Implied Assumptions: Reverse-Engineering What a Price Believes

An implied assumption is what a valuation model yields when it is run backwards: fix the value, solve for an input. At an enterprise value of Rs 22,40,00,00,000 and a 12.00 per cent cost of capital, Sankalp Industrial Systems Limited, invented, needs terminal growth of 5.80 per cent rather than 5.00. The technique states what a figure implies, and then it stops.

The technique begins at a sweet shop three weeks before a wedding, without a spreadsheet and without anybody using the word valuation. A customer asks for two hundred kilos of laddoo for the reception. The man behind the counter looks at his book, thinks for a moment, and says Rs 1,20,000. The customer knows his shelf rate, having bought from him every Diwali: Rs 500 a kilo. Two hundred kilos at Rs 500 is Rs 1,00,000. He has quoted Rs 1,20,000.

Nobody asked him what he was assuming. He did not say. And yet, standing there with nothing but his number and a knowledge of his shelf rate, the customer now knows something he never said out loud: this order carries a premium of Rs 100 a kilo, being 20 per cent, for doing two hundred kilos in three weeks in the middle of the season. The subtraction is a reverse solve, and the premium recovered was never a fact about the world, only a fact about his price and the customer's arithmetic held together. Change the view of his shelf rate to Rs 550 and the premium recovered falls to about 9 per cent. The number came out of the pair, not out of him.

The second half of the technique is where the discipline lives. Having recovered 20 per cent, the customer is entitled to say the quote carries a 20 per cent premium over his shelf rate. Nobody is entitled, from that arithmetic, to say the premium is steep, or fair, or greedy. Nothing in the subtraction produced any of those words. Saying one of them requires a different reason, found somewhere else, with its source named. The arithmetic stops where it stops.

Those two moves are the entire technique, and the entire discipline attached to it. Done properly, on a company and with a model, most of the work lies in where it has to stop.

What is an implied assumption, and how is one obtained?

An implied assumptionThe value of an input obtained by fixing a model's output and solving backwards. is the value of an input obtained by fixing a model's output and solving backwards for it. The number is not observed, it is not forecast, and nobody hands it over. Arithmetic produces it, out of two things already in hand: a value, and a model that would produce that value if one of its inputs took the right number.

The technique that produces one is called reverse valuationRunning a valuation model backwards, from a known value to an unknown assumption., and the reading of a traded figure this way is associated above all with Aswath Damodaran, whose valuation material treats a market value as a set of assumptions to be recovered rather than a verdict to be argued with. Reverse valuation rests on a symmetry that is easy to state and easy to underuse. A valuation model is one equation. On one side sits a value. On the other side sits a set of inputs. Run the equation left to right and it turns assumptions into a value. Run it right to left and it turns a value into an assumption.

A value and a set of assumptions are the same object seen from two ends, and the reverse run is worth trusting precisely because it adds nothing. No new information enters. No second model is built. No judgement is inserted. The arithmetic is the arithmetic already in place, solved for a different letter. Adding nothing is why a reversed answer is exactly as reliable, and exactly as fragile, as the forward answer it came from. An analyst who did not believe the forward model is not rescued by the reverse of it.

Think about what that means for a moment. Every price is already a full set of assumptions, whether or not anybody wrote them down. The number Rs 22,40,00,00,000 is not an opinion floating free of arithmetic. Put it into a model and it becomes a constraint: given these cash flows, given this rate, given this reinvestment, there is exactly one terminal growth rate that produces that number and no other. The rate was always in there. Reversing the model is just the act of reading it out.

What does the model say forwards, before anything is reversed?

Sankalp Industrial Systems Limited makes industrial valves, precision castings and the aftermarket parts and service that go with them. The company is listed, so it has an observed share price, and a discounted cash flow model already exists for it. Building that model is covered separately. A reversal needs something to reverse, so the model's output comes first, in three lines.

Line one. Free cash flow to the firm over the explicit periodThe years forecast individually, before the terminal value takes over. of five years runs Rs 98,00,00,000, 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 the company's own cost of capitalThe rate at which the firm's cash flows are discounted, weighted across its funding. of 12.00 per cent, using year-end discountingTreating each year's cash flow as arriving on the last day of that year., those five years are worth Rs 4,68,41,43,564 today.

