Levered and Unlevered Beta: Stripping Out Capital Structure

An unlevered beta measures the risk of the business on its own. A levered beta measures the risk of the shares in that business, and it is higher because borrowing lands the same business variation on a smaller equity base. For Sankalp Industrial Systems Limited, an invented manufacturer, an unlevered beta of 1.00 becomes a levered beta of exactly 1.25 once the funding is put back on at debt to equity of one third.

All of it rests on one dull fact about a lender: the interest is fixed, and it is paid first. Whatever the year does to the business, the lender takes the same amount. So every rupee by which the results move lands on the shareholders and on nobody else. Borrowing does not make the business riskier. Borrowing makes the shares riskier, by concentrating an unchanged quantity of variation onto a smaller base. Concentration is the whole content of the relation that follows, and it is why the adjustment turns on the ratio of borrowing to equity rather than on the size of the borrowing. The step rests on a proposition Modigliani and Miller put into the American Economic Review in 1958: what the business itself is worth does not depend on how it was funded. The tax deduction on interest is added to that afterwards.

Levered vs Unlevered Beta: two measurements of two different objects

The mistake that trips people is not a matter of precision. The two betas are not a rough beta and a careful beta, or an old beta and a fresh one. The two betas measure two different objects, and a number carrying no label saying which object it belongs to has said almost nothing.

An unlevered beta is about the business. Valves get made, castings get poured, aftermarket parts get shipped, and how much that trade wobbles when the wider economy wobbles is a fact about the trade. The wobble has nothing to do with who paid for the machines. A levered beta is about the shares. A levered beta carries the same business wobble plus whatever the funding arrangement does to it, and the funding arrangement is a choice a board made on a Tuesday.

All four names are in circulation, and they are only two ideas. The unlevered beta describes the assets doing the work, so it is also called the asset beta. The levered beta describes the shares, so it is also called the equity beta. Nothing changes when the name changes.

Notice what did not change between the two panels. The total is Rs 24,00,00,00,000 in both. The same valves are made in both. The customers are the same, the plants are the same, the order book is the same. Only the labelling of who has a claim on the output moved, and the beta moved with it.

Why does borrowing change the beta when it changes nothing about the business?

Everyone has lived the household version, so start there. Two households earn the same salary, and the salary varies a little with the bonus. One has no home loan. The other pays a fixed instalment on the first of every month, and that instalment does not shrink in a thin month or grow in a fat one. Ask each household how much its spending money moves from month to month. The salaries move by the same rupees. The spending money does not. In the second household the same rupees of variation land on a smaller amount of money left over. Nothing about either job got riskier. The claim sitting in front of the money did.

Financial risk is exactly that concentration, and it is entirely separate from business risk. Business risk is the wobble the trade produces on its own. Financial risk is what a fixed prior claim does to whatever is left after it. A beta measured on the shares picks up both and cannot separate the two. The stripping-out step exists for exactly that reason.

Put it on the case. Sankalp Industrial Systems Limited earned an operating profit of Rs 2,40,00,00,000 in the last completed year and paid Rs 48,00,00,000 of interest on its gross debtThe whole of what is owed to lenders, counted before any cash the company happens to be sitting on is netted off against it. of Rs 6,00,00,00,000. Now imagine two versions of the following year, one better and one worse, with the operating profit Rs 30,00,00,000 either side of where it started. Which of the two years arrives is unknowable in advance, and the relation does not need to know: it turns only on where the swing lands.

How far that concentration goes, measured as the amplification of the return on equity, is worked out separately. Only the direction and the reason matter at this step: fixed first claim, unchanged variation, smaller base, higher sensitivity in the shares.

What does the relation actually say, line by line?

One expression carries everything above. Read it once, then read the sentence underneath it. The sentence says the same thing without notation.

Every term inside the bracket is a fact about how the company is funded, and none of the three says anything whatever about the business. The separation is what makes the relation useful. A beta measured on one balance sheet can be moved to another, and that is needed far more often than might be guessed.

Two readings are worth pinning down before the arithmetic. First, at a ratio of zero the bracket collapses to one, so an unborrowed company's two betas are the same number and the distinction between them stops mattering. Second, the bracket is a straight line in the ratio, with a slope of one less the tax charge. Nothing curves, nothing accelerates, and there is no level at which the relation itself does something surprising.

Why a ratio, and not the amount borrowed?

People expect the size of the borrowing to matter, and it does not. Rs 6,00,00,00,000 sounds like a great deal of money, and against a Rs 18,00,00,00,000 equity base it produces a lift of exactly one quarter. The same Rs 6,00,00,00,000 against a Rs 60,00,00,00,000 equity base would barely register. Shareholders feel not the rupee bill but how much fixed claim stands in front of each rupee of their own money.

