How to Estimate a Cost of Capital When There Is No Share Price

The rate is built bottom up. An asset beta is taken from listed peers and relevered on a target capital structure, a premium is added to a base rate for the cost of equity, the debt is priced on what this borrower would pay today, and the two are weighted at target rather than at book. For Sankalp Coatings Private Limited, invented, that gives exactly 11.25 per cent.

Think about how a bank manager prices a loan to a tailoring workshop on a street where no tailoring workshop has ever borrowed before. There is no history to look at, so she does not look for one. She looks instead at the four workshops two streets over that did borrow, asks what each of them was charged, notices that two of them had premises to pledge and one did not, adjusts for that, and arrives at a number she can defend in a sentence. Nobody measured the tailoring workshop. Somebody measured things that resemble it, and then reasoned across the gap.

The bank manager's reasoning is the whole of the procedure below, moved into a spreadsheet and given decimal places. A business that does not trade has no price, so it has no measured risk of its own, so the rate at which its cash flows are discounted cannot be read off anything at all. The rate has to be built. An estimate for a business with no share price is never a measurement; it is a chain of stated judgements, and what the procedure produces is a rate together with the paragraph that defends it.

What is actually being estimated here, and why is a formula not enough?

The formula for a weighted average cost of capital has not been in dispute for decades and takes four lines to write down. Multiply the cost of equity by the share of the capital that is equity, multiply the after-tax cost of debt by the share that is debt, add the two, stop. None of that arithmetic is in dispute. How the blend is built, and what each input means, are set out separately.

Every hard question lives one level down, in where the inputs come from. And for a business that does not trade, two of the inputs the formula asks for do not exist at all. There is no share price, so there is no market value of equity to weight with. There is no series of returns, so there is nothing to measure a beta against. The formula asks for two numbers the world has declined to supply, and it does not care how they are obtained.

The procedure below exists to replace both with something defensible, and the defence always has the same shape. Find businesses whose risk is genuinely close to this one, take what the market prices for them, then adjust for the ways this business differs. Every replacement input is a judgement, so every step of this procedure ends in a decision somebody has to write down and sign, rather than in a formula that produces itself.

A signed decision at every step changes what a good answer looks like. A rate presented on its own is not an estimate. A rate on its own is a number. An estimate is that number plus six sentences. The six sentences say which entity the rate belongs to, which currency it is in, which peers produced the asset beta, which capital structure it was relevered on, what the borrower would pay to borrow today, and when the whole thing was last rebuilt. Take those six sentences away and nobody downstream can tell whether the rate is right for the thing they are about to use it on.

Step 1: whose cost of capital is this, and in what currency?

The business being estimated here is Sankalp Coatings Private Limited, the subsidiary in which Sankalp Industrial Systems Limited, also invented, holds 75.0 per cent. The subsidiary makes industrial coatings and is unlistedNot traded on an exchange, so there is no observable share price and no series of returns to work from., so nobody quotes a price for it on any morning, and the quarter of it the group does not hold sits in the group's own bridge at Rs 60,00,00,000 rather than at anything a market said.

Two questions have to be answered before a single input is chosen, and both of them are about matching rather than about measuring. Whose cash flows are being discounted, and what are those cash flows counted in?

The answer to the first is the coatings business's own cash flows, not the group's. The answer to the second is rupees, running for five explicit years and then into perpetuity. Put together, those two answers say the rate must be a rupee rate built for a long horizon. Everything chosen after this point has to be consistent with that one sentence, and an error made here cannot be repaired by any amount of care later on.

The reason is a piece of plumbing worth understanding once. A rate silently carries an inflation expectation inside it, and so a rate and a cash flow have to be denominated the same way. Currency matchingUsing a discount rate denominated in the same currency as the cash flows it is applied to, so that both carry the same inflation expectation. is not a nicety. Discounting a rupee cash flow at a rate built on another country's base rate subtracts one inflation expectation from a stream that was built with a different one. Nobody can see this in the output. The valuation looks entirely normal and is wrong in a fixed direction.

The horizon matches for the same reason. A short rate is a price for a short wait, and this is not a short wait. A stream that runs into perpetuity is therefore discounted with a rate built on a long-dated base rate rather than a short one. On this worked example that long base rate is taken as 7.75 per cent, and that figure is this worked example's assumption rather than any current market level. The published source for a current one is named below.

Why does the entity matter, if it is all one group anyway?