Line two. A terminal valueThe value of everything after the explicit forecast, in this model 77.99 per cent of the answer. is built on a terminal growth rateThe rate at which cash flows are assumed to grow forever after the explicit forecast ends. of 5.00 per cent a year in nominal rupees, with new capital assumed to earn 18.00 per cent. Growing at 5.00 per cent forever on 18.00 per cent returns requires reinvesting 5 over 18, being 27.78 per cent of profit, forever. Year 6 operating profit after tax is Rs 2,83,50,00,000, so Year 6 free cash flow is Rs 2,04,75,00,000. Capitalised at 12.00 less 5.00 per cent, that gives a terminal value of Rs 29,25,00,00,000. Discounted five years, that is Rs 16,59,72,35,530 today.

Line three. Add the two and the enterprise value is Rs 21,28,13,79,094. Of that answer, Rs 16,59,72,35,530 sits in the terminal block, being 77.99 per cent, so the five years anybody actually forecast do about a fifth of the work. Hold on to that share. The terminal share decides which input a reverse run is usually pointed at.

Two small honesties about that column before it gets used for anything. Footing exactly to Rs 4,68,41,43,564 as printed is a convenience rather than a rule. And every figure here is rounded to the rupee from a longer number: on an answer that is 78 per cent terminal value, the last two digits carry no information whatever, and they appear only so that a reader rebuilding the arithmetic lands on the same place rather than wondering about a stray rupee.

What is the other figure sitting on the table?

The company is listed, so there is a second number in the room and it did not come out of any model. Twenty crore shares at Rs 90.00 gives a market capitalisation of Rs 18,00,00,00,000. Walking the bridge in reverse gives the traded enterprise valueThe enterprise value computed from the observed share price and the balance sheet.: add gross debt of Rs 6,00,00,00,000, add minority interest of Rs 60,00,00,000 for the quarter of Sankalp Coatings Private Limited, invented, that the group does not hold, then take out cash of Rs 1,20,00,00,000 and non-operating assets of Rs 1,00,00,00,000, being surplus land and the holding in Aruna Tooling Private Limited, invented. The bridge lands on Rs 22,40,00,00,000 exactly.

So there are two enterprise values on the table. The model says Rs 21,28,13,79,094. The traded figure is Rs 22,40,00,00,000. The difference is Rs 1,11,86,20,906, being 5.26 per cent of the model's answer. The traded figure also sits inside the locked range of Rs 21,28,13,79,094 to Rs 27,36,00,00,000 produced by the four valuation methods, and that is the whole of what can be said about it. Not high, not low, not right, not wrong. Inside the range, and then the sentence ends.

The technique does not argue about the gap. The technique converts it. Because rupees of enterprise value are not a thing anybody has a view on, a difference of Rs 1,11,86,20,906 is very hard to have a useful conversation about. An assumption is. So the reverse run takes the gap out of rupees and puts it into the units of an input, where somebody can actually push back.

How is the model run backwards?

Mechanically, nothing changes except which letter is being solved for. Everything that was an input stays an input. The output, once the unknown, becomes the known. And exactly one of the old inputs becomes the unknown.

Written out for this case, the forward statement is: given cash flows of Rs 98,00,00,000 through Rs 1,70,00,00,000, given a rate of 12.00 per cent, given 18.00 per cent on new capital, given year-end discounting, and given terminal growth of 5.00 per cent, the enterprise value is Rs 21,28,13,79,094. The reverse statement is: given cash flows of Rs 98,00,00,000 through Rs 1,70,00,00,000, given a rate of 12.00 per cent, given 18.00 per cent on new capital, given year-end discounting, and given an enterprise value of Rs 22,40,00,00,000, the terminal growth rate is 5.797592 per cent.

Read those two sentences again and notice how little moved. Five of the six items are word for word identical. The sixth swapped ends. Reversing the model added nothing whatever to it. The reversed answer therefore carries the same weight as the forward one and not a gram more.

In practice it is solved the way any equation with one unknown that cannot be isolated cleanly is solved: a rate is tried, the value it produces is read off, and the search closes in. At 5.00 per cent the model gives Rs 21,28,13,79,094, short of the mark. At 6.00 per cent it gives Rs 22,72,83,17,576, an overshoot. Somewhere between them is the rate that lands on Rs 22,40,00,00,000, and closing in on it gives 5.797592 per cent. There is no cleverness in the search and no judgement in it either. The equation had one answer the whole time.

Why can only one input be freed at a time?

Because there is one equation. One equation is the whole reason, and nothing else separates a reverse valuation from a guess wearing its clothes. The rule deserves to be said slowly.

