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Enterprise-to-Equity Value Bridge: Every Line, With Its Reason

The bridge is five lines. Start at the enterprise value of the operating business, add cash, add anything the forecast did not produce, deduct gross debt, deduct minority interest, then divide by the share count. For Sankalp Industrial Systems Limited, an invented manufacturer, Rs 21,28,13,79,094 becomes an equity value of Rs 16,88,13,79,094, or Rs 84.41 a share.

Begin with a flat. Most people have done this arithmetic without ever calling it a valuation. A cousin has agreed to buy a two bedroom flat for Rs 80,00,000. The Rs 80,00,000 is the value of the flat itself, and it does not change according to who is buying it or how they are paying. But the cousin is not going to receive Rs 80,00,000 of anything. There is a home loan of Rs 52,00,000 still outstanding against the flat, the seller has left Rs 1,50,000 of unpaid maintenance dues with the society, and there is a fixed deposit of Rs 3,00,000 lying in the seller's name that has been pledged against the loan and comes across with it.

So how much does the seller actually walk away with? Rs 80,00,000, plus the Rs 3,00,000 deposit that travels with the deal, less the Rs 52,00,000 loan, less the Rs 1,50,000 of dues. Rs 29,50,000. Nobody in that conversation used the word valuation and nobody drew a diagram, but every one of them understood the two ideas the bridge is built on. The value of the thing and the value of the claim somebody has on it are two different numbers, and the walk between them is a list of who is owed what.

The walk from one number to the other has a name. The name is the bridge from enterprise valueThe value of the operating business, before any question of who funded it. to equity valueWhat is left for shareholders once every prior claim is met., and every line of it is walked below on one invented company, before the treatment of the four items that turn up on a real balance sheet and are not on this one.

Sankalp Industrial Systems Limited makes industrial valves and precision castings and sells the aftermarket parts and service that go with them. Everything about it is made up for teaching, including every rupee figure. Sankalp Coatings Private Limited, an invented subsidiary, is consolidated in full but not wholly held. Aruna Tooling Private Limited, an invented associate, is a quarter held and reported differently. Both of those facts turn into bridge lines, and that is exactly why they are there.

What decides whether something belongs in the bridge at all?

One question, asked twice, and there is nothing else to memorise. Is this a claim on the business or an asset of it, and was it already inside the earnings the enterprise value was built on? Answer both and the item's treatment falls out of the answers without any list.

Take the second half of that first. People skip it. The enterprise value did not arrive from nowhere. Somebody forecast a stream of operating cash and discounted it, and that forecast had a boundary: certain things were inside it and certain things were not. Anything that was inside it has already been paid for in the enterprise value. Putting it in the bridge as well counts it twice. Anything that was not inside it has been paid for nowhere, so it belongs in the bridge at its own value, with the sign that says whether it is something the company has or something the company owes.

Think about the flat again. The Rs 80,00,000 was a price for the flat. The society dues were not inside that price, so they came off. The pledged deposit was not inside it either, so it went on. If the seller had tried to deduct the society dues twice, once by lowering the asking price and once again at the registration table, the cousin would have spotted it in about four seconds. In an ordinary conversation about money nobody lets a claim be settled twice. People stop noticing only when the same arithmetic is written in a column with headings.

ONE TEST, ASKED TWICE, DECIDES EVERY LINE AND EVERY HARD CASEQUESTION 1Is it a claim on the business,or an asset of it?QUESTION 2Was it already inside the earnings theenterprise value was built on?Two answers, four outcomes, and no memorised list anywhere.Everything already inside stays out of the bridge, or it iscounted twice. Everything not inside goes in at its own value,with the sign the second answer hands it.THE SAME TEST SETTLES THE HARD CASESa lease, a pension deficit, employee optionsand a convertible are four applications of it.NO to question 1Neither a claim on the business nor an asset of it.NOT A BRIDGE LINE AT ALL.YES, and it WAS already insideThe forecast earnings already carried it.STAYS OUT. Putting it in counts it twice.YES, an ASSET, and it was NOT insideThe forecast never earned a rupee from it.ADD IT, at its own value.YES, a CLAIM, and it was NOT insideThe forecast never paid a rupee of it.DEDUCT IT, at its own value.
Two questions decide every bridge line and every hard case, which is why no list has to be memorised.
Try it out

What is the single test that decides whether something belongs in a bridge?

What is line 1 actually carrying?

The value of the operating business, and only that. For Sankalp Industrial Systems Limited that figure is Rs 21,28,13,79,094, and it is worth being precise about what sits inside it and what does not.

Inside it: the valve business, the castings business and the aftermarket parts and service business. Their revenue, their costs, the tax on their profit, the money they have to spend on machines to keep growing and the working capital they have to fund. Five years of that, forecast explicitly, and then a terminal value standing in for everything after. How that number was produced is covered separately. The bridge takes it as given and works on what happens next.

The whole subject of the four lines that follow is what is not inside it. The enterprise value is deliberately neutral to how the business was funded, so the forecast never counted the interest on any loan. The forecast never counted a rupee of the cash sitting in the company's bank accounts. The forecast never counted the rent on a piece of surplus land the company is not using. The forecast never counted the profit share of an associate that is reported below the operating lines. Four omissions, four bridge lines.

The figure itself is printed several times below, and it needs one note. Rs 21,28,13,79,094 is stated to the rupee here because the bridge is an arithmetic identity and the columns have to foot. The precision is not there because the last few digits mean anything. Roughly seventy eight per cent of that number is a terminal value, an assumption about what happens after Year 5 forever. The honest reading is Rs 2,128 crore give or take a good deal, and the trailing digits are there to make the addition checkable rather than to make a claim.

