The Cost of Capital: How the Weighted Average Is Built

A cost of capital is the return the people who funded a business could get elsewhere at the same risk, blended across them by how much each one put in. For Sankalp Industrial Systems Limited, invented, equity costs 14.00 per cent, debt costs 6.00 per cent after tax, and the market weights are 75.0 and 25.0 per cent, giving exactly 12.00 per cent.

The idea is older than the arithmetic and easier to feel outside a spreadsheet, so start with a shop. An aunt has Rs 5,00,000 sitting in a deposit that pays her something every year. Her nephew wants to open a second sweet counter and asks her for the money. She hands it over. Nobody now pays her the deposit interest, and no bill arrives at the counter every month saying so. But she has given something up, and everybody in the room knows it. If the counter cannot earn her at least what the deposit was earning, plus something for the fact that a counter can fail and a deposit rather less easily does, then the money should have stayed where it was.

The aunt's forgone deposit interest is the whole idea. She had an alternative, and the counter has to beat it. The rate a business has to beat is not a payment it makes; it is the best thing its funders gave up in order to fund it. A company simply has two kinds of funder instead of one aunt, so the same sentence has to be worked out to more decimal places.

What is a cost of capital, and whose cost is it?

The cost of capitalThe return the providers of a business's money could get elsewhere at the same risk. belongs to the people who put money in, not to the company that received it. Almost every confusion about this number comes from getting the direction backwards. Read that twice. The company is where the money went. The rate is what the money would have earned had it gone somewhere else of the same riskiness. The rate is an opportunity costThe value of the best alternative given up. in the strict sense of the phrase, and it is a cost only in that sense.

Two consequences fall straight out of that, and both of them surprise readers who have spent time with financial statements. The first is that the cost of capital never appears in a profit and loss account. Not in any line, not in any note. Interest appears. Interest is actually paid to somebody. The return that a shareholder could have earned elsewhere is not paid to anybody, so no accounting system on earth records it. A company can therefore report a profit every year for a decade and still have failed to cover its cost of capital every one of those years, and its own accounts will never say so.

The second consequence is stranger. The rate belongs to the risk of the thing the money is going into, not to the entity carrying it there. If Sankalp Industrial Systems Limited, invented, put money into a project as safe as a government security, the correct rate for that project would be close to a government security's rate, and not the 12.00 per cent that describes the company's ordinary operations. One rate is administratively convenient, so companies get this wrong constantly. Convenience has a price: a single rate quietly subsidises the risky projects out of the safe ones.

Cost of Equity vs Cost of Debt: why do two providers charge different prices for the same business?

Sankalp Industrial Systems Limited, invented, has taken money from two very different kinds of person, and on the same day, about the same factory, they want different returns. Its lenders have put in Rs 6,00,00,00,000. Its shareholders have bought equity now worth Rs 18,00,00,00,000 in the market. Nothing about the valves the company makes is different for one group than for the other. The difference is entirely in what each group was promised.

The lender was promised a number. A contract states an amount and a date, and if the amount does not arrive on the date the lender has remedies that do not require anybody's agreement. The shareholder was promised nothing whatsoever. A shareholder receives whatever is left after everybody else has been paid, and in a bad year that is nothing, and nobody has broken any promise. The cost of debtThe rate the company would pay to borrow today, taken after tax in the blend. is lower than the cost of equityThe return a shareholder requires for holding this company's shares rather than something safer. because a contract is worth more than a hope, and that is true before a single tax rule is considered.

There are three separate reasons, and it is worth keeping them apart because only one of them is about tax. First, certainty of amount: the lender's return is fixed and the shareholder's is a residual. Second, ranking: if the business is wound up, lenders are paid before shareholders see anything, so the shareholder absorbs the first losses. Third, and only third, the tax deduction: interest is subtracted before a company's tax charge is worked out and a dividend is not, so a rupee of interest costs the company less than a rupee. The first two reasons make debt cheaper for the lender's sake. The third makes it cheaper for the company's sake. The three reasons are different arguments and they compound.

So the company faces two prices at once. The cash the business throws off is available to both groups, and neither has a prior claim on all of it, so the company can neither discount at both prices nor pick one. One rate has to represent both, weighted by how much of the money each group actually supplied. The blended rate is the weighted average cost of capitalThe cost of equity and the after-tax cost of debt blended by their market weights., built below for one invented company, input by numbered input.

The Cost of Equity: how do four inputs reach exactly 14.00 per cent?