One sentence causes more wrong valuations than any other in this subject area: the group already has a cost of capital, and the group's rate will serve. Sankalp Industrial Systems Limited, invented, has a weighted average cost of capital of exactly 12.00 per cent. The group's own build is covered separately, and the figure is cited here for the comparison. The observed weights behind it are equity of Rs 18,00,00,00,000 against gross debt of Rs 6,00,00,00,000, and it is a listed company, so those weights were read off a market rather than chosen.

Now ask what that 12.00 per cent actually describes. The 12.00 per cent describes a blend of three industrial businesses, valves and precision castings and an aftermarket parts and service operation, funded on the group's own terms and priced by a market that can see all three at once. The blend does not describe a coatings business, and it never did. The rate belongs to the risk of the cash flows being discounted, not to the entity that happens to hold them, and a group rate is an average of businesses of which no single business is the average.

The household version runs the same way. A household with two earners has an average income. Neither earner is paid the average. The amount one earner can borrow against is a question about that one earner, and the household average settles almost nothing except that the truth is somewhere near it. A group rate is exactly that average, and a subsidiary is exactly that one earner.

On these locked assumptions the coatings business comes out at 11.25 per cent, seventy five basis points below the group. The difference is not a rounding and it does not run in a random direction. The difference runs against the safer business every time, and that is what makes the coatings business worth a figure of its own.

Step 2: where does a beta come from when nothing trades?

A beta is normally measured. A company's equity returns are set against a market's returns, and the measurement records how much the first moves when the second does. Where that measurement is done, and what it does and does not capture, is covered separately. The measurement needs a return series, and Sankalp Coatings Private Limited does not have one. There is nothing to regress. There never will be, unless somebody lists it.

So the beta is borrowed instead of measured. The technique has a name: a bottom-up betaA beta built up from the observed betas of comparable listed companies rather than measured on the company's own returns., built from listed companies doing something close enough to the same thing. A bottom-up beta is not a shortcut and it is not second best. For most businesses in the world it is the only route there is, and it has one real advantage over a measured beta: it draws on several companies rather than one noisy return series.

The step has a trap in the middle of it, and almost everybody walks into it once. A peer's published beta is a levered beta. A company with more debt has more volatile equity returns whatever its underlying business does, and a published beta carries that peer's own borrowing inside it. Taking a peer's published figure straight off a screen and applying it here imports somebody else's balance sheet along with their business risk. Every peer beta is stripped of its own borrowing first, producing an asset betaThe beta of the underlying business with the effect of borrowing stripped out, so that only business risk remains., and only the stripped figures are ever compared with each other.

Strip each peer using that peer's own capital structure and that peer's own tax rate, not this entity's. The choice of whose figures to unlever with matters, and is routinely fumbled. The unlevering is undoing what each peer's own borrowing did to its own beta, so it has to be done with each peer's own numbers. Only after every peer has been reduced to its underlying business risk are the figures on comparable footing.

Then take the middle one. The medianThe middle value of a set once the values are lined up in order, rather than the arithmetic average. is preferred to the average here for a reason that is easy to demonstrate anywhere in valuation: a single unusual peer moves an average and does not move a middle. A peer set of six may hold one company far faster growing and far less indebted than the rest. The odd company pulls the average away from anything typical and leaves the median exactly where it was. Since a peer set is chosen by judgement and there is always one company somebody argued about, the measure that shrugs at that company is the safer one.

For the invented coatings peer set the median asset beta is 0.80. The group's own asset beta, on its valve and castings peer set, is 1.00. Coatings is the steadier of the two businesses on these assumptions, and the whole point of doing the work separately is that the number is allowed to come out lower.

What makes a company a comparable, as opposed to merely similar

A comparableA company whose business risk is close enough to the one being valued that the market's pricing of it is informative here. is not a company in the same industry classification. Classification codes were built for statistics, not for risk. The question a peer set has to answer is narrower: does the same thing move this company's cash flows that moves ours? Three checks do most of the work, and each of them is answerable from a filed set of accounts rather than from an opinion.

The last row catches people and deserves a second look. Peers are not screened on leverage. The unlevering removes leverage already, so a heavily indebted peer in the right business is a perfectly good comparable once it has been unlevered, and a debt-free peer in the wrong business is not.

How is a peer set handled when its companies report in different currencies?