The ratio is why a large borrower and a small one can share a levered beta, and why growing both sides of the balance sheet together leaves the beta exactly where it was. A company that doubles in size and doubles its borrowing has changed two large numbers and changed nothing at all about the quotient between them.

The everyday version: a vendor who rents a cart for a fixed daily charge and takes Rs 900 of sales a day feels that rent very differently from a vendor paying the same rent on Rs 9,000 of sales. Same rent, different exposure, and the rent alone says nothing until the sales it sits against are known.

Why is there a tax term in it?

Interest is deducted before tax is worked out. The deduction does something specific to the argument above: the fixed charge sitting in front of the shareholders is not the full interest bill. Part of it is borne by the tax authority in the form of tax not paid.

A rupee of interest costs the shareholders only seventy five paise once the smaller tax bill is counted. This company assumes an effective tax rateThe tax charge a company actually bears, put as a share of its profit before tax. It is that company's own figure and never a published rate. of 25.0 per cent on its profit. So the fixed charge concentrates only three quarters as hard as it looks, and the bracket carries a 0.75 rather than a 1.00. Drop the term altogether and at a ratio of one third the bracket would read 1.3333 instead of 1.25, and the levered beta would come out at 1.3333.

The gap of 0.0833 of a beta sounds small and is not. The gap separates a figure saying the shares carry a quarter more risk than the business from a figure saying they carry a third more.

The worked relevering: 1.00 becomes exactly 1.25

Every input below is read off the record for Sankalp Industrial Systems Limited.

The bracket works out in three steps, none of which needs to be taken on trust. One less 0.25 is 0.75. Multiplying 0.75 by one third lands on 0.25. Adding one lands on 1.25. Multiplying the unlevered 1.00 by 1.25 gives a levered beta of 1.25 exactly, with no rounding anywhere in the chain.

One multiplication is worth more than it looks beyond this step. On a combined equity risk premiumWhatever holders of shares are assumed to want each year over and above what lending to a government would pay them. of 5.00 per cent, a rate this example assumes rather than reads off any market, the relevering adds 0.25 times 5.00 to the cost of equity. The addition is 1.25 percentage points. The figure shares its digits with the levered beta and is a completely different quantity. The cost of equity is 14.00 per cent rather than 12.75 because of that single step.

Unlevering a peer: the same relation run backwards

Now the reason any of this is practical. Suppose a beta is needed for a business that has no share price, or the beta produced by the company's own shares is not trusted. The place to look is similar listed businesses. Every beta available for them was measured on their shares, and their shares sit on their borrowings, so every one of those betas is a levered beta.

Used as it stands, a published beta carries that peer's funding arrangement into the valuation under the other company's name, with nothing about the number looking any different for it. The danger is exactly that: the arithmetic runs, the answer is plausible, and there is no line in the working to point at.

So the relation runs the other way. Dividing the published beta by one plus one less that company's tax charge times that company's own debt to equity ratio leaves business risk with the funding effect taken out. The borrowing sitting inside the number belongs to the company whose beta is being stripped, and so does every term in the reverse step.

The full procedure for a set of comparison companies

Three steps, in this order, and every failure in this area is a step skipped or a step done out of turn.

Two details in that picture earn their place. Take the median rather than the average. A set of six with one unusual member gets dragged badly by an average and not at all by a middle reading. And relever once, at the end. Stripping six balance sheets out arrives at business risk that no longer carries anybody's funding, ready to have exactly one funding arrangement put back on.

One observation from this peer setThe short list of comparison companies picked because they earn their money in a broadly similar way. turns on borrowing alone. The sixth company, Nallamala Components Limited, an invented parts maker, carries net cash of 0.4 times its earnings before interest, tax, depreciation and amortisation (EBITDA) rather than net borrowing. The second, Satpura Engineering Works Limited, also invented, carries net borrowing of 2.2 times. Whatever their observed equity betas turn out to be, the stripping step moves the two in opposite directions: the sixth company's asset beta will sit close to its equity beta, and the second company's will sit materially below its own. Each company's borrowing is on the record and its observed equity beta is not, so the direction of each move can be stated and the size of it cannot.

The static reading, for anyone who never touches the control: at the default setting the ratio is one third, being Rs 6,00,00,00,000 over Rs 18,00,00,00,000, the group's levered beta is exactly 1.25, the cost of equity is exactly 14.00 per cent, and the subsidiary's levered beta is exactly 1.00. The four figures at the default setting restate the worked example.

The cost of equity column shows two decimal places, and every figure standing in it is worked out on the unrounded beta beside it rather than on anything already rounded. The column runs off a base rate of 7.75 per cent and an equity risk premium of 5.00 per cent in total, both assumed in this example and neither read off any market.