Why is cash added at its full amount when it earns almost nothing?

Because it is an asset the company has and the forecast never earned a rupee from it. Both halves of the test are answered. Line 2 adds Rs 1,20,00,00,000.

Cash is the easiest line to get right and the easiest to talk yourself out of. A bank balance is not doing anything: it does not have a margin, it does not grow at five per cent forever and it will not still be there in three years. Adding one at its full amount to a business valuation feels wrong. But none of that matters. The valuation of the operating business is a valuation of the operating business. The cash is a separate thing the company happens to have, in the same way that the pledged deposit was a separate thing the seller of the flat happened to have, and a separate thing gets added at what it is, not at what it earns.

Now the question a sharp reader asks immediately, and it is a good one. The record for this company splits the cash: Rs 40,00,00,000 is operating cash the business needs in order to run, meaning float in tills and current accounts that never really goes away, and Rs 80,00,00,000 is surplus. If Rs 40,00,00,000 of it is genuinely part of running the business, is it not already inside the operating value? Added again, it has been counted twice.

Apply the test rather than the instinct. Net working capital in this forecast is receivables of Rs 2,16,00,00,000 plus inventory of Rs 1,44,00,00,000 less payables of Rs 1,80,00,00,000. There is no cash inside that definition at all, so the operating Rs 40,00,00,000 is nowhere in the model and adding the whole balance counts nothing twice.

Treating the whole balance as addable is a convention, not a law of nature, and the honest thing is to name it. A house that defines its working capital to include an operating cash float would add only the Rs 80,00,00,000 of surplus and would land Rs 40,00,00,000 lower, at Rs 16,48,13,79,094, or Rs 82.41 a share rather than Rs 84.41. Two rupees a share, on one definitional choice, and both houses are being perfectly careful. Every figure below uses the gross cash convention and names it wherever it appears. What matters is not which convention was chosen but whether the person reading the work can tell which one it was.

What goes into line 3, and how is each item tested?

Two items here, tested one at a time, and together they add Rs 1,00,00,00,000. A non-operating assetSomething held by the company that produces none of the earnings being forecast. is anything the company has that produces none of the earnings the valuation was built on.

The first is a surplus land parcel carried at Rs 45,00,00,000. Ask the two questions. Is it an asset of the business? Yes, the company has it and could sell it. Was it inside the earnings? No: it is surplus, nothing is made on it, no revenue passes over it and not one rupee of the forecast earnings before interest, tax, depreciation and amortisation (EBITDA) came from it. So it is added, at its own value.

The second is a 26.0 per cent holding in Aruna Tooling Private Limited, carried at Rs 55,00,00,000, and this one is more interesting because it is a real business that really does earn money. The holding still gets added, and the reason is about where its earnings are reported. A quarter-sized holding is not consolidated. The holding is equity accountedReported as a share of an associate's profit below the operating lines, and therefore outside EBITDA.. The group's share of its profit appears as a single line below the operating result, well below EBITDA. The forecast that produced Rs 21,28,13,79,094 was built on EBITDA. Aruna Tooling's contribution was never in it.

Notice how much work the accounting treatment is doing there, and how little the business is doing. If the group held enough of Aruna Tooling to consolidate it, its EBITDA would be inside the forecast and adding the holding in the bridge would count it twice. Same business, same profit, opposite treatment, and a shareholding percentage is all that flipped it. The test is therefore asked about the earnings the valuation used and not about the business in the abstract.

Try it out

Why is the 26.0 per cent holding in Aruna Tooling Private Limited added rather than left inside the forecast?

Why does line 4 say gross debt and not net debt?

Because line 2 already added the cash, and a rupee of cash may only be counted once. Line 4 deducts gross debtEverything owed to lenders, before deducting any cash. of Rs 6,00,00,00,000.

Sankalp Industrial Systems Limited owes that across three tranches: a secured rupee term loan of Rs 3,00,00,00,000, listed unsecured debentures of Rs 2,00,00,00,000 and a working capital facility of Rs 1,00,00,00,000 drawn against receivables and inventory. Three lenders, three sets of terms, one arithmetic treatment. Every rupee of it is a claim that ranks ahead of the shareholders and none of it was inside the forecast, so all of it comes out.

Net debtGross debt less cash, a figure used in leverage ratios and not in a bridge that already adds cash. is a different figure that exists for a different purpose. Net debt is gross debt less cash, Rs 4,80,00,00,000 on this company, and it is the number that turns up in leverage ratios, in covenant tests and in almost every conversation a credit officer has. The familiarity of net debt is the whole problem. Net debt is the figure sitting in the front of most people's minds when they reach the debt line of a bridge, and it is the wrong one to reach for once cash has already gone in on line 2.

There are two correct bridges here and the error is building half of each. Either add cash on line 2 and deduct gross debt on line 4, or add nothing on line 2 and deduct net debt on line 4. Both settle the same cash balance exactly once, so both land on exactly the same equity value. The first arrangement is used below. Each claim is then visible as its own line, and a bridge whose lines can each be argued with separately is easier to check than one that has quietly combined two of them.

What happens when the cash is added and net debt is then deducted?

The answer is Rs 90.41 a share, the traded price is Rs 90.00, and the model looks rather good. Deducting net debt after adding the cash is the most instructive failure in the whole bridge.

Work it through. 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 net debt of Rs 4,80,00,00,000, less minority interest of Rs 60,00,00,000, gives Rs 18,08,13,79,094. Divided by 20,00,00,000 shares that is Rs 90.41. The correct bridge gives Rs 16,88,13,79,094 and Rs 84.41. The gap between them is exactly Rs 1,20,00,00,000, the cash balance, counted once as a positive line and a second time inside net debt. As a share of the correct answer that is 7.11 per cent.