An equation with one unknown has an answer. An equation with two unknowns does not have an answer; it has a set of pairs, and every pair in that set satisfies it equally well. The point is not a subtlety of finance but the arithmetic every reader met at school and then stopped thinking about. If two numbers add to ten, neither number has been learned. A line has been learned, not a number.

Try it on this case. Hold the five cash flows, hold the 18.00 per cent on new capital, hold year-end discounting, and free both the terminal growth rate and the cost of capital together. Ask which pair reproduces Rs 22,40,00,00,000. The honest answer is that a great many pairs do. Terminal growth of 4.50 per cent with a rate of 11.468046 per cent does it. So does 5.00 with 11.671057. So does 5.50 with 11.876520. So does 5.797592 with exactly 12.00. So does 6.50 with 12.295062, and 7.00 with 12.508269. Every one of those pairs lands on Rs 22,40,00,00,000 to the rupee.

Reporting any single pair off that line as the implied answer is reporting the analyst's own choice as a finding, and a reader cannot tell the difference because the arithmetic behind it is genuinely correct. That is what makes the mistake so durable. The mistake survives a review. The sums check out. Nothing implied that pair rather than any of the others, so the word implied does not survive scrutiny.

There is a household version of this and it is worth carrying around. A couple says they spend Rs 60,000 a month. From that one sentence neither their rent nor their school fees can be recovered. Either one can be recovered, if they state the other. Asking the single number for both is no longer arithmetic but choosing, and the choosing is the part that will not be visible in the answer.

Which input should be freed, and does the choice matter?

The choice matters enormously, and the uncomfortable part is that most people who run a reverse valuation never notice they made one. Such an analyst frees the input the template happens to have a cell for, gets a number, and writes it up. The choice was made by the template.

Because the choice determines what the note will be about, it is better made openly and made before the solving starts. Freeing the terminal growth rate produces a note about growth. Freeing the cost of capital produces, from the identical gap, a note about the rate. Freeing the Year 3 margin produces a note about margins. The gap does not change. The subject of the note changes completely.

Every other input was held exactly where it was, so whichever input is freed absorbs the entire difference. The choice of input is the choice of what the reader will end up arguing about.

Having said the choice is free, there is a practical reason the terminal growth rate is where a reverse run is usually pointed in a model shaped like this one. Of the Rs 21,28,13,79,094 answer, Rs 16,59,72,35,530 sits beyond Year 5. A small movement in the terminal growth rate therefore shifts more of the answer than a large movement anywhere inside the explicit five years. The pull towards the terminal rate is a property of this model rather than a rule about models, and a model with a longer explicit period and a smaller terminal share would point somewhere else.

The figure below makes that concrete, and it carries a second point that is easy to miss. Move the terminal growth rate from 5.00 per cent to the 5.797592 per cent that reproduces the traded figure and the left-hand block, the present value of the five forecast years, does not move by a single rupee. The block stays at Rs 4,68,41,43,564. The whole of the change lands in the terminal block, rising from Rs 16,59,72,35,530 to Rs 17,71,58,56,436.

What are the two answers on the same gap?

Here they are, both solved on the same model, both reproducing the same Rs 22,40,00,00,000, and neither preferred over the other.

The first branch frees the terminal growth rate and holds everything else: the five cash flows, the 12.00 per cent cost of capital, the 18.00 per cent on new capital and year-end discounting all stay where they were. The rate that returns Rs 22,40,00,00,000 is 5.797592 per cent, printed here as 5.80. Against the 5.00 per cent the forward model used, that is a difference of 80 basis pointsOne hundredth of a percentage point. of growth.

The second branch puts the growth rate back at 5.00 per cent and frees the cost of capital instead, holding everything else exactly as before. The rate that returns Rs 22,40,00,00,000 is 11.671057 per cent, printed here as 11.67. Against 12.00 per cent, that is a difference of 33 basis points of rate.

Either one on its own closes the whole of the Rs 1,11,86,20,906, so to reach Rs 22,40,00,00,000 this model needs either 80 basis points more terminal growth or 33 basis points less cost of capital, and not both. The word between the two answers is or. The word is never and. Each was solved on the strict condition that everything else stayed exactly where it was, so applying both at once would sail well past the figure.

