Firm Value and Equity Value: The Bridge Between Them

Firm value is what the operating business is worth to everybody who put money into it. Equity value is what is left for the shareholders once every claim ahead of theirs has been met. Two questions, two correct answers, one business. On Sankalp Industrial Systems Limited, invented, the operating business is worth Rs 21,28,13,79,094 and the shareholders' claim on it is worth Rs 16,88,13,79,094.

Almost everybody has had this conversation at a dining table, so a flat is the place to start. A neighbour bought a two bedroom flat some years ago. She paid part of the price from savings and borrowed the rest from a bank, and about Rs 28,00,000 of that loan is still outstanding. One evening somebody asks her what the flat would fetch.

In ordinary speech the question has one meaning, so she will answer without thinking about it. The flat is worth what a flat like it changes hands for. Say Rs 90,00,000. The Rs 90,00,000 is a fact about the building, the floor, the road it sits on and what people are paying in that locality this year. The price has nothing whatever to do with her loan.

Now ask her a different question. What is her stake in the flat worth? Her stake is Rs 90,00,000 less the Rs 28,00,000 she still owes, being Rs 62,00,000. And notice what has just happened. Two different numbers came out of one flat, on one evening, and neither of them is wrong. The first answers a question about the property. The second answers a question about her.

Notice the second thing too, the one people miss. If she went to the bank tomorrow and took a top up loan of Rs 10,00,000 against the same flat, her stake would fall to Rs 52,00,000. The flat would still be worth Rs 90,00,000. The bricks do not know about the loan. Nobody would say the flat became cheaper because she borrowed against it.

A company is the same shape as that flat, only with more claimants and bigger numbers. The reason a business carries two correct values on the same day is not a modelling convention and not an accounting choice: it is that more than one set of people has a claim on one set of cash flows, and those claims sit in an order. Once that is true, asking what the operating business is worth and asking what the shares are worth become two different questions, and each has its own right answer.

The company throughout is Sankalp Industrial Systems Limited, invented, a listed manufacturer of industrial valves, precision castings and the aftermarket parts and service that go with them.

Who actually has a claim on this business?

Three sets of people, and the order they sit in is what produces the two values, so they are worth naming one at a time rather than treating them as a lump.

First, the lenders. Sankalp Industrial Systems Limited, invented, has gross debtEverything the company owes to lenders, counted before any cash it happens to be holding is deducted. of Rs 6,00,00,00,000 spread across three tranches. The tranches matter less than their character. The Rs 6,00,00,00,000 is a fixed amount. The company owes it whether the business has a magnificent year or a wretched one. Lenders do not get more when the valve order book fills up and they do not get less when it empties. Lenders get their contracted amount, and they get it before anybody else gets anything.

Second, and this one surprises people the first time, there is a claimant who does not appear on any share register of the parent company at all. Sankalp Industrial Systems Limited holds 75.0 per cent of Sankalp Coatings Private Limited, invented, and the group accounts consolidateAdd a subsidiary's figures into the group accounts line by line, as though the group held the whole of it. that subsidiary in full. Every rupee of the coatings business's revenue, cost and cash flow sits inside the group numbers. But the group does not hold every rupee of it. Somebody else holds the other 25.0 per cent, and that somebody's share of the business is carried at Rs 60,00,00,000. The outside holding is the minority interestThe share of a consolidated subsidiary that the group does not hold, and which therefore belongs to somebody outside the group., and it is a genuine claim on cash flow that the group's own forecast counted in full.

Third, the ordinary shareholders. There are 20,00,00,000 shares in issue. Between them they get whatever is left after the first two are satisfied, and nothing else. There is no contracted amount for them. There is no schedule. There is only a remainder.

Hold the three side by side and the asymmetry is obvious. Two of the three claims are fixed amounts and the third is not an amount at all. The asymmetry is exactly why a single number cannot describe both the worth of the operating business and the worth of the shares. Forcing one figure to do both jobs silently answers one of the two questions while appearing to answer the other.

The same structure appears at a scale anybody can walk past. A man runs a small tempo hire business with two vehicles. He bought the second one on a loan and still owes Rs 3,20,000 on it. His brother put in some money years ago and takes a fifth of whatever the business clears. Asking what the tempo business is worth is asking about routes, contracts, the condition of two vehicles and what a buyer would pay for the lot. Asking what the man's own share is worth means taking the lender out first, then the brother, and only then looking at what is left. Same business, two questions, two answers, and an order that decides who is taken out first.