THE ERROR THAT VALIDATES ITSELF, IN RUPEES A SHARETwo bridges off the same enterprise value. One deducts gross debt. The other deducts net debt and still adds the cash.the traded share price, Rs 90.00THE CORRECT BRIDGEadd cash, then deductGROSS debt of Rs 6,00,00,00,000Rs 84.41THE COMMON ERRORadd cash, then deductNET debt of Rs 4,80,00,00,000Rs 90.41Rs 5.59 away from the market41 paise awayRs 80Rs 82Rs 84Rs 86Rs 88Rs 90Rs 92Rs 94The wrong answer counts the same Rs 1,20,00,00,000 of cash twice, once as a positive line and once inside net debt.It lands nearer the market than the correct one. Agreement with a traded price is never evidence that a bridge was built correctly.
Deducting net debt while still adding the cash lands 41 paise from the market, while the correct bridge lands Rs 5.59 away from it.

Now look at where the two answers land. The shares of Sankalp Industrial Systems Limited trade at Rs 90.00. The wrong answer sits 41 paise from the market and the right answer sits Rs 5.59 away. Somebody who double counts the cash produces a number that appears to confirm itself, and somebody who does the arithmetic correctly produces one that appears not to.

Sit with how bad that is for a moment. Every instinct a careful person has says that a model landing within half a rupee of the traded price has probably been built correctly, and every instinct says that one landing more than five rupees away needs another look. Here both instincts point at the defective bridge and away from the sound one. The error is self-validating; the correct answer is not.

The consequence is a rule about checking, and it is the single most important sentence in this guide. Agreement with a market price is never evidence that a bridge was built correctly. A bridge is checked structurally, by reading each line and confirming that every claim appears exactly once, and never by glancing at whether the answer looks about right. The meaning of a difference between a computed value and an observed price is covered separately.

The mechanical check takes a second and costs nothing. If cash appears anywhere in a bridge as a positive line, the debt line must say gross. If the debt line says net, there must be no cash line above it. There is no third arrangement.

Try it out

A bridge adds Rs 1,20,00,00,000 of cash and then deducts Rs 4,80,00,00,000 of net debt. What has gone wrong, and by how much?

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Why does consolidation create a minority interest line?

Because the forecast counted cash flow the group does not entirely have a claim on. Line 5 deducts minority interestThe share of a consolidated subsidiary the group does not hold. of Rs 60,00,00,000.

Sankalp Industrial Systems Limited holds 75.0 per cent of Sankalp Coatings Private Limited and consolidates it in full. Full consolidation means every rupee of the coatings subsidiary's revenue, cost and EBITDA appears in the group numbers, not three quarters of it. So when somebody forecast the group's operating cash flow and discounted it into Rs 21,28,13,79,094, the whole of that subsidiary went into the numerator.

But the group does not have a claim on the whole of it. A quarter of that subsidiary belongs to somebody else, and that somebody else has a prior claim on a quarter of everything it produces. The forecast counted one hundred per cent of the cash flow and the shareholders of the parent have a claim on seventy five per cent of it, so the missing quarter has to come out somewhere, and line 5 is where.

Here is the household version, and it is closer than it looks. Two brothers run a shop together, one holding three quarters and the other a quarter. The elder brother, doing the sums at home, writes down the shop's whole monthly takings because that is what the shop takes, then remembers at the foot of the sheet that a quarter of it is not his. He does not go back and rewrite every line of the takings; he puts one deduction at the bottom for his brother's share. Line 5 is that deduction, and it exists for the same reason: it is far easier to count the business once and adjust at the end than to count three quarters of everything all the way down.

Why the figure is Rs 60,00,00,000 rather than some other figure, whether that amount reflects book value or something closer to what the stake would fetch, and what becomes of it when a company changes hands, are all covered separately. The bridge gives the line its sign and its reason and hands the rest on.

What does the finished walk look like from end to end?

Five lines, in this order, with no line optional and no line able to move.

LineItemSignAmountRunning total
1Enterprise value of the operating businessstartRs 21,28,13,79,094Rs 21,28,13,79,094
2Cash and cash equivalentsaddRs 1,20,00,00,000Rs 22,48,13,79,094
3Non-operating assets: surplus land Rs 45,00,00,000 and the associate holding Rs 55,00,00,000addRs 1,00,00,00,000Rs 23,48,13,79,094
4Gross debt across the three tranches, not net debtlessRs 6,00,00,00,000Rs 17,48,13,79,094
5Minority interest in the consolidated subsidiarylessRs 60,00,00,000Rs 16,88,13,79,094
Equity value, and on 20,00,00,000 shares=Rs 16,88,13,79,094Rs 84.41 a share
THE WALK, LINE BY LINE, ON ONE RUNNING TOTALThe value axis starts at Rs 16,00,00,00,000, not at zero, so the four adjustments are legible against each other.16,00,00,00,00018,00,00,00,00020,00,00,00,00022,00,00,00,00024,00,00,00,000rupeesLINE 1enterprise valueof the operating business+ Rs 1,20,00,00,000LINE 2add cashand cash equivalents+ Rs 1,00,00,00,000LINE 3add non-operatingassetsless Rs 6,00,00,00,000LINE 4less gross debtless Rs 60,00,00,000LINE 5less minorityinterestRs 21,28,13,79,094Rs 16,88,13,79,094equity value,Rs 84.41 a shareEvery line has a sign and a reason. Nothing between the five levels is interpolated: the track is flat across each line and moves only at the joins.Divided by 20,00,00,000 shares, Rs 16,88,13,79,094 is Rs 84.41 a share, rounded from Rs 84.4068 for display.
The five lines carry a running total from Rs 21,28,13,79,094 down to Rs 16,88,13,79,094, each with its own sign and reason.