A careful reader will find one more thing and wonder whether something is broken, so it is worth stating rather than hiding. Put 5.80 per cent into the model, exactly 5.80 and not 5.797592, and the answer is Rs 22,40,37,86,022. The answer is Rs 37,86,022 above the traded figure. Nothing has gone wrong. 5.80 is the printed form of 5.797592, and the overshoot is the rounding becoming visible when the rounded figure is put back through the machine. The same thing happens on the other side: put exactly 11.67 per cent in and the model gives Rs 22,40,37,72,515, being Rs 37,72,515 above. Both are the rounding seen from the two directions, and saying so is better than quietly printing a figure that does not rebuild.

Read the bottom row, then read the two columns above it again. The same Rs 1,11,86,20,906 has been described twice, in two different units, and both descriptions are complete. Neither is half of anything, and neither is a check on the other.

Getting a prediction wrong is the fastest way to feel what an implied figure actually is, so here is one.

What is an implied figure conditional on?

On everything else in the model, without exception, and this is the sentence that most often gets dropped when an implied figure travels from a spreadsheet into an email.

The answer 5.797592 per cent was not produced by the traded figure. The rate was produced by the traded figure and a particular model, standing together. Change any part of that model and the figure moves. Change the five cash flows and it moves. The 18.00 per cent on new capital sets how much of terminal profit has to be ploughed back rather than paid out. Change it and the figure moves. Change year-end discounting to mid-year and it moves. Extend the explicit forecast from five years to ten and it moves. None of that is a weakness in the technique. Conditionality is the technique.

An implied assumption is a property of a model and a value taken together, never of the value on its own, so a figure quoted without the model behind it cannot be checked by anybody. That makes it a conditional figureA number that is only meaningful while every other input behind it is held exactly as stated., and a conditional figure printed without its conditions is not a smaller version of the truth. A figure stripped of its conditions is a different kind of object altogether.

The prediction above turned on the discounting convention, so that condition comes first. Under year-end discounting the five forecast years are worth Rs 4,68,41,43,564. Under mid-year discounting, treating each year's cash as arriving halfway through rather than on the last day, the same five cash flows at the same 12.00 per cent are worth Rs 4,95,72,31,590. The uplift is 5.8301 per cent, exactly the square root of 1.12 less one, and it is not a small number. Everything else being equal, a model that starts Rs 27,30,88,026 further along needs less implied growth to reach the same Rs 22,40,00,00,000. The timing convention for the terminal block under mid-year discounting is not settled, so no mid-year implied rate can be stated.

There is a second condition worth naming, and it is about what 5.80 per cent even means as a rate. Inflation of 5.00 per cent is assumed throughout the worked example. Terminal growth of 5.00 per cent in nominal rupees against 5.00 per cent assumed inflation is therefore no real growth at all after Year 5: the business is assumed to keep pace with prices and do nothing more. Read that way, the 5.80 per cent solve is about 76 basis points of real growth, being 1.058 over 1.05. Real growth of 76 basis points is a more useful way to hold the number than 5.80 on its own, and the 5.00 per cent inflation figure is this example's assumption rather than a statement about anybody's economy.

Is an implied assumption a forecast?

No. Each of three neighbouring ideas gets confused with an implied assumption in a slightly different way, so the three are worth separating.

An implied assumption is not a forecast. A forecast is a statement about what somebody expects to happen. An implied assumption is a statement about what a model would have to assume in order to return a particular figure. Nobody has predicted anything, nobody has committed to anything, and the person who ran the reverse solve may hold no view whatever about the business. Saying that Sankalp Industrial Systems Limited is implied to grow at 5.80 per cent forever attributes a belief to somebody. Saying that this model needs 5.80 per cent to reproduce Rs 22,40,00,00,000 attributes nothing to anybody.

An implied assumption is not a probability. The reverse run produces one number, not a distribution, and there is nothing anywhere in the arithmetic that says how likely that number is. A figure can be implied and extremely unlikely; it can be implied and thoroughly ordinary. Likelihood was never an input, so the solving cannot tell the two apart.

And it is not a verdict. The confusion with a verdict does the damage. The answer to a reverse valuation is never what the market believes; it is what the market would have to believe about this one input if it agreed with the analyst about every other input in the model. That conditional clause is the whole of the difference between a technique and an assertion. Dropping it is a quiet claim to know somebody else's thinking, on the strength of a model they have never seen and did not build.

The same claim would be plainly strange if it were said out loud about the sweet shop. Recovering a 20 per cent premium from his quote says nothing about what he was thinking. He may have had a wholesale rate nobody outside the shop knows about, or a cousin doing the packing for nothing, or a completely different arithmetic in his head. All that was recovered is what the customer's arithmetic would need in order to land on his number. The recovery is genuinely useful and genuinely modest, and the modesty is not optional.