What are the two values, and whose claim is each one?

Most explanations of these two start with how each is computed. Computation is the wrong end to start from, and the reason is practical rather than pedantic: the computation is what people remember and the claim is what they actually need. Somebody who can recite the arithmetic but cannot say whose money a figure represents will still put the wrong number into the wrong comparison. Somebody who knows whose claim it is will get it right even if they have to look the arithmetic up.

So define them by the question each one answers.

Firm value answers: what is the operating business worth to everybody who put money into it, taken together? Firm value makes no distinction between the money that came from lenders and the money that came from shareholders, and it does not ask who they are. The figure looks at a set of operations, forms a view about the cash those operations will produce, and puts a value on that. In practice this is called enterprise valueThe same thing as firm value, and the term used far more often in day to day work. more often than it is called firm value, and the two mean the same thing. On Sankalp Industrial Systems Limited, invented, that figure is Rs 21,28,13,79,094. How it was produced is covered separately; what matters here is whose claim it represents.

Equity value answers a different question: what is the shareholders' claim on that same business worth? Equity value is the residual claimThe claim that is settled last and receives whatever remains once everything ahead of it has been met., and residual is the operative word. The shareholders get what is left. On the same invented company, on the same day, that figure is Rs 16,88,13,79,094, being Rs 84.41 for each of the 20,00,00,000 shares.

Neither of those is more real than the other. Neither is a rough version of the other. The two figures answer two different questions, and the gap between them is not an error term. The gap is the value of everybody else's claim, plus the value of everything the company holds that the operating forecast never counted.

There is one habit worth building here and it costs nothing. Whenever a value arrives on a desk, the first thing to state is whose claim it is. Not what it was computed from. Whose claim. A rupee figure attached to a business is not a complete statement until it names the claimant it belongs to, and a figure quoted without that clause has not finished the sentence. Once that question is asked routinely, a surprising proportion of the numbers people quote at each other in meetings do not survive it.

Why is the shareholders' claim settled last, and what does that do to it?

The shareholders' claim is settled last because that is what was agreed when the money went in, and it is worth being precise about what that agreement buys each side. A lender accepted a fixed return and a place at the front of the queue. A shareholder accepted no fixed return and a place at the back. Neither arrangement is generous or harsh; they are two different trades, and each side knew which one it was making.

The consequence of standing at the back is arithmetic rather than sentiment. A prior claimA claim that has to be met in full before the claim behind it receives anything at all. that is fixed does not move when the business does. So when the value of the business moves, the whole of that movement has to land somewhere, and the only place left for it to land is on the claim that was defined as a remainder.

Take the numbers on this invented company. Everything Sankalp Industrial Systems Limited holds, being the operating business at Rs 21,28,13,79,094 plus Rs 1,20,00,00,000 of cash plus Rs 1,00,00,00,000 of assets that produce none of the operating earnings, comes to Rs 23,48,13,79,094. The claims ahead of the ordinary shareholders come to Rs 6,60,00,00,000, being the Rs 6,00,00,00,000 of debt and the Rs 60,00,00,000 belonging to the minority. The remainder, Rs 16,88,13,79,094, is theirs.

Now suppose the operating business turned out to be worth Rs 2,00,00,00,000 less than the model says. The total falls to Rs 21,48,13,79,094. The claims in front are fixed amounts, so they do not move at all. So the shareholders' remainder falls to Rs 14,88,13,79,094, and it has absorbed the entire Rs 2,00,00,00,000. Suppose instead the business turned out to be worth Rs 2,00,00,00,000 more. The remainder rises to Rs 18,88,13,79,094, and again it has absorbed the entire movement.

A fixed claim standing in front of a remainder passes every rupee of change straight through to the remainder, in both directions. The equity figure is therefore always the more volatile of the two. The volatility is not a statement about risk appetite or market mood but about the shape of a queue.

A picture cannot be cited and a sentence can, so two things about that drawing are worth saying in words. The dark segment is the same width in all three rows: the prior claims are Rs 6,60,00,00,000 whatever happens to the business. And the arithmetic is symmetrical: the model has no view on which way the business will move, only that whichever way it goes, the remainder is where the movement arrives.

Why is firm value neutral to how the business is funded?

Because the operations do not know how they were paid for. The claim sounds glib, so it is worth making concrete before making it general.