Nobody sends the company an invoice for the cost of equity, so it has to be inferred. The standard route is Sharpe's capital asset pricing modelThe route from a base rate, a premium and a beta to a cost of equity., set out in Capital Asset Prices, Journal of Finance, 1964. The shape of it is simple enough to say in one sentence. Start with what a default-free government security pays. Add what shareholders as a group require for holding equities at all rather than that security. Then scale that addition up or down according to how much this particular company's equity moves when the whole market moves. Three ingredients, one addition and one multiplication.

Here are the first four inputs for Sankalp Industrial Systems Limited, invented, at Year 0. Every figure in this table is an assumption, chosen so the arithmetic comes out round; a live market would move each of them, and the answer with it. Where a current figure is needed, the estimation of every input of this kind is set out in Aswath Damodaran's valuation material at pages.stern.nyu.edu.

Inputs 5 and 6 are the two that readers find hardest, and they are hardest because the second one undoes and then redoes something the first one did. BetaHow much a company's equity return moves when the market's return moves. arrives here as an input with a stated meaning: it is how much this company's equity return moves when the market's return moves. Where a beta is measured, and what makes only part of a company's risk worth pricing at all, are covered separately; the number is taken as given here.

The unlevered betaThe beta of the business with the effect of borrowing stripped out. for Sankalp Industrial Systems Limited is 1.00, taken as the median asset beta of a numbered set of six invented peer companies. Read what that number is describing: the riskiness of making and selling industrial valves, with no borrowing anywhere in the picture. The unlevered beta is a statement about the business, not about the balance sheet.

But this company does borrow. The lenders take their fixed amount first whatever happens, so borrowing does not change how risky the valve business is; borrowing changes how much of that risk lands on the shareholders. The reasoning is Modigliani and Miller's, from The Cost of Capital, Corporation Finance and the Theory of Investment, American Economic Review, 1958, and the relevering step is their proposition turned into one multiplication. Putting the borrowing back on gives a levered betaThe beta after the company's own borrowing is put back on. of exactly 1.25. The debt to equity ratio is Rs 6,00,00,00,000 over Rs 18,00,00,00,000, which comes to one third. One plus 0.75 times one third is 1.25, and 1.00 times 1.25 is 1.25. The 0.75 is one less the 25.0 per cent effective tax rate that this company assumes for itself, invented, and it sits there because the tax deduction on interest softens what borrowing does to the shareholder.

Input 7 is now one line of arithmetic. The levered beta of 1.25 times the total premium of 5.00 per cent is 6.25. Add the base rate of 7.75 per cent and the total comes to 14.00 per cent exactly. The 14.00 per cent is the cost of equity for Sankalp Industrial Systems Limited, invented, at Year 0: the price the shareholders charge for leaving their money in this business rather than taking it somewhere else of the same riskiness.

A stack makes visible something a formula hides: how much of the finished rate came from each source. Just over half of the 14.00 per cent is the base rate, and nobody argues about that part very much. The rest is premium, and every rupee of that premium is contested.

The Cost of Debt: why does 8.00 per cent enter the blend as 6.00?

The debt branch is shorter and it is easier, for a reason worth naming out loud: the lender already wrote the price down. No part of the debt price has to be inferred from how a share price moves. Sankalp Industrial Systems Limited, invented, has three borrowings at Year 0 and each one carries a contracted rate, so the only real question is how to turn three rates into one.

The three rates are blended by amount, in exactly the way a household would find the average rate on two loans. If a household owes Rs 20,00,000 on a home loan and Rs 2,00,000 on a two-wheeler loan, the average rate it pays is not halfway between the two rates. Almost all the money sits in the home loan, so the average sits much closer to the home loan's rate. The same weighting applies here, and it is why the largest tranche pulls the blend towards its own coupon.

The rate-times-amount column is in units of rupees crore times percentage points, a bookkeeping device rather than a meaningful quantity; the column exists so the division at the bottom is visible. Four thousand eight hundred over six hundred is exactly 8.00. Input 8 is therefore a pre-tax cost of debt of 8.00 per cent for this invented company, and each of the three contracted rates behind it is an assumption of this example rather than a market quotation.

Now the step that gets skipped. Interest is deducted before a company's tax charge is worked out, so a rupee of interest does not cost the company a full rupee. At this company's own assumed effective rate of 25.0 per cent, invented, a rupee of interest reduces the tax bill by a quarter of a rupee, so the company genuinely bears three quarters of it. Eight times 0.75 is 6.00. Input 9 is an after-tax cost of debt of exactly 6.00 per cent, and this is the only place in the entire model where that tax deduction is allowed to appear.

The restriction to one place matters more than it looks. The free cash flow being discounted was already computed after a full tax charge on operating profit, with no credit given for interest. The deduction on interest is therefore not inside the cash flow, and that is precisely why it has to be inside the rate. Taken in both places, one tax saving is counted twice; taken in neither, a real saving is thrown away. One place, and this is the place.

