written in crore it is 607.86 crore. The crore figure is already rounded, so multiplying 607.86 by a crore lands on a different number from the rupee figure. Every figure here is computed on the full value and rounded once at the end, never rounded twice and never rebuilt out of another figure that was already rounded. On an answer of this size the last two digits carry no information whatever, and they are printed only so that a reader who wants to check the subtraction can.

Where does the traded figure sit against the range?

Inside it.

The traded enterprise value of Sankalp Industrial Systems Limited is Rs 22,40,00,00,000, being 7.78 times the same Year 0 EBITDA. The traded figure goes onto the drawing as a reference lineAn observed figure drawn across a range for comparison, rather than one of the estimates in it. and not as a fifth marker, and the distinction is real. The four markers were each constructed: somebody chose a forecast, a rate, a peer set, a debt package. The traded figure was not constructed by anybody in this exercise. The traded figure records a market in action, observed rather than estimated. Putting it in among the four would quietly suggest the market is running a fifth method, and it is not running any method at all.

An observed figure and an estimated figure are different objects, and mixing them on one drawing is how a reader ends up thinking a price is an opinion somebody defended.

There is a drawing problem here worth naming rather than hiding. The traded figure of Rs 22,40,00,00,000 and the trading comparables figure of Rs 22,46,40,00,000 are Rs 6,40,00,000 apart. On a scale wide enough to hold all four values that distance is under five pixels. At five pixels the drawing appears to be saying the two are the same number. The two are not the same number. Rs 6,40,00,000 is 0.29 per cent of the traded figure and 0.28 per cent of the trading comparables figure, and the two percentages differ in the second decimal for exactly the reason the last section gave: two bases, two answers. Rather than let a picture assert something the arithmetic does not, the next figure pulls twenty crore of that scale open.

Why does each of the four sit where it sits?

Because each answered a differently worded question, and this is the part that turns a span into a range. An unexplained spread is a defect rather than a range, so every marker gets one line saying what it was allowed to assume that the others were not.

The discounted cash flow is the lowest at Rs 21,28,13,79,094 because it grants the fewest permissions. The model values the standalone cash flows of the business as it stands, at the company's own 12.00 per cent weighted average cost of capital, with no control, no synergy, no change of funding and nobody else's plan for it. The model measures the worth of the business to itself. Aswath Damodaran is the source of the argument that a terminal value has to be consistent with the reinvestment the same forecast assumes, and that consistency is one of the reasons this figure lands where it does.

Trading comparables sit Rs 1,18,26,20,906 above the model, being 5.56 per cent of the model figure. The comparables figure is what the market pays today for a minority stake in businesses that look similar, and it carries whatever the market currently assumes about the sector. Koller, Goedhart and Wessels make the case for treating a multiple as a check on a cash flow model rather than as a replacement for one, and a gap of this size is exactly the kind of check they mean: small enough to be a difference of view, large enough to be worth naming.

The leveraged buyout entry sits above both at Rs 24,48,00,00,000, and this is the one that surprises people. A financial buyer can pay more than the standalone model says the business is worth without believing anything different about the business. Nothing in that case assumes better revenue or a better margin. The extra comes from funding a large part of the price with debt, deducting the interest against tax, and paying the debt down out of the same cash flows the model was already forecasting. Leverage and a tax deduction, not a better opinion. Reading a higher offer as a stronger view of the business is one of the most common misreadings in this subject.

Precedent transactions are the highest at Rs 27,36,00,00,000. Every price in that set was paid by a buyer taking control, and most were paid by buyers who had their own reasons for wanting that particular business. Against the trading median of 7.80 times, the precedent median of 9.50 times is 1.70 turns higher. In percentage terms that is 21.79 per cent.

Now the refusal, and it travels with that 1.70 turns wherever the figure appears. The gap cannot be split into a share for control and a share for synergy. The control share and the synergy share are the pair of numbers most commonly made up in this subject, usually offered as a confident round figure for one and another for the other, and nothing in a completed transaction record separates them. A buyer paid one price. The reasoning inside that buyer's head was not disclosed, was probably not agreed on inside their own room, and certainly cannot be recovered from a multiple. A division nobody can check is worse than no division at all, so naming the gap and refusing to divide it is the correct treatment.

What happens to the range when one method is dropped?

Something very uneven, and it is the most useful thing in this guide. Removing one of four methods might be expected to take roughly a quarter out of the range. Removing one does nothing of the sort.

Remove the precedent transactions and the range runs Rs 21,28,13,79,094 to Rs 24,48,00,00,000. The spread falls from Rs 6,07,86,20,906 to Rs 3,19,86,20,906, from 28.56 per cent of the low end to 15.03 per cent of the same low end. The precedent set was holding the high end on its own, so deleting one method out of four cut the width roughly in half.