The valve shop at Sankalp Industrial Systems Limited, invented, runs a line of machine tools. The machines cut metal at a certain rate, to a certain tolerance, with a certain number of people around them, and they produce a certain quantity of valves that get sold at a certain price. Nothing in that sequence changes if the machines were paid for with borrowed money rather than with shareholders' money. The castings still get poured. The aftermarket business still earns the margin it earns. The tolerance does not widen because a term loan was drawn.

So a value built on those operations, being a view of the cash the operations will produce, does not change when the funding mix changes. Holding still like that is what neutral to fundingUnchanged when the mix of debt and equity moves, provided the rate the business is discounted at is held still. means. Neutrality is a property of what the firm value measures, not a convenient assumption somebody made.

The equity value is defined as what is left after the claims, so it is a different animal entirely. Changing the size of a claim in front of it changes the equity value by construction. So the two behave differently under exactly the same experiment, and watching them behave differently is the fastest way to feel the distinction rather than merely agreeing with it.

Here is the experiment. Hold the operating business completely still, including the rate it is discounted at. Change only how much of it was funded with borrowing. With no debt at all, the shareholders' claim would be Rs 22,88,13,79,094, being Rs 114.41 a share. With Rs 3,00,00,00,000 of debt, Rs 19,88,13,79,094, being Rs 99.41. With Rs 6,00,00,00,000, the company's own position, Rs 16,88,13,79,094 and Rs 84.41. With Rs 9,00,00,00,000, Rs 13,88,13,79,094 and Rs 69.41. With Rs 12,00,00,00,000, Rs 10,88,13,79,094 and Rs 54.41.

Across all five of those cases the value of the operating business reads Rs 21,28,13,79,094 and never moves by a single rupee. The shareholders' claim falls by the whole of the Rs 12,00,00,00,000. The valves still get made, the castings still get poured, and the aftermarket business still earns what it earns, whoever paid for the machines.

One oddity in that drawing is worth explaining rather than leaving to look like a mistake. The two lines cross, and they cross at Rs 1,60,00,00,000 of borrowing. Below that level the shareholders' claim is worth more than the operating business itself. The oddity disappears once the rest of what the company holds comes into view. The cash of Rs 1,20,00,00,000 and the non-operating assetsThings a company holds that produce none of the earnings being forecast, such as surplus land or a stake in an associate. of Rs 1,00,00,00,000 belong to the shareholders too, and against them sits Rs 60,00,00,000 of minority interest. Netting those three gives Rs 1,60,00,00,000 exactly, and that is where the lines meet. The crossing is an accident of this balance sheet and it carries no general lesson.

Does borrowing really leave the business worth the same?

Not once the rate is allowed to respond.

The demonstration above holds the weighted average cost of capitalThe single rate at which a business's cash flows are discounted, blended across what its lenders and its shareholders each require. still at 12.00 per cent. Holding the rate still is a teaching device. The device isolates one effect so that the effect can be seen, in the same way that a physics demonstration ignores air resistance in order to show what gravity does. A demonstration is not a claim about how the world behaves.

In the world, the funding mix and the rate are connected. Borrowing is cheaper than equity to begin with, partly because a lender's claim is safer and partly because interest is deductible against tax, so adding some borrowing pulls the blended rate down. But every rupee of additional borrowing also makes the remaining shareholders' position riskier, so what they require goes up, and past some point lenders start charging more too. One force pulls the blended rate down and the other pushes it up. The blended rate therefore turns somewhere between the two. Because the rate moves, the value of the operating business moves with it, and that movement is precisely what the control above holds still.

Where it turns is worked out separately. Two things should travel with the idea wherever it appears, though, and both of them cut against reading too much into a turning point. The first is that the curve is very flat near its bottom, so the precision of any stated best level of borrowing is far lower than the precision of the table it came out of. The second is that the whole shape of that curve depends on an assumed schedule of what borrowing would cost at each level, and a different schedule puts the turn somewhere else.

So what survives from the demonstration, once the honest qualification is applied? Something narrower and still worth having. At a given cost of capital, the value of the operating business is a fact about the operations, and the value of the shareholders' claim is a fact about the operations and about the claims on them together. Moving a claim moves the second and leaves the first alone. The narrower statement is true, it is useful, and it is what the control was built to show.

What links the two values, and why does every line carry a sign?