What does this do to an argument between two people?

The reverse solve ends the kind of argument that cannot end, and starts the kind that can.

Two people who disagree about a value are stuck. One says Rs 21,28,13,79,094 and the other says Rs 22,40,00,00,000, and there is nothing inside that disagreement to examine. Neither figure is evidence. Neither figure has parts. A conclusion is not the sort of thing two people can look at together, so they can repeat themselves at each other for an hour and finish exactly where they started.

Run the reverse solve and the disagreement changes shape. The disagreement is now about whether a manufacturer of industrial valves grows at 5.00 per cent forever or at 5.80 per cent forever. A growth rate is a question with evidence attached to it, and evidence is the only thing that has ever settled an argument. How fast has the aftermarket business been adding revenue. How much does the plant capacity allow. Does 5.80 per cent nominal, being about 76 basis points of real growth on this example's assumed inflation, look like the kind of business this is. None of those questions is easy, but all of them are questions two people can stand in front of together and point at.

The same move appears outside finance. A household arguing about whether to take a home loan gets nowhere while the argument is about the loan. An instalment can be compared with a payslip and a loan cannot, so the argument moves the moment somebody works out what monthly instalment the loan implies and everyone starts talking about that instead. The reverse solve does the same job: it converts a conclusion into a quantity, and quantities can be examined.

What is the last sentence the analyst is entitled to write?

The refusal is a mechanism rather than a caution, not something bolted on at the end to keep a compliance officer happy. Refusing is part of how the technique works, in the same way that holding five inputs still is part of how the technique works.

Here is the last sentence, in full, with nothing removed. To reach an enterprise value of Rs 22,40,00,00,000 at a cost of capital of 12.00 per cent, holding the five forecast cash flows, the 18.00 per cent return on new invested capital and year-end discounting exactly as stated, this model requires terminal growth of 5.80 per cent rather than 5.00.

Then the note stops.

The note does not say that 5.80 per cent is demanding. The note does not say that the figure is comfortable, or heroic, or a stretch, or reasonable, or modest. The note does not say that the model is wrong, and it does not say that the market is wrong. The note does not say that the company is cheap or expensive. Every one of those sentences would be a different exercise, resting on evidence the arithmetic never touched, and every one of them would arrive wearing the authority of a calculation that did not produce it.

One last point is the sharp one, so it is worth putting plainly. The reverse solve is a genuinely mechanical procedure. Anybody with the same model and the same value gets 5.797592 per cent, every time, with no room for taste. A reader who has just watched a machine produce a number will read the adjective that follows as though the machine produced that too, and the mechanical reliability is exactly what makes the next sentence dangerous. It did not. A person put it there.

When does an implied figure mean nothing at all?

There are four conditions, and a reader who can name them will not be caught out by an implied figure quoted in a meeting.

The first is when the held inputs are not stated. A figure of 5.80 per cent with no model behind it cannot be checked, cannot be reproduced and cannot be argued with. The figure is not a weak claim. Nothing in it is available for anybody to disagree with, so it is not a claim at all.

The second is when more than one input was freed. If both the growth rate and the rate moved, the reported pair is one point off a line and the word implied has been misused. Ask which inputs were held. If the answer is vague, the figure is vague.

The third is when the model behind it is one the reader would not have used anyway. A reverse solve inherits every weakness of the forward model exactly. If the five cash flows are wrong, the implied growth rate is wrong by a matching amount, and its mechanical precision does nothing whatever to rescue it. A reverse valuation is never better than the model it reverses, and the six-decimal answer it produces can make a shaky model look settled.

The fourth is when the figure is being used to carry a conclusion it cannot carry. If the sentence after the number contains an adjective, the number has stopped being an implied assumption and started being a rhetorical device. The problem is not a failure of arithmetic. The failure is in the sentence, and the fix is to delete the sentence rather than to re-run the model.

Set against those four, notice what a properly stated implied figure survives. Somebody who disputes it can rebuild it, so the figure survives disagreement. A reader who prefers a different cost of capital can solve it again with their own, so the figure survives a change of view. The figure never claimed to know what would happen, so it survives time. An implied assumption is one of the few outputs in valuation that is fully checkable by a reader who has nothing but the working in front of them, and it is checkable precisely because it refuses to conclude anything.