The per share figure deserves one sentence of care. Rs 16,88,13,79,094 divided by 20,00,00,000 shares is Rs 84.406895 and a bit, printed as Rs 84.41. Every printed figure is rounded for display and every identity is computed on the unrounded value. The rupee column therefore foots exactly, and the per share figure carries the word rounded whenever it matters. Multiplying Rs 84.41 back by the share count gives Rs 16,88,20,00,000, or Rs 6,20,906 away from the starting figure. The difference is not an error in either figure; it is what rounding does, and work that quietly adjusts one of its own numbers so that the round trip comes out has stopped being checkable.

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Which of the two equity figures is the one meant?

There are two, they are both correct, and confusing them is a genuine hazard rather than a pedantic one.

The first is the one just built: Rs 16,88,13,79,094, all five lines, everything the company has and everything it owes. The figure is the equity value of Sankalp Industrial Systems Limited, full stop.

The second is Rs 15,88,13,79,094, and it is the equity value of the operating business alone. Lines 1, 2, 4 and 5 build it, with line 3 left out. The non-operating assets are not in it, so the land and the associate holding are not in it, and on 20,00,00,000 shares it is Rs 79.41 rather than Rs 84.41. The two figures differ by exactly Rs 1,00,00,00,000, the amount of line 3 and nothing else.

TWO EQUITY FIGURES, ONE LINE APARTBoth are correct. They answer different questions, and any work printing either one has to say which it is.THE WHOLE EQUITY VALUElines 1 to 5, non-operating assets includedRs 16,88,13,79,094THE OPERATING EQUITY VALUElines 1, 2, 4 and 5, non-operating assets left outRs 15,88,13,79,094Rs 1,00,00,00,000, exactly line 303,00,00,00,0006,00,00,00,0009,00,00,00,00012,00,00,00,00015,00,00,00,00018,00,00,00,000Rs 84.41 a share on the upper figure and Rs 79.41 a share on the lower one. The Rs 5.00 between them is the surplus land and the associate holding.The lower figure exists so that the firm route and the equity route can be compared on the same basis, since the equity route excludes those assets too.
Two correct equity figures for one invented company differ by exactly the Rs 1,00,00,00,000 of non-operating assets and by nothing else.

Why keep a second figure at all? Because there is a second route to an equity value that discounts the cash flow available to shareholders directly, at the cost of equity, rather than valuing the whole business and walking down. Neither the surplus land nor the associate holding produces cash flow to shareholders inside the forecast, so that route never sees them either. For the two routes to be compared at all, they have to be set on the same basis, and Rs 15,88,13,79,094 is the figure prepared on that basis. Why the two routes do not in fact agree on this company is covered separately.

The hazard is small and specific. Somebody runs two valuations, gets Rs 16,88,13,79,094 from one and something near Rs 15,88,13,79,094 from the other, and reports a Rs 1,00,00,00,000 discrepancy as a finding. The gap is not a finding. The gap is a difference in what was included, and it is exactly the size of a line one of them left out. Any work printing either figure says which one it is, in the same sentence.

Try it out

What is the difference between the Rs 16,88,13,79,094 and the Rs 15,88,13,79,094 equity figures for this invented company?

Building a Discounted Cash Flow teaches you to build a model, say where its answer comes from, and defend the two assumptions carrying it.

How does the same walk run backwards from a share price?

Before reading on, predict what comes out. The question is a better one than it looks.

Try it out

Run the bridge backwards from the Rs 90.00 share price of Sankalp Industrial Systems Limited. What should come out at the far end?

A bridge is an identityA relationship that holds by construction and can be run in either direction. rather than a method. An identity is a statement about which claims exist, and a statement like that does not have a preferred direction any more than the fact that a shop's takings less its costs equals its profit has a preferred direction. So it can be entered from whichever end happens to be observable.

From the market end the share price is observable. Rs 90.00 on 20,00,00,000 shares is a market capitalisation of Rs 18,00,00,00,000. Every sign then flips. Gross debt of Rs 6,00,00,00,000 is added. The buyer of the whole business would take that on. Minority interest of Rs 60,00,00,000 is added, for the same reason. The cash of Rs 1,20,00,00,000 is deducted. The buyer gets that back on day one. The non-operating assets of Rs 1,00,00,00,000 are deducted. The buyer of the operating business is not paying for a surplus land parcel. Rs 22,40,00,00,000 exactly.

THE SAME FIVE ITEMS, RUN IN BOTH DIRECTIONSEvery sign flips and nothing else changes, which is what makes this an identity rather than a technique.FORWARDS, FROM THE MODELstart at a computed enterprise value, finish at a value per shareBACKWARDS, FROM THE MARKETstart at an observed market capitalisation, finish at an enterprise valueenterprise valuestartRs 21,28,13,79,094cash and cash equivalents+Rs 1,20,00,00,000non-operating assets+Rs 1,00,00,00,000gross debtlessRs 6,00,00,00,000minority interestlessRs 60,00,00,000EQUITY VALUE=Rs 16,88,13,79,094market capitalisationstartRs 18,00,00,00,000gross debt+Rs 6,00,00,00,000minority interest+Rs 60,00,00,000cash and cash equivalentslessRs 1,20,00,00,000non-operating assetslessRs 1,00,00,00,000TRADED ENTERPRISE VALUE=Rs 22,40,00,00,000Rs 22,40,00,00,000 is 7.78 times the Year 0 EBITDA of Rs 2,88,00,00,000 for Sankalp Industrial Systems Limited, invented.The two columns are not two methods. They are one identity entered from whichever end happens to be observable.
Every sign flips and no amount changes between the two directions, which is what makes these five lines an identity rather than a method.