Should the ratio be taken at book value or at market value?

At market. Always at market. Careful people go wrong on this point more often than on anything else in the procedure.

Sankalp Industrial Systems Limited has a book value of equityThe shareholders' side of the balance sheet, built up from what was paid in plus what has been kept back since. of Rs 9,00,00,00,000, being Rs 45.00 a share on 20,00,00,000 shares. Its market capitalisationWhat the whole of a listed company's shares would fetch at today's quoted price, being the price multiplied by the number of shares issued. is Rs 18,00,00,00,000, being Rs 90.00 a share. Same company, same day, two equity figures, and the borrowing of Rs 6,00,00,00,000 is the same in both.

The ratio is asking one question: how much of the money at risk in this business today is borrowed. Book equity records what shareholders paid in and what has been kept back since. Book equity is a careful historical record, and a historical record cannot answer a present-tense question. Market capitalisation answers it, being what the equity claim is actually worth right now.

There is a second error in this area, smaller and worth its own paragraph. Unlevering a comparison company on the VALUING company's tax charge and ratio rather than on its own. The reverse step is removing the borrowing that is actually sitting inside the observed number, and the observed number belongs to the comparison company, so every term in the reverse step has to be that company's own. The valuing company's figures belong to the third step and nowhere else.

The same business risk at two different capital structures

The group holds 75.0 per cent of an unlisted subsidiary, Sankalp Coatings Private Limited, an invented coatings arm. Its asset beta is 0.80, taken as the median of an invented coatings comparison set, and it is lower than the group's 1.00 because coatings is a steadier trade than valves and castings. Its target borrowing is 25.0 per cent of its capital, the same one third debt to equity ratio the group sits at.

So the bracket is the same. The bracket only ever sees the tax charge and the ratio, and both are identical, so one plus 0.75 times one third is 1.25 for the subsidiary exactly as it was for the group. The multiplier is shared and only the business risk being multiplied differs: 0.80 times 1.25 is exactly 1.00.

The coincidence makes one point better than any warning could. A number is not a beta until somebody says which kind of beta it is. A 1.00 written down with no label leaves a reader no way to know whether they are holding a business or a set of shares, and the two lead to answers a quarter apart. The most common consequence is a beta that gets levered twice: once by whoever published it and once by whoever used it.

What does the beta then do?

A beta is not a rate and it never becomes one on its own. A beta is a sensitivity: a pure number carrying no unit, no currency and no per cent sign. Nothing can be discounted with it, it cannot be compared to a coupon, and it does not state what a shareholder should expect to earn.

A beta becomes a rate through one further step, covered separately. The levered beta is multiplied by an equity risk premium, and a base rate is added. On this example's stated assumptions of a 7.75 per cent base rate and a 5.00 per cent total premium, 7.75 plus 1.25 times 5.00 is a cost of equityThe yearly return a shareholder is assumed to require before putting money into a particular company's shares. of 14.00 per cent. Where those two assumptions come from, and why estimates of the premium differ so much between careful people, is worked out separately.

A beta travels and a rate does not, and that is the practical use of the distinction. A beta has no currency in it, so a beta measured on one market can be discussed against a beta measured on another. Two costs of equity in different currencies cannot be compared at all without first taking apart what is inside each one.

The three assumptions the whole procedure rests on

Everything above is arithmetic, and arithmetic is easy to trust. The relation carries three assumptions that turn it back into a judgement. None of the three is written in it, and each needs naming every time the relation is used.

Assumption one is why no beta is ever assigned to the borrowing in this relation. Assumption one treats the lender's claim as something that does not move with the market at all. For investment gradeThe upper band of lender assessments, where repayment is treated as very likely and the borrowing trades close to the government curve. borrowing that is a reasonable approximation. For borrowing that has started to look doubtful it is a poor one. Doubtful borrowing begins behaving rather like equity and carries its own sensitivity, and the relation has no slot for that.

Assumption two is why the tax term is there at all. Assumption two treats the deduction as arriving as reliably as the interest goes out. A company with no taxable profit to set the interest against does not get the deduction that year, and the bracket has quietly over-corrected.

Assumption three is the one that bites most often and it is the subtlest. A beta levered on a 25.0 per cent borrowing share describes the shares in a company that has a 25.0 per cent borrowing share. If the company's actual funding drifts somewhere else, the beta is describing a company that no longer exists. On this record that drift is real and it is measurable: two different routes to the same equity value disagree by Rs 1,46,76,00,000 for exactly that reason, and why they disagree is worked out separately.

Sankalp Industrial Systems Limited, Sankalp Coatings Private Limited, Satpura Engineering Works Limited and Nallamala Components Limited are invented.
Educational material. Not advice on any investment, tax, budget or market position.