Which weights go into the average, and why not the balance sheet?

Two prices are settled. How much of each one to count is the remaining question, and the answer is each group's share of the money currently at risk in the business. The share is measured at what each claim is worth today, not at what it was recorded at when it was raised. The proportions are the market weightsThe proportions of equity and debt measured at what each is worth today, not at book value., and they are input 10.

Why market rather than book? Ask what question the weights are answering. The weights are asking: of the money that is exposed to this business right now, what proportion belongs to each group of funders? The question is about today. Book equity answers a different question entirely, and answers it perfectly well: it records what shareholders originally paid in plus what has been retained since. Book equity is a record of history and the weights need a measure of the present. The two therefore disagree, and only one of them is the right input.

Notice the asymmetry. The asymmetry is the part readers miss. Debt is largely immune to this problem: an ordinary term loan of Rs 3,00,00,00,000 is worth roughly Rs 3,00,00,00,000 to the lender unless something dramatic has happened. Equity is not immune at all. Sankalp Industrial Systems Limited's book equity is Rs 9,00,00,00,000 and its market capitalisation is Rs 18,00,00,00,000. So the error, when somebody makes it, is almost entirely an equity error, and it runs in one direction whenever equity trades above book. A one-directional error is the dangerous kind, and the failure block below prices it.

The Weighted Average Cost of Capital: how do the two branches meet at 12.00 per cent?

Input 11 is the shortest step of all. Each price is multiplied by its weight and the two products added. Three quarters of 14.00 per cent is 10.50. One quarter of 6.00 per cent is 1.50. Ten and a half plus one and a half is twelve. The weighted average cost of capital for Sankalp Industrial Systems Limited, invented, at Year 0 is exactly 12.00 per cent, and that single number is what every cash flow this business produces gets divided by.

Now sit with the answer for a moment. Something in it catches almost everybody. The two prices are 14.00 and 6.00. The number halfway between them is 10.00. The answer is 12.00, nowhere near halfway. The gap of two full points above the midpoint has one cause: three quarters of the money is equity, so the expensive price dominates the average.

There is a neat way to hold this. The blended rate always sits along the line between the two prices, and the fraction of the way it sits is exactly the equity weight. The shareholders supplied 75.0 per cent, so the rate sits 75.0 per cent of the way from 6.00 up to 14.00. If the company were half funded by each, it would sit halfway, at 10.00. The weights are not a detail applied at the end; they are the whole of where the answer lands.

CAPM vs WACC: what does each one actually produce?

The two get confused constantly, usually by people who are otherwise doing everything right, so the difference is worth stating in one line before anything else. Sharpe's capital asset pricing model produces the price of equity and stops there; the weighted average cost of capital takes that price as one of two inputs and blends it with the price of debt. The model and the average are not two ways of doing the same job. One feeds the other.

On this company, the model produced 14.00 per cent and went no further. The model knows nothing about the three debt tranches, nothing about the tax deduction, nothing about the Rs 24,00,00,00,000 of total capital. The model answered exactly one question: what a shareholder requires. The weighted average then took that 14.00 per cent, put it beside 6.00 per cent, weighted the two at 75.0 and 25.0 per cent, and produced 12.00 per cent.

The practical consequence of confusing them is specific and expensive. The cash flow this model discounts is free cash flow to the firm: the cash available to lenders and shareholders together, before any interest is paid. Discounting that at 14.00 per cent, the cost of equity, divides everybody's cash by the shareholders' required return alone. On this company's own stream that swaps 12.00 for 14.00 in the denominator. Value is very sensitive to the rate, so the error is not small. How sensitive is the whole subject of the section below.

How the Cost of Capital Affects Firm Value: what do 33 basis points buy?

The rate is only interesting because of what it does downstream, so here is the downstream. The model that turns cash flows into a value is covered separately; what follows restates its output and then moves one thing, the rate, holding every other assumption exactly where it was.

Sankalp Industrial Systems Limited, invented, is forecast to produce free cash flow to the firm of Rs 98,00,00,000 in Year 1, rising by Rs 18,00,00,000 a year to Rs 1,70,00,00,000 in Year 5, followed by a terminal value of Rs 29,25,00,00,000 built on 5.00 per cent growth forever. Discounted at 12.00 per cent with year-end discounting, that gives an enterprise value of Rs 21,28,13,79,094, stated here to the rupee because the arithmetic supports it. The same company's shares and debt are together priced in the market at an enterprise value of Rs 22,40,00,00,000.