Coatings businesses of the right size do not all sit in one country, so a serious peer set usually spans markets. Three peers report in one currency, three in another, each pays tax at its own rate and each carries its own borrowing. Which figures, exactly, get averaged?

The rule is short, and it is the single most useful rule in the procedure for anybody who has to do the work in practice. Every peer is unlevered where it stands, in its own setting, using its own tax rate and its own capital structure. The median of the asset betas is taken. The relevering then happens once, and the cost of equity is built once, in the currency of the cash flows being valued. A beta carries no currency inside it and a rate does, so unlever in many places and relever in exactly one.

The reasoning is worth holding on to. A beta is a sensitivity, a statement about how much one thing moves when another does. A sensitivity is a pure number with no units attached, so a median taken across four markets is a meaningful object. A cost of equity is not a pure number. A cost of equity is a rate, and every rate carries an inflation expectation folded inside it. The same business in two economies genuinely does have two different costs of equity, both correct. Average two of those and the result belongs to no economy at all. The average is not a compromise between two answers. The average is an answer to no question.

Step 3: what capital structure carries the weights, when nothing is observed?

For a listed company this step is almost free. The market capitalisation is read, the debt is read, and one is divided by the other. For Sankalp Industrial Systems Limited that gives the 75.0 and 25.0 per cent weights already mentioned, and nobody had to argue about them because a market produced them.

For the coatings business there is nothing to read. Its equity has no price, so it has no market value, so there is no observed mix at all. In its place goes a target capital structureThe mix of debt and equity a business is assumed to fund itself with over the long run, rather than the mix it happens to carry today.: the funding mix this business is assumed to sustain over the long run. On these assumptions that target is 25.0 per cent debt and 75.0 per cent equity. Debt to equity is then exactly one third.

A target is a choice, and the only thing that separates an estimate from an assertion is that the choice is written down with the reasoning beside it. Three kinds of evidence carry that reasoning, and a good estimate names which of them it leaned on. The listed comparables carry structures of their own, and those can be looked at rather than assumed. A lender would advance a fairly narrow range against this asset base and this cash flow. And for a subsidiary, what the owner intends to fund the business with is a real and answerable question rather than a guess.

Something slightly awkward sits underneath this step. Even for a listed company the weights carry a circularity: the market value of equity is both an input to the rate and, in a sense, an output of the valuation. For an unlisted business that circularity is not resolved so much as replaced, by a target that was chosen instead of observed. The substitution is a genuine weakness of the method and is better stated than hidden. The weakness is tolerable because the answer is not very sensitive to the target, and dangerous because an undefended target is invisible to everybody downstream.

Now releverTo put a chosen capital structure's financial risk back onto an asset beta, so the beta once again describes an equity holding rather than the underlying business. the asset beta on that target. The arithmetic of levering and unlevering is covered separately and is used here in a single line: one plus 0.75 times a debt to equity ratio of one third is 1.25, so the multiplier is 1.25, and 0.80 times 1.25 is a levered beta of exactly 1.00. The reasoning underneath it, that the underlying business risk and the financing risk can be separated and then recombined, is Modigliani and Miller's, in The Cost of Capital, Corporation Finance and the Theory of Investment, American Economic Review, 1958.

The two figures that follow look like a coincidence and are not. The coatings business is a demonstrably less risky business than the group, with an asset beta of 0.80 against 1.00. Relevered on a 25.0 per cent target it arrives at a levered beta of exactly 1.00. The group's business started at that same number before relevering. Two different quantities, the same numeral, and confusing them is one of the easier mistakes to make in a line of working. An asset beta and a levered beta are different objects and the label matters more than the digits.

And if the target moved? A 30.0 per cent debt target puts debt to equity at three sevenths, so the multiplier becomes 1.32143 and the levered beta becomes 1.05714, taking the cost of equity to 13.04 per cent. A different structure would also carry a different borrowing cost, and what this business would pay to borrow at a 30.0 per cent target is not something this worked example fixes. No blended rate follows at that target. Inventing it to complete a table would be exactly the kind of plausible wrong number the whole procedure exists to avoid.

Step 4: how is the cost of equity built once the beta is settled?

The route from a base rate, a premium and a beta to a required return on equity is the capital asset pricing model, and it is Sharpe's, from Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk, Journal of Finance, 1964. The model asserts, in one sentence, that the extra return an equity holder requires above a base rate is the market's premium scaled by how much this particular equity moves with the market.

