Estimating the cost of capital: formula sheet

Key formulas

Constant dividend (no growth)

Ke = D ÷ P0

P0 is the ex-div price. The dividend is the same every year forever.

Dividend valuation model with growth

P0 = D0 × (1 + g) ÷ (Ke − g), so Ke = D0 × (1 + g) ÷ P0 + g

D0 is the dividend just paid. D1 = D0 × (1 + g). P0 must be ex-div. Needs Ke > g.

Growth from past dividends

g = (latest dividend ÷ earliest dividend)^(1 ÷ n) − 1

n is the number of years of growth, which is the number of dividends minus one.

Gordon growth model

g = b × r

b = proportion of earnings retained. r = return on retained funds, often taken as ROE.

Cum-div to ex-div

Ex-div price = cum-div price − dividend about to be paid

Use the ex-div price in the DVM.

CAPM cost of equity

Ke = Rf + β × (Rm − Rf)

Rm − Rf is the equity risk premium. If the question gives Rm (market return) rather than the premium, subtract Rf first.

Equity risk premium

Equity risk premium = Rm − Rf

Do not add Rf a second time when the premium is given.

Dividend growth model (cost of equity)

Ke = [D0 × (1 + g) ÷ P0] + g

P0 must be ex-dividend. D0 is the dividend just paid. Use for comparison with CAPM.

Geometric growth rate

g = (latest dividend ÷ earliest dividend)^(1 ÷ n) − 1

n is the number of years of growth, which is one fewer than the number of dividends listed.

After-tax cost of irredeemable debt

Kd(1 – T) = i(1 – T) ÷ P0

i = annual interest in $, P0 = current market price ex-interest per bond, T = tax rate. Use the same unit for i and P0 (for example per $100 nominal).

After-tax interest cash flow

Interest × (1 – T)

Use this for each year's interest in the IRR calculation. Assume tax relief arrives in the same year unless told otherwise.

IRR by interpolation

IRR = L + [NPV at L ÷ (NPV at L – NPV at H)] × (H – L)

L and H are the low and high trial rates. NPV at L is positive and NPV at H is negative, so the bracket must contain the answer.

Conversion value

Current share price × (1 + g)^n × number of shares per bond

g = expected annual share price growth, n = years to conversion. Compare with the redemption value and use the higher.

Cash flows for the IRR

Year 0: – market price; Years 1 to n: after-tax interest; Year n: redemption or conversion value

Cash flows for the bond holder's return; the same rate is the company's cost of debt.

Cost of irredeemable preference shares

Kp = D ÷ P₀

D is the fixed annual dividend (nominal value × dividend rate). P₀ is the ex-div market price. No tax adjustment.

Cost of bank loan (repaid at par)

Kd = i × (1 − T)

i is the interest rate on the loan. T is the corporate tax rate. Use when the loan is repaid at par and has no market price.

Cost of redeemable debt or preference shares

Find the rate r where: Present value of cash flows = current price

Use IRR with two trial rates and interpolation. For debt use after-tax interest. For preference shares use the dividend with no tax.

Interpolated IRR

IRR = L + [NPV_L ÷ (NPV_L − NPV_H)] × (H − L)

L and H are the low and high trial rates. NPV_H is usually negative.

Annual tax saving on interest

Tax saving = interest × T

Tax relief may be delayed by a year if the question says tax is paid with a lag.

WACC

WACC = [E × ke + P × kp + D × kd(1 − T)] ÷ (E + P + D)

E, P and D are the market values of equity, preference shares and debt. T is the tax rate. For debt that is redeemable or convertible, use the IRR-based after-tax cost instead.

Cost of equity (CAPM)

ke = Rf + β × (Rm − Rf)

Rf is the risk-free return, Rm the market return. (Rm − Rf) is the equity risk premium.

Cost of equity (dividend growth model)

ke = [D0 × (1 + g) ÷ P0] + g

P0 must be the ex-dividend price. D0 is the dividend just paid.

Irredeemable debt after tax

kd = Interest × (1 − T) ÷ Market value

Use the market value of the debt per $100 nominal and the interest on the same $100.

Preference shares

kp = Preference dividend ÷ Market price

No tax adjustment. Treat as irredeemable unless told otherwise.

Redeemable debt

kd = IRR of: −Market price now, + after-tax interest each year, + redemption value at the end

Use two trial rates and interpolate. Tax relief on interest is included in the cash flows.

Convertible debt

kd = IRR using the higher of redemption value and conversion value at the conversion date

Conversion value = expected future share price × conversion ratio. Compare it with the redemption value.

