Base, Bull and Bear: Building a Valuation Range

A scenario case moves several assumptions together because they move together in the world. Sankalp Industrial Systems Limited, invented, is worth Rs 21,28,13,79,094 in the base case, Rs 26,26,89,00,000 in the bull and Rs 16,54,94,00,000 in the bear. A company growing more slowly reinvests less, so the bear has more free cash flow in Year 1 than the base does, and the difference lands beyond Year 5.

A tea stall makes the shape of the arithmetic easier to feel before any spreadsheet appears. The woman who runs the stall outside an office block is thinking about the year ahead. If the building next door finishes and fills up, she sells more cups, she can lift her price a little because the queue is longer, the lender who has been circling starts talking about a better rate, and the whole stall looks like it will still be standing in ten years. If the building stalls, none of those four things happens. Not one of them.

The reader linked those four changes without instruction. Nobody pictures a world in which the queue doubles but the lender still charges the old rate and the stall still looks like it might close next year. A world in which the queue doubles and nothing else moves is not impossible. The four things are wired to each other, so a reader's head does not produce that world. Nobody has to be shown the wiring. A case is a version of the world in which every assumption is consistent with every other assumption, written down so that somebody can argue with the world rather than with the arithmetic.

What is a scenario case, and what makes a set of them a range?

A scenario caseOne internally consistent state of the world, with every assumption in it written down. is one state of the world with every assumption in it written down. Not a nudge to a spreadsheet. Not a number somebody thought looked better in a meeting. A state of the world, described completely enough that a reader can disagree with the world rather than with the sums.

Run the same model three times at three of those states and three answers come out. The three answers together are a range. A range differs from a scatter of guesses because each answer stays attached to a written down state of the world, and each state of the world holds together on its own terms. Cut that attachment and what remains is three numbers with no story behind any of them. Three storyless numbers are worse than one number. More work appears to have been done when in fact rather less has.

Sankalp Industrial Systems Limited, invented, is the company used throughout. The company makes industrial valves, precision castings, and the aftermarket parts and service that go with them. Sankalp has one subsidiary, Sankalp Coatings Private Limited, invented, and one associate holding, Aruna Tooling Private Limited, invented. Year 0 is its last completed year, and in Year 0 it turned over Rs 12,00,00,00,000. Every rupee figure that follows is Sankalp's own.

The discounted cash flow model those three answers come out of is built separately in this subject area, and a reader arrives already knowing that a value is the output of a named set of assumptions, discounted at a rate, with a block at the end standing in for everything after the forecast stops. Running that settled machine three times, at three settings, is a narrower exercise and a more useful one.

Which four assumptions move, and why exactly these four?

Four, and they are named before anything is computed. The annual rupee increase in revenue. The margin on earnings before interest, tax, depreciation and amortisation (EBITDA). The cost of capital. The terminal growth rate. Each case sets all four, and a case is not a case until every one of the four has both a value and a reason.

The first is the revenue incrementThe fixed number of rupees added to revenue each year, Rs 1,20,00,00,000 in the base case., a number of rupees added to revenue every year rather than a percentage. The forecast is built in rupees deliberately, so the growth rate falls year after year while the rupees added never move. The second is the EBITDA marginEBITDA as a share of revenue, held flat within each case., held flat inside each case and set differently between them. The third is the cost of capital, the rate at which every future rupee is pulled back to today. The fourth is the terminal growth rate, the pace at which the business is assumed to keep growing forever after Year 5.

The four move together because a story about the world changes all four at once. A world in which this company sells more valves is a world in which it fills its plants better, borrows a little more cheaply and looks likely to last longer. Selling more, filling plants, borrowing cheaply and lasting longer are four separate sentences about four separate parts of a model. The four are one sentence about the world.

Why do the four move together rather than one at a time?

Because a reason for one of them is usually a reason for the others, and refusing to follow the reason through is how a model ends up describing a world that could not exist. Take each link on its own.

A company adding Rs 1,50,00,00,000 of revenue a year rather than Rs 80,00,00,000 is filling capacity it has already paid for. The rent, the supervisors and the tooling do not rise in step with the volume going through them, so more revenue over the same fixed base lifts the margin. The first link runs from the revenue increment to the EBITDA margin.