Three numbers go in. A base rate of 7.75 per cent, standing in for the yield on a long-dated government security. A mature market equity risk premium of 3.50 per cent. A country risk premium of 1.50 per cent, giving a total premium of 5.00 per cent. All three of those figures are this worked example's assumptions and not one of them is a current market level, a current premium, a current rating or a current spread.

Assumed figures are not a formality. A base rate and a premium move continuously. Any figure written down ages the moment it is written. A stale number stated confidently is worse than no number at all, and somebody will always use it. A current base rate, a current mature market premium and a current country premium are estimated and published by Aswath Damodaran, at pages.stern.nyu.edu.

Put the three together with the levered beta of 1.00. Seven point seven five plus one times five is exactly 12.75 per cent. Because the levered beta here happens to be exactly 1.00, both premium components arrive at full size. The build is unusually easy to read: 7.75 of base, 3.50 of mature market premium, 1.50 of country premium, and nothing left over.

Step 5: what does the debt cost, and which debt is being priced?

Step 5 is the one people rush, and it has a specific wrong answer that is very easy to reach. The wrong answer is the coupon printed on the borrowings already sitting on the balance sheet. The printed coupon is a historical fact. The coupon records the price of money borrowed at some point in the past, on terms agreed under conditions that have since moved.

A cost of capital is not asking about that money. A cost of capital asks what the next rupee costs. The figure wanted is the marginal cost of debtWhat this particular borrower would have to pay to borrow the next rupee today, as opposed to what it is paying on money borrowed earlier.: the rate this borrower would be offered today, for debt of the kind and the term it would actually raise.

And it is this borrower, not the group. Sankalp Coatings Private Limited is a smaller entity with a narrower asset base and a weaker covenant package than its parent. The subsidiary does not borrow on the parent's terms unless the parent stands behind the borrowing, and on these assumptions the parent does not. Its own pre-tax cost of debt is taken as 9.00 per cent, against the group's blended 8.00 per cent. The subsidiary is the safer business and the worse borrower at the same time, and both facts are true because they are answers to different questions.

Where does a 9.00 per cent come from when nothing is quoted? From the same kind of evidence the bank manager used on the tailoring workshop. The first is what this entity has actually been offered recently, if anything. The second is what lenders are advancing to comparable borrowers of this size against this kind of security. The third is the increment over the parent's own cost that a lender would attach to the weaker covenant, and whether somebody can defend that increment in a sentence. Every one of those is a source that can be named and dated in the paragraph that travels with the rate.

Then take it after tax. Interest is deducted before the tax charge is computed, so the company bears only part of it. At the company's own assumed effective tax rateThe company's own assumed tax charge as a share of profit before tax, used here as a stated assumption rather than as any statutory rate. of 25.0 per cent, 9.00 times 0.75 is 6.75 per cent. The 25.0 per cent is this company's own assumed rate rather than any statutory rate.

Step 6: how do the two prices become one rate?

The last step is the arithmetic everybody already knows, and it is deliberately the shortest part of the procedure. Weight the cost of equity by the equity share of the target, weight the after-tax cost of debt by the debt share, add them.

Nine point five six two five plus one point six eight seven five is 11.25, exactly, with nothing hiding in a fourth decimal. The clean figure is a property of the chosen inputs rather than of the world, and it is convenient rather than meaningful.

Which brings up the last decision in the procedure, and it is a decision about how to write the answer rather than how to compute it. The arithmetic will produce 11.250000 per cent if asked to. Every digit printed is a claim about how well the thing is known, so state the answer to the precision the inputs carry rather than to the precision the division produces. The beta is a middle value taken from a handful of chosen comparables. The premium is contested by more than a point depending on how it is estimated. The target structure was adopted rather than observed. Two decimal places on top of that is already generous, and the honest presentation is a rate with a stated range around it and a sentence saying what the range came from.

What do the two hardest judgements actually cost, in basis points?

An estimate built from six judgements invites an obvious question: which of them is worth arguing about? The answer is not obvious from the outside, and running two side checks on this worked instance settles it in a way no amount of discussion will.

Side check one: what the weaker covenant is worth

Suppose the coatings business could borrow on its parent's terms after all, at 8.00 per cent rather than 9.00. After the assumed 25.0 per cent tax rate that is 6.00 per cent rather than 6.75, and the blend becomes 0.75 times 12.75 plus 0.25 times 6.00, which is 9.5625 plus 1.5000, being 11.0625 per cent. Against the actual 11.25 per cent, a whole extra percentage point of borrowing cost turned out to be worth 18.75 basis points of the answer.