Every sign has flipped, no amount has changed, and that is the clearest demonstration available that the five lines are an identity and not a technique. The traded enterprise value of Rs 22,40,00,00,000 is 7.78 times the Year 0 EBITDA of Rs 2,88,00,00,000 for this invented company, and that is a useful thing to be able to compute. A share price on its own cannot be compared with anything that has a different amount of debt behind it.

Two things do not follow from those numbers. Which of the two enterprise values is right is not settled by the arithmetic, and neither of them makes the company cheap, expensive, undervalued, overvalued or fairly valued. Both are outputs of stated assumptions. The consequence of a gap between a computed value and an observed price is covered separately.

What happens to the equity value when the business is worth more?

The answer is slightly surprising, and it is worth predicting first.

Try it out

The enterprise value rises 5.26 per cent, from Rs 21,28,13,79,094 to Rs 22,40,00,00,000. By what percentage does the equity value rise?

Four of the five lines are fixed amounts. Cash is Rs 1,20,00,00,000 whatever the operating business turns out to be worth. Non-operating assets are Rs 1,00,00,00,000, gross debt is Rs 6,00,00,00,000 and minority interest is Rs 60,00,00,000. Netted together they give a constant: less Rs 4,40,00,00,000, at every setting, always. So the equity value is simply the enterprise value less Rs 4,40,00,00,000. Every rupee the operating business gains or loses lands on the shareholders undiluted, and lands on a smaller base.

The fixed block is why the percentages differ. A 5.26 per cent rise in the enterprise value is a 6.63 per cent rise in the equity value. The rupees are identical, Rs 1,11,86,20,906 in both cases; only the denominators differ, Rs 21,28,13,79,094 against Rs 16,88,13,79,094. Move the control below and watch the two percentages come apart while the four fixed lines refuse to budge.

Play with it

Move the enterprise value and watch the four fixed lines refuse to move with it

One control, ninety two stops. The upper panel holds the walk, with only the first bar and the last bar redrawing. The lower panel is a scale in rupees a share, carrying two fixed marks: this model's own Rs 84.41 and the traded Rs 90.00.

The whole reading in static text, so it survives without the picture. The four lines after the first never move at any setting: add cash Rs 1,20,00,00,000, add non-operating assets Rs 1,00,00,00,000, less gross debt Rs 6,00,00,00,000, less minority interest Rs 60,00,00,000, netting to less Rs 4,40,00,00,000. The share count is 20,00,00,000 throughout. So: an enterprise value of Rs 18,00,00,00,000 gives an equity value of Rs 13,60,00,00,000 and Rs 68.00 a share. Rs 20,00,00,00,000 gives Rs 15,60,00,00,000 and Rs 78.00. Rs 21,28,13,79,094, this model's own answer, gives Rs 16,88,13,79,094 and Rs 84.41. Rs 22,40,00,00,000, the traded enterprise value, gives Rs 18,00,00,00,000 and exactly Rs 90.00, the traded share price, and that agreement is the identity checking itself rather than a valuation being confirmed. Rs 24,00,00,00,000 gives Rs 19,60,00,00,000 and Rs 98.00. Rs 27,00,00,00,000 gives Rs 22,60,00,00,000 and Rs 113.00. The figure this control exists to show: moving from Rs 21,28,13,79,094 to Rs 22,40,00,00,000 is a rise of 5.26 per cent in the enterprise value and a rise of 6.63 per cent in the equity value. The Rs 4,40,00,00,000 of net fixed claims does not move. None of the four hard cases below is present on this invented company's balance sheet and none is in this arithmetic.
Rs 18,00,00,00,000Rs 21,28,13,79,094Rs 27,00,00,00,000
1 THE FIVE LINES AT THE CHOSEN SETTINGLINE 1 enterprise valueof the operating businessRs 21,28,13,79,094LINES 2 TO 5. THESE FOUR DO NOT MOVE AT ANY SETTING OF THE CONTROLLINE 2 add cashand cash equivalents+ Rs 1,20,00,00,000LINE 3 add non-operatingassets+ Rs 1,00,00,00,000LINE 4 less gross debtnot net debtless Rs 6,00,00,00,000LINE 5 less minorityinterestless Rs 60,00,00,000The four net to less Rs 4,40,00,00,000 at every setting, so the equity value is simply the enterprise value less that amount.EQUITY VALUEwhat is left for shareholdersRs 16,88,13,79,09405,00,00,00,00010,00,00,00,00015,00,00,00,00020,00,00,00,00025,00,00,00,000rupees2 WHAT THAT IS PER SHARE, ON 20,00,00,000 SHARESRs 70Rs 75Rs 80Rs 85Rs 90Rs 95Rs 100Rs 105Rs 110Rs 115Rs 84.41, the model's own answerRs 90.00, the traded priceRs 84.41value per shareBoth bars are drawn on the same scale. Neither the four lines above nor the share count moves at any setting of the control.
Enterprise value
Rs 21,28,13,79,094
The four fixed lines, net
less Rs 4,40,00,00,000
Equity value
Rs 16,88,13,79,094
Value per share
Rs 84.41
Move in the enterprise value
this is the model's own answer
Move in the equity value
this is the model's own answer

At an enterprise value of Rs 21,28,13,79,094 for the operating business of Sankalp Industrial Systems Limited, invented, the four lines below it still net to less Rs 4,40,00,00,000, so the equity value is Rs 16,88,13,79,094 and the value per share is Rs 84.41. This is the enterprise value this model produces, and the setting this guide is built on.