The second runs from there to the cost of capital. A business filling its plants and lifting its margin is a steadier borrower than one that is not, and steadier borrowers are funded at finer rates. The third link runs to the terminal growth rate: a business winning share in a market it can keep serving is a business a forecaster is willing to assume will still be growing after Year 5. Each link is a sentence somebody can dispute. None of them is a coincidence, and that is the whole reason the four assumptions travel as one object rather than four.

Which assumption is held still, and why is that deliberate?

One assumption reads exactly the same in all three cases: the return on new invested capitalWhat a rupee of new capital earns. It is held at 18.00 per cent in all three cases here, as an assumption of the forecast., at 18.00 per cent. A rupee of newly invested capital earns that much for this invented company. The 18.00 per cent is an assumption of the forecast rather than a measured fact about anything, and it is stated as an assumption everywhere it appears.

Holding it still is a design decision and it costs something. Real businesses do not earn the same return on new capital in every state of the world, and a case set in which returns on new capital also moved would be perfectly legitimate. The reason it is pinned here is that a range is only interpretable when what produced it is known, and with the fifth assumption held, every rupee of the spread between the three answers is the work of the four that moved. Let it move too and the spread carries five effects with no way to separate them.

There is a second, quieter payoff, and the rest of the argument runs on it. Because 18.00 per cent is identical everywhere, the amount of new capital each case has to put in is not a free input at all. The amount of new capital falls straight out of how much profit that case is adding. The single link produces the surprise in the Year 1 cash flows.

How is a case built, step by step?

In one direction, and skipping a step is how an incoherent case gets made. Set the revenue increment. Set the margin. Work down to profit after tax on operations. Work out how much new capital that year of extra profit needed, given that new capital earns 18.00 per cent. Take the difference. The difference is the cash left over. Then discount the five years at the case's own rate and add a block for everything after Year 5, built at the case's own terminal growth.

The fourth step is the one people skip, and it is the important one. Net new invested capitalCapital expenditure less depreciation plus the movement in net working capital. It is what a year of growth costs. is capital expenditure less depreciation plus the movement in working capital, and it is what a year of growth costs. If new capital earns 18.00 per cent and this year's operating profit after tax is Rs 18,00,00,000 higher than last year's, then the capital that produced the rise was Rs 18,00,00,000 divided by 18.00 per cent. The division gives Rs 1,00,00,00,000. Free cash flow to the firmProfit after tax on the operations, less the net new capital that year of growth required. is what is left after that capital has gone back in.

What does the base case actually produce?

The base caseThe case built on the assumptions the forecast already uses. is the forecast as it already stands, and it is worth walking once so the other two have something to be different from. Revenue rises by Rs 1,20,00,00,000 a year from Rs 12,00,00,00,000 to Rs 18,00,00,00,000 by Year 5. EBITDA runs at 24.0 per cent of it. After depreciation of 4.0 per cent of revenue and tax at 25.0 per cent, operating profit after tax rises by exactly Rs 18,00,00,000 every year. The new capital each year is therefore Rs 18,00,00,000 over 18.00 per cent, or Rs 1,00,00,00,000, in every single year.

Free cash flow is therefore that year's operating profit after tax less Rs 1,00,00,00,000: Rs 98,00,00,000 in Year 1, then Rs 1,16,00,00,000, Rs 1,34,00,00,000, Rs 1,52,00,00,000 and Rs 1,70,00,00,000. Discounted at 12.00 per cent, year end, those five come to Rs 4,68,41,43,564. The terminal valueThe value of everything after Year 5, which is where the three cases differ most. is built so that the growth it assumes is paid for. At 5.00 per cent growth on 18.00 per cent returns, the company must put back 5 over 18 of its profit forever, or 27.78 per cent. The argument belongs to Aswath Damodaran. The terminal block comes to Rs 29,25,00,00,000, worth Rs 16,59,72,35,530 today. The base case answer is Rs 21,28,13,79,094, or 7.39 times the Year 0 EBITDA of Rs 2,88,00,00,000, and 77.99 per cent of it sits in the terminal block.

How do the bull and the bear differ from it?