The arithmetic behind that generalises, and is worth carrying around. One point of pre-tax borrowing cost becomes 0.75 of a point after the tax deduction, and only a quarter of the capital is debt, so 0.75 times 0.25 is 0.1875 of a point. A whole extra point on the borrowing rate moves the blended answer by less than a fifth of a point, and anybody about to spend a week negotiating over that point should know as much first. Step 5 is not a rounding step and it should still be done properly, but it is not where the estimate lives.

Side check two: what the lower business risk is worth

Now hold the borrowing cost at this entity's own 9.00 per cent and take away the lower business risk that made the coatings business different. Suppose its peer set had produced an asset beta of 1.00, exactly the same business risk as the group. The levered beta becomes 1.25, the cost of equity becomes 7.75 plus 1.25 times 5.00, being 14.00 per cent, and the blend becomes 0.75 times 14.00 plus 1.6875, or 12.1875 per cent.

Look at where that lands. Twelve point one nine per cent is above the group's own 12.00 per cent, on identical business risk. Nothing has gone wrong. The subsidiary borrows worse, so with the same business risk it must end up costlier. Two entities with the same business risk and different funding terms get different rates, and a procedure that returned the same number for both would be broken rather than consistent.

The two side checks together give the ranking. Move the borrowing cost by a whole point and the answer moves 18.75 basis points. Move the asset beta by 0.20 and the answer moves 93.75 basis points, five times as much. The judgement that carries this estimate is the one made in step 2, on a number nobody measured on this company at all.

How often should a rate like this be rebuilt?

Every input in the procedure moves. Base rates move daily. Premiums are re-estimated continuously by the people who publish them. Peer betas move as peers report and as their own borrowing changes. Somebody who wanted to could rebuild this rate every morning and would get a slightly different number every morning.

Rebuilding every morning would also destroy the usefulness of the work, and that is the least intuitive rule in the subject. The difference between two valuations built at two different rates mixes a change in the business with a change in the analyst's own inputs, so a rate rebuilt every quarter makes two valuations of the same business impossible to compare. Value the coatings unit in one period at one rate and in the next period at a rate built on different inputs, and the movement between the two answers cannot be attributed to anything. Did the business get better, or did the base rate drift? The output no longer answers the question it was built to answer.

The shopkeeper version is quicker. A man who changes his prices every hour cannot tell whether Tuesday was slow because of his price or because it rained. He has removed his own ability to learn anything from his own shop. Holding one thing still is not laziness; it is the only way a comparison means anything.

So the discipline has three parts and all three are about writing things down rather than about calculating. Fix the rate for a stated period and say what that period is. State the date and the full set of inputs beside every valuation that uses it, so anybody comparing two valuations can see immediately whether they were built on the same rate. And rebuild on a schedule, or on a named event, rather than continuously.

A named event has to be defined narrowly, or everything counts as one. A change in the target capital structure. A change in what the business actually does, such as a purchase or a disposal that moves its business risk. A material change in what this borrower would be offered today. The stated period ending. Notice what is not on that list: an input moving. Inputs move all the time, and that is precisely why moving with them is not a policy.

What has to travel with the rate when it is handed over?

An estimate that leaves the analyst's desk as a single figure will be used by somebody who does not know how it was made, on a business the analyst did not have in mind, in a period when half the inputs have moved. Everything in the procedure that made the number defensible is invisible to that person unless it is written next to it.

Six lines do it, and they fit in a paragraph. Anybody who has all six can rebuild the rate; anybody who is missing one of them is guessing.

Notice how much of that is provenance rather than arithmetic. The arithmetic in this whole procedure is four multiplications and one addition; everything else that makes the answer usable is a record of where each input came from. That is not a weakness of the method. Provenance is what an estimate is, in any discipline whose subject cannot be measured directly.

One last discipline goes in the same paragraph. Carry the answer as 11.25 per cent with a range around it and a sentence saying what produced the range, rather than as a figure with four decimals. The figure that moves the range most is the asset beta from step 2. Every 0.05 of asset beta is worth 23.44 basis points, and the peer set that produced it was chosen by somebody who could reasonably have chosen a slightly different one.