One identity links them. The line by line walk is a substantial subject of its own and is covered separately in full, including what to do about a lease, a pension deficit, employee options and a convertible.

The identity in words: the equity value is the value of the operating business, plus everything the company holds that the operating forecast never counted, less every claim standing in front of the ordinary shareholders. On this invented company that is Rs 21,28,13,79,094, plus Rs 1,20,00,00,000 of cash and Rs 1,00,00,00,000 of non-operating assets, less Rs 6,00,00,00,000 of gross debt and Rs 60,00,00,000 of minority interest. The walk lands on Rs 16,88,13,79,094. Stated once, as promised, and not worked.

Two things about the shape of that identity are principle rather than mechanism.

The first is that every line carries a sign, and the sign is decided by whose claim the item is. Something the shareholders have that the operating forecast never counted is added, or they are not credited with it. Something somebody else has a claim on is deducted, or the shareholders are credited with a claim that is not theirs. There is no line whose sign is a matter of taste.

The second is that no line is optional. Leaving a line out does not make an answer approximate; it makes it an answer to a different question, and usually one nobody asked. Leaving out the minority interest on this company credits the ordinary shareholders with a quarter of a coatings business that belongs to somebody else. Leaving out the non-operating assets quietly gives away a land parcel and a stake in an associate. The answer that comes out still looks like an answer. The arithmetic still foots. The answer simply describes a different company from the one in front of you.

Do the traded figures behave the same way?

The traded figures do, and this is the part that turns the pair from a modelling habit into something that can be pointed at.

The market puts Rs 90.00 a share on Sankalp Industrial Systems Limited, invented. Multiplied by the 20,00,00,000 shares in issue, that gives a market capitalisationThe traded equity value, being the share price multiplied by the number of shares in issue. of Rs 18,00,00,00,000. The recurring question applies again: whose claim is that? The market capitalisation is the shareholders' claim, and it is what their position changes hands for. An equity value, observed rather than computed, it is the only one of these four figures that anybody quotes.

Run in the other direction, adding back the borrowing and the minority interest and taking out the cash and the non-operating assets, the same identity arrives at the value the market implicitly puts on the operating business: Rs 22,40,00,00,000. The Rs 22,40,00,00,000 is a firm value. Nobody quotes it, nobody publishes it, and it is constructed rather than observed, but it is the same kind of object as the Rs 21,28,13,79,094 the model produced.

So there are four figures here and they arrange themselves in a two by two grid. Two firm values, one from a model and one implied by a price. Two equity values, one from a model and one observed directly. The same identity connects the pair on each side, and the only difference is which end it was entered from.

One discipline goes with that grid and it matters more than it sounds. Which of the four figures was observed and which was built has to be known. The Rs 18,00,00,00,000 is an observation: two honest people looking it up will find the same figure. The other three are constructions, each carrying whatever went into building it. Being constructed does not make them worse, but it does mean that when two of them disagree, what was assumed is what has to be examined, rather than what was measured. The difference between a computed value and an observed price is a separate subject, treated separately.

So which of the two is this company worth?

Neither, and the honest answer is that the question as put has no answer at all. The refusal sounds like a dodge, so it is worth being exact about why it is not one.

The question is not hard, only incomplete. Asking it is the same shape as asking how far away something is without saying from where, or asking what time it is without saying in which country. Add one clause and it becomes answerable immediately. Worth to whom?

Worth to everybody who funded the operations: Rs 21,28,13,79,094. Worth to the ordinary shareholders: Rs 16,88,13,79,094, being Rs 84.41 a share. Both of those are answers. The unqualified question is not a third, harder question sitting behind them; it is those two questions with the identifying clause knocked off.

A valuation that does not say which of the two questions it answered has not finished, and the missing clause is one line long. This matters far beyond tidiness. Every downstream use of a valuation, every comparison, every ratio, every negotiation, depends on knowing which claim the number describes, and none of those uses gives a warning when it has been got wrong.

There is a small pleasure in watching somebody learn this, because they usually resist it for about a minute and then stop resisting it entirely. The resistance is that ordinary speech does say things like that company is worth two thousand crore, and everybody nods. The moment it stops is when they realise they have never once said that flat is worth ninety lakh less my home loan without knowing perfectly well which of the two they meant. In the flat conversation the clause is automatic. In the company conversation it gets dropped, and dropping it is where the trouble starts.