Estimating the cost of capital

The cost of capital is the return a company's investors require, and so the minimum return its investments must earn. It is the discount rate in every NPV in area D, and the link between the investing and financing decisions. This lesson shows how to estimate the cost of each source of finance from market data and combine them into the weighted average cost of capital (WACC). It is among the most heavily examined calculations in FM: expect objective test questions on single components (a cost of equity, a cost of redeemable debt) and Section C questions that build a full WACC and then ask you to discuss whether it is the right rate for a project. The formulae sheet gives you the dividend growth model, Gordon's growth approximation, the CAPM and the WACC formula; you must know how to use each and when.

The principle

An investor who buys a share or a loan note pays today's market price for the future cash it will bring: dividends, or interest and repayment. The cost of that finance is the rate of return that makes the present value of those future receipts equal today's market price. So the cost of capital is always found from market values, and it is higher for riskier securities, because investors demand more for bearing more risk.

The cost of equity: the dividend growth model

If dividends are expected to grow at a constant rate g forever, the share price is the present value of that growing stream. The formulae sheet gives the model as P0 = D0(1 + g) ÷ (re − g), and rearranged for the cost of equity:

re = D0(1 + g) ÷ P0 + g

where D0 is the dividend just paid, D0(1 + g) is next year's dividend and P0 is the ex-dividend share price. If the price is quoted cum dividend (just before a dividend is paid), deduct the dividend about to be paid to get P0.

There are two ways to estimate g:

• From past dividends. g = (latest dividend ÷ earliest dividend)^(1 ÷ n) − 1, where n is the number of years of growth. A dividend of $0.20 four years ago and $0.27 now gives (0.27 ÷ 0.20)^(1/4) − 1 = 1.35^0.25 − 1 = 7.8%. Count carefully: five years of dividends contain four years of growth.
• Gordon's growth approximation, on the formulae sheet: g = b × re, where b is the proportion of earnings retained and re here is the return earned on those reinvested earnings (often the accounting return on equity), not the cost of equity. A company retaining 40% of its earnings and earning 12.5% on them grows at 0.40 × 12.5% = 5%.

The model is simple and uses market data. Its weaknesses are its assumptions: dividends grow at a constant rate forever, the growth rate is below the cost of equity, and the share price reflects all investors know. The answer is very sensitive to g, the model cannot be used for a company that pays no dividends, and it does not consider risk directly.

Systematic risk, portfolio theory and the CAPM

The risk of holding one share has two parts. Unsystematic (specific) risk comes from events particular to the company, such as a failed product or a strike. Systematic (market) risk comes from factors that affect all companies to some degree, such as interest rates, inflation and the economic cycle. Portfolio theory shows that by holding a well-diversified portfolio, an investor can remove almost all unsystematic risk, because the specific surprises of different companies cancel out. Systematic risk cannot be diversified away.

The capital asset pricing model (CAPM) builds on this: if investors are diversified, they need to be rewarded only for systematic risk. A share's systematic risk is measured by its beta (β), relative to the market as a whole, which has a beta of 1. A beta of 1.2 means the share's returns tend to move 1.2 times as much as the market's. The formulae sheet gives:

E(ri) = Rf + βi(E(rm) − Rf)

where Rf is the risk-free rate, E(rm) is the expected market return and E(rm) − Rf is the market risk premium. With Rf of 5%, E(rm) of 12% and β of 1.2, the cost of equity is 5% + 1.2 × (12% − 5%) = 13.4%. If the question gives the market risk premium itself, do not subtract Rf again.

The CAPM's strength is that it prices risk explicitly and, used with a beta for a particular business, can give a project-specific discount rate (lesson 4 of this module). Its assumptions are demanding: investors hold diversified portfolios, can borrow and lend at the risk-free rate, and face a perfect capital market with no taxes or transaction costs, all over a single period. In practice, betas are estimated from past data and change over time, and the risk-free rate and the market return are themselves hard to pin down.

The cost of debt and preference shares

The company's cost of debt is after tax, because interest is tax-deductible. Market prices are quoted ex interest, per $100 nominal value.

For example, 8% irredeemable loan notes trading at $105, with tax at 30%, cost 8 × 0.70 ÷ 105 = 5.33% after tax.

The redemption value in the IRR is not taxed: only the interest attracts tax relief. Investors' own required return, the pre-tax yield, is found the same way with interest before tax; that is the rate used to value loan notes in area F.

For a convertible, investors will convert if the shares they receive are worth more than the cash redemption. Take 5% convertible loan notes trading at $98, convertible in four years into 20 shares or redeemable at $100. The share price is $4.50 and expected to grow at 4% a year, so in four years it will be $4.50 × 1.04⁴ = $5.264, a conversion value of 20 × $5.264 = $105.29. That beats $100, so assume conversion. With tax at 30%, the after-tax interest is $3.50:

Cost ≈ 5% + [1.06 ÷ (1.06 + 2.48)] × 1% = 5.3% after tax.

The weighted average cost of capital

The WACC weights the cost of each source by its share of the company's total finance. The formulae sheet gives the two-source version:

WACC = [Ve ÷ (Ve + Vd)]ke + [Vd ÷ (Ve + Vd)]kd(1 − T)

Here kd is the pre-tax cost of debt, which the formula converts with (1 − T). If you have already found an after-tax cost, as with the IRR method using after-tax interest, use it as it is: never deduct tax twice. With more sources, such as preference shares and several kinds of debt, add one term for each.