In four numbers each, and in nothing else. The machine is identical. The bull caseA case in which the linked assumptions all move in the favourable direction together. adds Rs 1,50,00,00,000 of revenue a year, runs a 25.0 per cent margin, discounts at 11.50 per cent and grows at 5.50 per cent forever. The bear caseA case in which the same linked assumptions all move the other way together. adds Rs 80,00,00,000 a year, runs 22.5 per cent, discounts at 12.50 per cent and grows at 4.00 per cent. The word bull and the word bear name those two sets of assumptions here and nothing else. The two labels are not views and not forecasts of what will happen. Neither carries a probability, and no probability was ever estimated for either.

Notice a consequence of the margin moving that trips readers up constantly. Year 0 revenue is the same Rs 12,00,00,00,000 everywhere but the margin applied to it is not, so each case has its own Year 0 EBITDA. Bear Year 0 EBITDA is Rs 2,70,00,00,000, base is Rs 2,88,00,00,000, bull is Rs 3,00,00,00,000. Each case's multiple is computed against its own Year 0 EBITDA and never against the base case's, and a note that divides all three by Rs 2,88,00,00,000 has printed three wrong multiples.

An argument rests on the difference between two of these figures, so the printing convention matters. The base column carries the three figures this record fixes to the rupee. Everything else is rounded to the nearest lakh, and each of those columns adds up at that rounding. Every derived figure below is worked out on the unrounded values and rounded only at the end, never rebuilt by subtracting one printed figure from another. On an answer that is seventy per cent terminal value, the last five digits of any of these carry no information at all.

Which case has the most cash in Year 1?

The question is worth answering before reading further. Three cases, one of them describing a company selling much more and one describing a company selling much less. Which of the three throws off the most free cash flow in the very first year of the forecast? Most people answer in under a second, and most people are wrong, so the answer is worth committing to before the control below is touched.

Why does the bear case throw off more cash than the base?

Because free cash flow is what is left over after the growth has been paid for, and the bear case has much less growth to pay for. In Year 1 the bear's operating profit after tax is Rs 11,10,00,000 higher than Year 0. At an 18.00 per cent return, producing that rise took Rs 61,67,00,000 of new capital. The base's profit rises Rs 18,00,00,000 and took Rs 1,00,00,00,000. The bull's rises Rs 23,62,50,000 and took Rs 1,31,25,00,000.

Now subtract. The bear starts with Rs 1,77,60,00,000 of operating profit after tax and gives up Rs 61,67,00,000 of it, leaving Rs 1,15,93,00,000. The base starts with Rs 1,98,00,00,000 and gives up Rs 1,00,00,00,000, leaving Rs 98,00,00,000. The bull starts highest of all at Rs 2,12,62,50,000 and gives up the most, Rs 1,31,25,00,000, leaving Rs 81,37,50,000. The bear case has the highest Year 1 free cash flow of the three and the bull case has the lowest, the exact reverse of their ranking by value.

Say it as a household would. A couple running a small tiffin service can take home everything the kitchen earns this year, or they can take home less and use the difference to rent a second kitchen. The year they rent the second kitchen, there is visibly less money in the house. Nothing has gone wrong. The couple have swapped cash now for a bigger kitchen later, and any measure that looks only at the cash in the house this year will report the expansion as a setback. A growing company spends its own cash flow on growing, and free cash flow is the measure that shows it did.

Where does the difference between the three answers land?

Almost entirely beyond Year 5, and the arithmetic of that is startling once it is laid out. The present value of the explicit five years is Rs 4,82,58,00,000 in the bear, Rs 4,68,41,43,564 in the base and Rs 4,50,79,00,000 in the bull. The three present values are within Rs 31,79,00,000 of each other, and the direction matters: the bear is the largest of the three, for exactly the reason set out above.

Now the terminal block. Rs 11,72,36,00,000 in the bear, Rs 16,59,72,35,530 in the base, Rs 21,76,10,00,000 in the bull. The terminal blocks span Rs 10,03,75,00,000, computed on the unrounded values. The terminal block carries a difference roughly thirty two times the size of the difference the whole explicit period carries, and the explicit period carries its difference the wrong way round. Five years of carefully forecast revenue, margin and capital spending do almost nothing to separate the three answers.

The two comparisons are worth sitting with before the next forecasting week begins. If a Year 3 revenue line is worth arguing about for two days, the two days are being spent on the part of the model that contributes about three per cent of the spread between a bear and a bull answer. The terminal assumptions usually get the last twenty minutes of a Friday, and they contribute the rest.