Market value weights

Weight = Market value of source ÷ Total market value of all sources

Shares: number of shares × share price. Debt: nominal × market price ÷ 100.

Pre-tax cost of debt from a spread

Kd (pre-tax) = risk-free yield for the maturity + credit spread

Convert basis points to a percentage first: 150 basis points = 1.50%.

After-tax cost of debt

Kd (after tax) = Kd (pre-tax) × (1 − T)

Use only if the company pays tax and interest is deductible. For redeemable debt, find the IRR using after-tax interest.

Yield to maturity (YTM)

Market price = Σ interest ÷ (1 + r)^t + redemption value ÷ (1 + r)^n

Solve r by interpolation between two trial rates: r = L + [NPV at L ÷ (NPV at L − NPV at H)] × (H − L).

Basis points

1 basis point = 0.01%

100 basis points = 1%.

Spread implied by a bond

Credit spread = YTM of corporate bond − yield on government bond of the same maturity

Compare bonds with the same maturity and currency.

Quick revision

• Dividend valuation model: Ke = D1 ÷ P0 + g, with P0 ex div.
• Dividend growth: estimate g from past dividends as g = (D_latest ÷ D_earliest)^(1 ÷ n) − 1, where n is the number of years of growth, meaning the number of intervals between the earliest and latest dividend, not the number of dividends listed.
• CAPM: Ke = Rf + β × (Rm − Rf).
• CAPM gives the required return for systematic risk only.
• Irredeemable debt after tax: Kd = i × (1 − t) ÷ P0, where i is annual interest.
• Redeemable debt: Kd is the IRR of price, after-tax interest and redemption value.
• Convertible debt: Kd is the IRR of the market price at time 0, the after-tax interest each year, and the higher of the cash redemption value and the conversion value at the conversion date. Conversion value = expected future share price × number of shares per bond. Estimate the future share price using the growth rate given in the question, for example P0 × (1 + g)^n. Compare the conversion value with the redemption value, and use the higher figure as the final cash flow in the IRR.
• Preference shares: Kp = dividend ÷ price, with no tax adjustment.
• Bank debt: after-tax cost = interest rate × (1 − t), if tax relief is available and the loan is at par.
• WACC weights use market values of each source.
• Use WACC for projects only if business risk and gearing stay unchanged.
• A credit spread is the extra yield over a risk-free rate, which reflects default risk.

Common mistakes

• Using D0 instead of D1 in Ke = D1 ÷ P0 + g. Fix: Always ask: is this the dividend just paid? If so, multiply by (1 + g) first.
• Using a cum-div price without adjusting. Fix: Subtract the dividend due to be paid shortly to get the ex-div price before calculating.
• Using Rm as the equity risk premium. Fix: Always subtract Rf from Rm before multiplying by beta, unless the question states the premium itself.
• Forgetting to add the risk-free rate at the end. Fix: Write the full formula first and tick off each part. Ke can never be below Rf when beta is positive.
• Using the nominal value instead of the market price as the year 0 figure. Fix: Interest comes from the nominal value. The investment (year 0) is always the market price.
• Forgetting to deduct tax from interest, or taking tax off the redemption value as well. Fix: Multiply only the interest by (1 – T). Redemption and conversion values are not taxed in this method.
• Adjusting the cost of preference shares for tax. Fix: Dividends come from after-tax profit. No relief, so Kp = D ÷ P₀ with no (1 − T).
• Using the nominal value as the denominator. Fix: Divide by the market price. Nominal value only sets the dividend amount.
• Using book values for weights when market values are given Fix: Always search the question for share prices and bond prices first. Shares are number × price, not share capital plus reserves.
• Forgetting to adjust debt for tax Fix: Write kd(1 − T) in your table heading. Only debt interest gets tax relief, not preference or equity dividends.

Exam tips

• Circle ex-div or cum-div in the question before you start. Examiners test this often.
• In OT questions, only one option is right and no partial marks are given, so check D0 versus D1 before you pick.
• In Section C, show the growth calculation and D1 as separate lines so you earn method marks even if a number slips.
• When asked to discuss the model, give three or four precise limitations, not a general comment.
• Cross-check Ke with the CAPM figure if both are given. Large differences are worth commenting on.
• Read carefully whether the question gives Rm or the equity risk premium. Examiners often test this.
• In objective questions, a quick sense check helps: beta above 1 means Ke is above the market return.
• In written parts, comparing CAPM with the DVM needs both strengths and weaknesses, not just a statement of formulas.