Market values are the right weights, because they reflect what investors would pay today and so the returns they require now: equity is the number of shares × the ex-dividend price, loan notes are the nominal value × the ex-interest price ÷ 100, and bank loans are usually taken at their book value. Book values are easier to find but historical; for a profitable company the book value of equity is usually well below its market value, so book weights overstate the weight of cheap debt and understate the WACC.

The WACC is an average cost of the finance the company already has. The marginal cost of capital is the cost of the next dollar of new finance, which may differ: a large new loan, for instance, may raise the cost of both debt and equity. Using the existing WACC to appraise a project is valid only in particular conditions, discussed in the next lesson.

Worked example

Wuse Holdings Co's finance is as follows. Tax is 30%.

• 20 million ordinary shares of $0.50 nominal, trading at $3.60 ex dividend. A dividend of $0.30 has just been paid. The company retains 40% of its earnings and earns a return of 12.5% on reinvested equity.
• 5 million 7% preference shares of $1, trading at $0.80 ex dividend.
• $25 million of 6% loan notes, redeemable at nominal value in five years, trading at $95 ex interest.
• A bank loan of $5 million at 8% interest.

Step 1: cost of equity. g = b × re = 0.40 × 12.5% = 5%. ke = $0.30 × 1.05 ÷ $3.60 + 0.05 = 0.0875 + 0.05 = 13.75%.

Step 2: cost of preference shares. kp = $0.07 ÷ $0.80 = 8.75%.

Step 3: cost of the loan notes. After-tax interest = $6 × (1 − 0.30) = $4.20 a year.

kd ≈ 5% + [1.58 ÷ (1.58 + 2.61)] × (6% − 5%) = 5% + 0.38% = 5.38% after tax.

Step 4: cost of the bank loan. 8% × (1 − 0.30) = 5.6%.

Step 5: WACC at market values.

WACC = $11.808m ÷ $104.75m = 11.3%. Notice that the loan notes' after-tax cost is used as it stands, with no further tax deduction, and that the nominal value of the shares ($10m) plays no part.

Common mistakes

• Using a cum dividend share price or a cum interest loan note price without adjusting it.
• Putting D0 instead of D0(1 + g) in the dividend growth model.
• Counting n wrongly when estimating growth: four years of growth separate five dividends.
• Applying (1 − T) to preference dividends, or applying it a second time to a cost of debt that is already after tax.
• Weighting the WACC by nominal values instead of market values.
• Using the market return instead of the market risk premium in the CAPM.

Exam technique

Calculate each component separately and label it (ke, kp, kd), then bring them together in a WACC table like the one above, which shows every market value and weight in one place. For redeemable debt, choose a first rate near the after-tax coupon yield (here 4.20 ÷ 95 = 4.4%, so 5% is sensible) and a second rate one or two points away on the other side of the IRR. In discussion parts, the marks are for the assumptions of the dividend growth model and the CAPM, and for explaining the conditions under which the WACC may be used, so have three or four specific points ready for each.

Study with AI

I am studying ACCA Financial Management (FM), the cost of capital. Give me a company
with ordinary shares (ex-dividend price, last dividend and either past dividends or a
retention ratio and return on equity), preference shares, redeemable loan notes with
an ex-interest price, and a bank loan, with a tax rate. Ask me for ke, kp, the
after-tax kd by IRR and the WACC at market values, one at a time. Wait for each
answer, then show the correct working using three-decimal factors from the ACCA tables.

AI assistants can make arithmetic mistakes and can misstate the rules: check every answer against these notes and the ACCA tables before you rely on it.

Lecturer's note: Every cost of capital calculation is the same idea in a different form: find the rate at which the future cash an investor will receive is worth today's market price. For an irredeemable security that rate is a simple division, for a growing dividend it is the growth model, and for redeemable or convertible debt it is an IRR. Once you see that each one is just an IRR in disguise, you stop memorising formulas and start understanding them, and the traps (cum prices, D0 versus D1, tax applied twice) become obvious.

Key points

• The cost of capital is the return investors require, found from market values; riskier securities cost more.
• Dividend growth model: re = D0(1 + g) ÷ P0 + g, with P0 ex dividend; g from past dividends or Gordon's g = b × re.
• CAPM: E(ri) = Rf + βi(E(rm) − Rf); beta measures systematic risk, the only risk a diversified investor is paid to bear.
• After-tax cost of debt: I(1 − T) ÷ P0 for irredeemable debt; the IRR of after-tax interest and redemption for redeemable debt; for convertibles, the higher of redemption and conversion value. Preference shares get no tax relief.
• WACC weights each after-tax cost by market value; it is an average cost and is not automatically the marginal cost of new finance.

Quick check

1. A share trades at $2.50 ex dividend. A dividend of $0.20 has just been paid, and dividends are expected to grow at 6% a year. What is the cost of equity?
2. The risk-free rate is 3%, the expected market return is 10% and a share's beta is 0.8. What is its cost of equity under the CAPM?
3. 10% irredeemable loan notes trade at $125 ex interest, and tax is 25%. What is their after-tax cost?

Answers

1. ke = $0.20 × 1.06 ÷ $2.50 + 0.06 = 0.0848 + 0.06 = 14.5%.
2. 3% + 0.8 × (10% − 3%) = 3% + 5.6% = 8.6%.
3. kd = $10 × (1 − 0.25) ÷ $125 = $7.50 ÷ $125 = 6.0%.