How much of each answer sits beyond the forecast?

Not the same share in each, and the pattern runs the way the previous block implies. The terminal block is 70.84 per cent of the bear answer, 77.99 per cent of the base and 82.84 per cent of the bull. The bull grows faster forever and is discounted at a lower rate, and both of those push value out past Year 5.

The case with the largest number attached to it is also the case resting most heavily on the part of the model nobody forecasts line by line. The observation is not a criticism of the bull case. The observation is a description of the bull case, and the kind of description that belongs next to the number rather than three slides later. A reader told only that the bull answer is Rs 26,26,89,00,000 has been told less than a reader told that Rs 21,76,10,00,000 of it is a perpetuity assumption.

How wide is the range, and against which end is it measured?

The three answers run from Rs 16,54,94,00,000 to Rs 26,26,89,00,000. The spreadThe distance between the ends of a range, which only means something once the end it was divided by is named. is Rs 9,71,96,00,000. The spread is 58.73 per cent of the bear case and 37.00 per cent of the bull case, and the two numbers describe the same distance. A spread quoted as a percentage means nothing until the end it was divided by is named, and the divisor here is the bear.

Against the base case the bull sits 23.44 per cent above and the bear sits 22.24 per cent below. The two percentages are not symmetric, and nobody should expect them to be: the four assumptions did not move by symmetric amounts, and a discounted cash flow answer is not linear in any of them anyway. A half point off the cost of capital does not do the same work as a half point on it.

One note on the arithmetic of that spread, given the rounding convention above. Worked on the unrounded case values the spread is Rs 9,71,95,53,742, and that rounds to Rs 9,71,96,00,000 at the nearest lakh. Subtracting the two enterprise values as they are printed to the nearest lakh instead gives Rs 9,71,95,00,000. Neither figure has been adjusted to make the other come out. The terminal block spread does the same: Rs 10,03,75,00,000 worked properly and Rs 10,03,74,00,000 when printed figures are subtracted. Both gaps are rounding rather than error, and a note that quietly nudges a printed figure so a subtraction lands neatly has said something false about its own precision.

What is a set of three cases not?

Not a minimum. Not a maximum. Not a most likely with an error bar either side. Not a probability distribution, and not something to average. Each of those four readings is common and each of them attaches a meaning to the three numbers that nothing in their construction supports.

Start with the minimum and the maximum. The two are the most natural mistakes. Nothing about the bear case makes it a floor. A fifth set of assumptions, in which the revenue increment is Rs 40,00,00,000 and the terminal growth is 3.00 per cent, would produce a fourth number below all of these, and it would be exactly as legitimate a case as the three here. The reason there are three rather than nine is that three can be held in a reader's head and argued with. The choice of three is a decision about legibility, not about arithmetic.

Then the averaging, a quieter and much more common mistake. Rs 16,54,94,00,000, Rs 21,28,13,79,094 and Rs 26,26,89,00,000 average to Rs 21,36,66,00,000. The average corresponds to no state of the world and to no set of assumptions anybody wrote down, and it is not improved by sitting very close to the base case answer. An average of three numbers is only meaningful if something says how likely each of them is. No probability is attached to any of these three cases, and none was ever estimated. Three cases are three stories. There is nothing populating the space between them to average over.

What does a set of cases never tell?

How likely any of them is. Probability is exactly the thing a set of cases cannot deliver. Building a bear case is an act of description, not of estimation: somebody wrote down a slower world and ran the model in it. Nothing in that act produces a probability, and attaching one afterwards, whether as a number or as a word like probable, invents information the work never contained.

A set of cases also never says what a company is worth. A set of cases says what the model produces under each of three named settings, a smaller and more honest claim. A case set is a range with its assumptions printed next to it, and no case in one says that Sankalp Industrial Systems Limited is cheap, expensive, undervalued, overvalued, attractive or worth anything to anybody. The missing judgements are not omissions to be filled in later but the boundary of what this arithmetic can support.

And it never identifies which assumption was the one worth arguing about, though it points hard at the answer. The three cases move four assumptions at once, and that is precisely why the spread cannot be split between them. Separating the effect of one assumption from the effect of another needs a different exercise entirely, run one assumption at a time, and that exercise is covered separately.