Educational illustration. Not a valuation tool and not a decision aid. It says nothing about what Sankalp Industrial Systems Limited or any other company is worth, and it names no setting as right, likely or suitable for anybody. Cash of Rs 1,20,00,00,000, non-operating assets of Rs 1,00,00,00,000, gross debt of Rs 6,00,00,00,000 and minority interest of Rs 60,00,00,000 are held fixed at every setting. Holding them fixed is the point being made rather than a claim that they would hold still in life. The share count is 20,00,00,000 throughout and no option, convertible or new issue is modelled. The gross cash convention is used, so the whole cash balance is added rather than the surplus alone. Money is held in whole rupees throughout and each per share figure is rounded to the paise for display only. The control has ninety two stops: ninety one at Rs 10,00,00,000 intervals from Rs 18,00,00,00,000 to Rs 27,00,00,00,000, plus this model's own Rs 21,28,13,79,094. The odd figure does not sit on that interval and is carried as its own stop, so the default reproduces the worked example exactly.

One reading of that control is worth spelling out because it is the honest version of a thing people say loosely. The equity is the residual claim: it is what is left after the fixed claims are met, so it absorbs the whole of any movement in the value of the business while sitting on a smaller base than the business does. On this invented company that turns a 5.26 per cent movement into a 6.63 per cent one. The same arithmetic works identically in the other direction, and no bridge predicts which way the value of a business will move.

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How is a lease treated?

Here the ground changes. The four items that follow are the ones that turn up on real balance sheets and cause real arguments, and none of them is on this invented company's balance sheet at all. Sankalp Industrial Systems Limited has no lease, no pension deficit, no employee options and no convertible, and its own bridge has exactly five lines. Each of the four is therefore taught as a treatment, a sign and a reason, with no rupee amount attached to any of them.

Start with the lease, and do not start by asking what a lease is. The definition of a lease is settled elsewhere and assumed here. Ask the only question the bridge cares about: does the earnings measure the enterprise value was built on already bear the cost of the lease?

There are two answers and each one determines the treatment completely. Where a lease obligation is recognised on the balance sheet and its cost appears below EBITDA, split into depreciation on the right-of-use asset and interest on the liability, then the EBITDA the valuation used has not paid for the leased asset at all. The obligation is a genuine outstanding claim that the forecast never settled, so it is a debt-like itemA claim that is not a loan but behaves as one for the purposes of a bridge. and it is deducted alongside the debt. Where instead the rental sits inside EBITDA as an operating cost, the earnings have already borne it, the forecast has already paid for the use of the asset, and deducting the liability as well counts the same cost twice.

A LEASE: THE SAME OBLIGATION, TWO TREATMENTS, DECIDED BY ONE FACTNo rupee figure appears here. Sankalp Industrial Systems Limited, invented, carries no lease in its record.WHERE DOES THE LEASE COST SITin the earnings the valuation used?BELOW EBITDAas depreciation and interestTHE EBITDA HAS NOT PAID FOR ITso the obligation is a debt-like claimDEDUCT IT on the debt lineINSIDE EBITDAas a rental charge in operating costsTHE EBITDA HAS ALREADY PAID FOR ITso deducting the liability as well counts it twiceLEAVE IT OUT of the bridgeThe test is the same one that decided every line above: was this already inside the earnings the enterprise value was built on?The damage is done not by choosing wrongly but by applying a multiple built on one basis to earnings prepared on the other.
Where the lease cost sits in the earnings decides the treatment completely, while the building and the rent stay exactly the same.

The test is the same one set out at the start, not a new one, and nothing about the lease itself changed between the two branches. The building is the same building and the rent is the same rent. The change is in which line of the accounts the cost landed on, and therefore in whether the earnings the valuation used had already absorbed it.

The damage in practice is done not by choosing the wrong branch but by mixing the branches across a comparison. A multiple built on the earnings of companies that keep their rentals inside EBITDA, applied to a company that reports its lease costs below EBITDA, with that company's lease liability then deducted as well, is a multiple that assumes the rent has been paid followed by a charge for the asset a second time. The mismatch is worth far more than the sign on any single line, and it is invisible in a spreadsheet because both halves look individually reasonable.

Try it out

A company's lease cost appears below EBITDA, as depreciation and interest. Does the lease liability come out in the bridge?

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What about a pension deficit?

Deducted, and there are two things about it that get muddled almost every time.

A pension deficitThe shortfall between a pension obligation and the assets held against it. is a claim on future cash that the operating forecast did not deduct. Nothing in the projected EBITDA of a valve business pays down a shortfall in a retirement scheme, so the shortfall is still outstanding at the moment the enterprise value is struck. Test passed on both halves: a claim, and not inside the earnings. The deficit comes out.

The first muddle is about which number. The deficit is deducted and never the gross obligation. The assets held against the obligation are already sitting there and settling part of it. A scheme owing a large amount with a large pot set aside against it may have a modest shortfall or none at all, and deducting the whole promise while ignoring the pot is the same double count in a new coat: it charges the company for money it has already put by.

The second muddle is about tax. Where relief against tax is available on the contributions that will eventually clear the shortfall, the true cost to the shareholders is less than the shortfall itself, and the usual treatment is to deduct it net of that relief. Whether such relief exists, on what terms and within what limits is a matter for the tax authority of the place the scheme sits in. The unvarying part is that the work doing the arithmetic has to say which of the two it did. A reader cannot tell a gross deduction from a net one by looking at the total.

What about employee options?

Two treatments, both correct, and the error is using both at once or neither.

Options held by employees are a claim on the equity itself rather than on the business, and that makes them different from every line above. Options do not reduce the value of the business; they divide it among more people. So the arithmetic can be done at either end of the division.