The two figures are Rs 1,11,86,00,000 apart, or 5.26 per cent of the model's answer. The gap poses one question: what rate, holding growth and every cash flow exactly where they are, would make the model produce the traded figure? The answer is 11.67 per cent. Thirty-three hundredths of one percentage point of rate is worth Rs 1,11,86,00,000 on this invented company, and no other fact about the rate is more useful than that one.

The arithmetic says what rate would close the gap, and it stops there. Whether the gap is large or small, what it means, and whether a model or a market is the better description of a business, are questions with their own treatment elsewhere.

The reason a third of a point moves so much money is worth seeing rather than asserting. Most of this value is a terminal value, and a terminal value is a cash flow divided by the rate less the growth. Here that divisor is 12.00 less 5.00, being 7.00. Take 33 basis points off the rate and the divisor becomes 6.67, nearly five per cent smaller, so the terminal value rises by nearly five per cent of a very large number. Small changes in the rate become large changes in value because the rate is competing against the growth rate in a subtraction, and the subtraction leaves a small number behind.

The weights use a value the model is trying to produce. Is that not circular?

The objection is fair, and every careful reader spots it, usually within about a minute of seeing input 10. Here is the objection stated as sharply as it deserves. The equity weight of 75.0 per cent came from a market capitalisation of Rs 18,00,00,00,000. But the model exists to work out what equity is worth. So a number the model is supposed to produce has been fed into the rate the model uses to produce it. The weight is an input and it is also an output.

The disagreement is not hypothetical on this company. Discount the cash flows at 12.00 per cent, take off the borrowings and add back the cash, and the equity value of the operating business comes out at Rs 15,88,13,79,090, not Rs 18,00,00,00,000. Weight the two branches on that instead and the debt ratio is 27.16 per cent rather than 25.0. The model and the market are describing the same company and disagreeing about its shape.

The ruling applied here, and applied everywhere: the headline weights are the observed market capital structure, taken as observed rather than re-solved. Sankalp Industrial Systems Limited, invented, is listed. Its equity has an observed price of Rs 90.00 a share across 20,00,00,000 shares. An observed price is a fact about the world in a way that the model's output is not, and using an observed structure keeps the rate independent of the answer it is being used to compute. The model's own equity figure is a separate estimate, and what to make of the distance between an estimate and a price is a subject of its own, treated elsewhere.

A ruling that cannot be priced is indistinguishable from a dodge, so here is the price. Running the loop properly means taking the model's equity output, relevering the beta on it, recomputing the cost of equity, recomputing the weights, recomputing the rate, discounting again, and repeating until the answer stops moving.

Read the numbers rather than the shape. The loop settles after four passes at an equity value of Rs 16,09,31,00,000, a levered beta of 1.27962, a cost of equity of 14.148 per cent, a rate of 11.935 per cent and an enterprise value of Rs 21,49,31,00,000. Against the headline Rs 21,28,13,79,094 that is a move of Rs 21,17,00,000, or 1.0 per cent. The objection is entirely correct and it is worth one per cent, and knowing that number is what lets an analyst decide not to iterate rather than merely hope it does not matter.

Pricing an objection is a genuinely useful habit, and it generalises well past this subject. When somebody raises a theoretical objection to a body of work, the first question is not whether they are right. Assume they are right. The question is how much being right is worth. An objection worth one per cent and an objection worth twenty per cent are different objects and deserve different amounts of an analyst's afternoon.

How precise is a cost of capital, really?

Less precise than the arithmetic makes it look, and that gap is what keeps everything above from being trusted further than it deserves. Every step above was exact. Nothing was rounded, nothing was fudged, and the answer came out at 12.00 per cent to the last decimal. But exact arithmetic on estimated inputs produces an exact number that is still an estimate, and the exactness of the working says nothing whatever about the reliability of the answer.

Look at where the 12.00 per cent actually came from. Three quarters of it is the equity branch, and the equity branch rests on three things nobody can observe directly.

The least observable inputs carry the most of the answer. The ordering is exactly the wrong way round, and it is the honest headline of the whole method. A point of disagreement about the premium, at a beta of 1.25 and an equity weight of 75.0 per cent, moves the rate by 0.94 points, and the section above has already shown what a fraction of a point is worth in rupees here.

So carry the rate the way it deserves to be carried. Quote 12.00 per cent, say which assumptions produced it, and put a range around it that reflects the disagreement about the premium and the beta rather than the precision of the division. A rate written as 11.9873 per cent is not more careful than one written as 12.00. Four decimals claim an accuracy that none of the inputs can support, and they invite exactly the reliance the estimate cannot carry. Write down the assumptions beside the number. The assumptions are the part a reader can argue with, and the number is not.