Remove the leveraged buyout entry instead and nothing happens at all. The range still runs Rs 21,28,13,79,094 to Rs 27,36,00,00,000 and the spread is still Rs 6,07,86,20,906. The buyout value sits inside the range, so it contributes exactly nothing to the width. The buyout value is the one marker that shows a buyer paying more without thinking anything different, so it contributes plenty to the understanding. To the width it contributes nothing.

Remove the discounted cash flow and the range runs Rs 22,46,40,00,000 to Rs 27,36,00,00,000, a spread of Rs 4,89,60,00,000, being 21.79 per cent of the new low end. The 21.79 per cent should look familiar, and the match is not a coincidence. Both surviving values are multiples of the same Year 0 EBITDA, so the percentage gap between the two values has to equal the percentage gap between the two multiples, and 9.50 over 7.80 is the same 21.79 per cent. When two figures share a denominator, a comparison of the figures and a comparison of the multiples are the same comparison written twice.

Remove both of the market based methods, leaving the model and the buyout, and the range runs Rs 21,28,13,79,094 to Rs 24,48,00,00,000 again, a spread of Rs 3,19,86,20,906. Two markers left. Calling that a range would be generous.

Why is the midpoint not the answer, and why is the average not either?

Because both of them answer a question that none of the four methods asked. The midpoint and the average are arithmetically correct, and neither is a valuation. Both are worth computing anyway. The reason neither is a valuation becomes clear only once the two sit beside the four.

The midpointThe figure halfway between the two ends of a range, being their average. of the range is Rs 24,32,06,89,547, or Rs 2,432.07 crore. The midpoint uses two of the four numbers and ignores the other two completely. The plain average of all four is Rs 23,89,63,44,774, or Rs 2,389.63 crore. The average uses all four and gives each the same weight. Equal weight is a strong claim about four methods built on completely different assumptions.

The two sit Rs 42.43 crore apart, and they agree only when the values are evenly spaced. These four are not evenly spaced. The three gaps between consecutive values are Rs 118.26 crore, Rs 201.60 crore and Rs 288.00 crore. The gaps widen towards the top, so the average is dragged towards the crowded bottom while the midpoint stays exactly halfway between the two extremes.

The average carries a rounding trap that a careful person walks straight into, so it is worth doing slowly. The four values sum to Rs 95,58,53,79,094. The sum is an odd rupee count, so dividing it by four genuinely lands on a half rupee: Rs 23,89,63,44,773.50. Rounded to the rupee it is written Rs 23,89,63,44,774, and a half rupee has been rounded away in the writing of it.

Now the trap. Take the four values as they are usually printed, in crore to two decimals, and average those instead: 2,128.14, 2,246.40, 2,448.00 and 2,736.00 give 2,389.635. Rounded up that is 2,389.64. The correct figure is 2,389.63. Never average a printed column; average the full values and round once at the end. One paisa of carelessness at the start became a lakh at the finish, and nothing in the arithmetic reveals that it happened.

The midpoint has no such problem. Rs 21,28,13,79,094 plus Rs 27,36,00,00,000 is Rs 48,64,13,79,094, and half of that is Rs 24,32,06,89,547 exactly. The midpoint reconciles to the rupee. Reconciling perfectly does not make the midpoint a valuation. The midpoint is a tidy number that answers nothing anybody asked.

How is a range written down so somebody else can argue with it?

In four lines, one to a marker, and each line carries four things: the number, the method that produced it, the one assumption that moves it most, and which way it moves when that assumption moves. Anything shorter is not usable by a second reader.

Take the bottom line as an example. Rs 21,28,13,79,094, discounted cash flow, most sensitive to the terminal growth of 5.00 per cent, and it rises as that growth rises. A colleague reading those four items can immediately do something with them. The colleague can say that growth forever should be lower, and knows which direction the number goes. The colleague can say the rate is wrong, and knows it is the same line being attacked. A colleague cannot nod at a number and move on. A bare figure invites exactly that.

Take the top line. Rs 27,36,00,00,000, precedent median, most sensitive to which completed deals are in the set, and it falls as the set is widened. Now a colleague can ask the two things that actually matter about a precedent set: what is in it and what was left out. People usually ask instead whether the median is the right statistic, and that matters less.

A range written down like this is not a conclusion, it is an invitation to disagree in a specific place. An invitation to disagree in a specific place is the highest form this work takes. Two people looking at four lines can find in about a minute the single assumption they actually differ on, and everything else in the written range turns out to be agreement they did not know they had.