The first treatment deducts the value of the outstanding options from the equity value and then divides by the existing share count. The second, sometimes called the treasury methodTreating option exercise proceeds as cash in and adding the resulting shares to the count., adds the exercise proceeds to the equity value because the company receives them, and then divides by the enlarged count that results from those options being exercised. Both settle the option holders' claim exactly once. The two treatments rest on slightly different views of what happens between here and exercise, so they will not give identical answers to the paise. Both are defensible and any work using one states which.

EMPLOYEE OPTIONS: TWO CORRECT TREATMENTS, AND TWO WAYS TO GET IT WRONGThey are alternatives. Using both, or neither, is the error, and no rupee figure is attached to any of the four.TREATMENT Adeduct the value of the outstandingoptionsdivide by the EXISTING share countCORRECTTREATMENT Badd the exercise proceeds to the equityvaluedivide by the ENLARGED share countCORRECTBOTH AT ONCEdeduct the option value AND enlarge thecountthe same claim comes out twiceTOO LOWNEITHERno deduction and no extra sharesthe claim is handed to the existingholdersTOO HIGHBoth correct treatments settle the option holders' claim exactly once. That is the whole of the rule, and it needs no memorised list.Any work states which of the two it used, because the two produce different share counts and therefore different per share figures.The same one-claim-once test runs through a pension deficit and a convertible as well.
Employee options carry two correct treatments and the error is doing both at once, which a column of figures still foots on.

Doing both deducts the same claim twice and understates what the ordinary shareholders have; doing neither hands them value that belongs to somebody else. Neither of those is a sign error and neither will be caught by re-adding the column, and that is what makes them worth naming separately. The column foots perfectly in both cases.

One practical note on which options count. Options a long way from being worth exercising at the computed value are a much smaller claim than options that are already worth exercising, and treatments differ on how much of that distinction to carry. Where an option is in the moneyWorth exercising at the value being computed. at the value reached, its claim is not in serious doubt and it belongs in the arithmetic. How an option itself is valued is covered separately.

Try it out

An analyst deducts the value of the outstanding employee options and also divides by the enlarged share count. What has that done?

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What about a convertible?

A convertible is debt or it is equity. A convertible is never half of each in the same bridge, and that is the whole rule.

A convertible is an obligation that its holder may turn into shares instead of taking repayment. So it sits in exactly one of two states at the value being computed, and the state has to be settled before the bridge can be built. Where the conversion is worth taking at that value, the instrument is treated as equity: it is left out of the debt line entirely and the shares it would become are added to the share count. Where the conversion is not worth taking at that value, it is treated as debt: deducted on the debt line at its amount, with the share count left alone.

The two failures are mirror images and they push the answer in opposite directions. Taking the amount out of the debt line while leaving the new shares out of the count flatters the answer, and leaving the amount in the debt line while also adding the shares punishes it twice. The first says the holders have given up their claim and asked for nothing in return; the second says they hold both a loan and the shares it converts into.

There is a circularity here that is worth naming rather than hiding. Whether the conversion is worth taking depends on the value per share, and the value per share depends on how the convertible was treated. On most balance sheets the answer is obvious in one direction or the other and the circularity never bites. Where it does bite, the honest treatment is to compute it both ways, show both answers and say which state each one assumes, rather than picking one quietly and presenting a single figure.

How do the four hard cases collapse into one rule?

Because they were never four rules. Read them back with the test in hand and each one is a single question asked about a different item.

A lease: is it a claim, and was it inside the earnings? A claim, and inside or outside depending on where the cost was reported, and that is why it has two branches. A pension deficit: a claim, never inside the operating earnings, so out it comes, at the shortfall rather than the promise. Employee options: a claim on the equity rather than the business, settled once at either end of the division. A convertible: a claim that is either a loan or a shareholding but cannot be both at once, so the state is settled first and the consequences follow from it consistently.

Four items, one test, and not a single one of them needs to be memorised as a rule of its own. The test comes before the lines rather than after them for exactly that reason. A reader who has the test can work out the treatment of an item nobody has taught them, and a reader who has only a list is stuck the first time something turns up that is not on it.

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How is a finished bridge checked in a few seconds?

Read it aloud as an ordinary sentence about who is owed what, and listen to whether it makes sense.

The operating business is worth Rs 21,28,13,79,094. The business holds Rs 2,20,00,00,000 of things the forecast did not produce, being the cash and the two non-operating assets. The business owes lenders Rs 6,00,00,00,000. Of everything it consolidated, Rs 60,00,00,000 belongs to somebody else. So the shareholders have Rs 16,88,13,79,094, or Rs 84.41 each on 20,00,00,000 shares.

THE FINISHED BRIDGE, READ BACK AS AN ORDINARY SENTENCE ABOUT CLAIMSThe operating business is worthRs 21,28,13,79,094It holds things the forecast did not produce, worthRs 2,20,00,00,000It owes lendersRs 6,00,00,00,000Of what it consolidated, somebody else's share isRs 60,00,00,000So the shareholders haveRs 16,88,13,79,094and dividing by 20,00,00,000 shares, that is Rs 84.41 each.If the sentence does not make ordinary sense read aloud, one of the five lines is wrong.It is a slower check than re-adding the column and it catches a different class of mistake, because a sign error still adds up.
A finished bridge read aloud as a sentence about claims catches sign errors that a column of figures adds up through perfectly.

The spoken version is doing something the column of figures cannot. A column with a sign error still adds up perfectly, and a sentence with a sign error stops making sense out loud. Say the defective version and hear what happens: the business is worth this much, it holds this much cash, it owes lenders this much after taking off the cash it has, and it also holds that same cash. Nobody says that sentence twice without stopping.

The spoken check is slower than re-adding the column and it catches a different class of mistake, so it is worth doing as well rather than instead. Three questions, in order. Does every claim appear exactly once? Does every asset appear exactly once? And if cash appears as a positive line anywhere, does the debt line say gross?

The failure: a wrong answer that agrees with the market

The error happens because both correct methods are taught, and somewhere between the two the halves get spliced. The analyst adds cash of Rs 1,20,00,00,000 on line 2. Cash is an asset and adding it is obviously right. Then, at line 4, they reach for the debt figure that is in front of them, and the figure in front of almost everybody is net debt. Net debt is what appears in the leverage ratios, in the covenant packs and in every conversation with a lender.

The answer comes out at Rs 18,08,13,79,094 rather than Rs 16,88,13,79,094, too high by Rs 1,20,00,00,000 and 7.11 per cent, and Rs 90.41 a share rather than Rs 84.41.

The danger is not the size of the error but where it lands. Rs 90.41 sits 41 paise from a traded price of Rs 90.00, so the defective bridge looks like a model that has come out near the market, and the sound bridge at Rs 84.41 looks like one that has not. Every review instinct a careful person has will wave the wrong answer through and send the right one back for another look.

The cost is a conclusion that survives review because it looks reasonable. Nobody argues with it, nobody re-derives it, and it goes into a report, and then into a note, and then into somebody's memory of the value of this company.

The check is mechanical and takes a second: if cash appears anywhere as a positive line, the debt line must say gross.

How this is actually used in a working week

An equity research associate uses the bridge twice on the same afternoon and in opposite directions. Forwards, to turn the enterprise value their model produced into a figure per share that can sit next to a traded price. Backwards, to turn the traded prices of six other companies into traded enterprise values. A share price on its own cannot be compared with anything that carries a different amount of debt. The backward use comes up more often, and it is the reason the identity has to run cleanly in both directions rather than only the way it was first taught.

A credit officer at a lender reads the same five lines with the middle three removed. The credit officer wants to know how much value sits above their own claim: if the operating business is worth Rs 21,28,13,79,094 and their loan sits inside Rs 6,00,00,00,000 of gross debt, there is a large cushion between the two, and the size of that cushion is the whole question. Lenders insist on gross debt and are sceptical about the cash line. Cash can be spent in a fortnight and a term loan cannot be repaid in one.

A person selling a small business they have run for twenty years uses it without ever writing it down, exactly as the seller of the flat did. Somebody has offered a price for the business. Out of that price comes the loan against the machinery, out comes the overdraft, out comes the deposit the landlord holds and back comes the money in the current account. The remainder is what reaches the household, and it is always a good deal less than the number they repeat to their relatives.

In all three the bridge is doing the same job. The bridge is turning a statement about a business into a statement about a person's claim, and there is no financial conversation of any size that does not need that translation at some point.

India

Where the raw material of these lines comes from

The arithmetic of a bridge is not specific to any country. Adding a cash balance and deducting a loan is addition and subtraction, and neither changes at a border. The publishing requirements do change, and they decide whether anybody outside the company can find the five figures at all. In India, what a listed company discloses sits under the framework of the Securities and Exchange Board of India at sebi.gov.in. A company's filings, its charges and its shareholding sit with the Ministry of Corporate Affairs at mca.gov.in, and the shareholding is where a minority interest becomes visible. Anything involving a lender or a cross-border flow sits with the Reserve Bank of India at rbi.org.in. Where a pension deficit is deducted net of relief against tax, the availability and terms of that relief are a matter for the relevant tax authority. All of these frameworks change, and a reader who needs a current condition reads the current text at the source rather than a summary of it.

Walking the bridge is one job, and several neighbouring ones are set out elsewhere. Why two values exist at all, whose claim each one represents and why one is neutral to how the business was funded are covered separately. Which of the two values pairs with which financial measure, and what goes wrong when they are mismatched, is covered separately. Why the minority interest is Rs 60,00,00,000, whether that figure is a book one or a fair one and what becomes of it when a company changes hands are covered separately. The argument about whether operating cash should be treated differently from surplus cash is covered separately, and the convention used here is named wherever the figure appears. How the enterprise value of Rs 21,28,13,79,094 is produced is covered separately, and so is the reconciliation between the firm route and the equity route. A lease, a pension obligation, an employee option and a convertible are each defined as instruments elsewhere; what each one does to a bridge is the subject of the four treatments above. A bridge settles claims and says nothing about whether a figure is cheap or dear. The five lines turn an enterprise value into an equity value, and both ends of that walk rest on assumptions somebody has to defend.
The bridge reads as a sentence about who is owed. See whether equity holds.

Sources

SourceDocumentSite
Aswath DamodaranValuation material on the treatment of cash, cross holdings and other non-operating assets in moving from the value of a business to the value of its equity, and on the debt-like items that belong in that walkpages.stern.nyu.edu
Koller, Goedhart and WesselsValuation, for the frame in which the value of operations, the value of non-operating assets and the claims against both are set out as one reconciliation, and for the treatment of leases and pension shortfalls as debt equivalentsWiley
Securities and Exchange Board of IndiaThe authority whose framework governs what a listed company in India discloses, and therefore what raw material a bridge can be built fromsebi.gov.in
Ministry of Corporate AffairsThe authority with which company filings, charges and shareholding are recorded in India, which is where the holding behind a minority interest line becomes visiblemca.gov.in
Reserve Bank of IndiaThe authority engaged wherever a lender or a cross-border flow is involvedrbi.org.in
Social Science Research NetworkA repository where working paper versions of academic work on valuation are held, for a reader who would rather read an original than a summary of itssrn.com

Sankalp Industrial Systems Limited, Sankalp Coatings Private Limited and Aruna Tooling Private Limited are invented.
Educational material. Not advice on any investment, tax, budget or market position.

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