What is the cost of equity (CoE)?
– The cost of equity is the return that equity investors require for holding a company’s stock. For investors it’s the minimum expected return on an equity investment. For a firm it’s the benchmark return a project or financing must deliver to justify using equity capital.
– Key point: equity does not have to be repaid like debt, so firms pay equity holders through dividends and/or capital gains; because interest on debt is tax-deductible, debt usually has a lower after‑tax cost than equity.
Two common ways to estimate the cost of equity
1) Dividend capitalization model (a form of the Gordon Growth Model)
– Use when the firm pays dividends and you can reasonably estimate dividend growth.
– Formula: CoE = (D1 / P0) + g
– D1 = expected dividend per share next year
– P0 = current market price per share
– g = expected annual growth rate of dividends (decimal form)
– Intuition: the required return equals the dividend yield plus expected dividend growth.
2) Capital Asset Pricing Model (CAPM)
– Useable for dividend and non‑dividend payers; relates required return to market risk.
– Formula: CoE = Rf + β × (Rm − Rf)
– Rf = risk‑free rate (e.g., Treasury yield)
– β (beta) = sensitivity of the stock’s returns to market returns (measure of systematic risk)
– Rm = expected market return (so Rm − Rf is the market risk premium)
– Intuition: required return = risk‑free return plus a premium for exposure to market risk, scaled by beta.
Worked numeric examples
1) Dividend model example
– Assume next year’s dividend D1 = $2.00, current price P0 = $50.00, expected dividend growth g = 4% (0.04).
– CoE = (2.00 / 50.00) + 0.04 = 0.04 + 0.04 = 0.08 → 8.0%
2) CAPM example
– Assume risk‑free rate Rf = 3.0% (0.03), expected market return Rm = 8.0% (0.08), beta β = 1.2.
– Market risk premium = 0.08 − 0.03 = 0.05.
– CoE = 0.03 + 1.2 × 0.05 = 0.03 + 0.06 = 0.09 → 9.0%
Using CoE to compute overall cost of capital (WACC)
– Weighted Average Cost of Capital (WACC) blends after‑tax cost of debt and cost of equity according to capital structure weights.
– Example: CoE = 9.0%, after‑tax cost of debt = 4.0%, equity weight = 60%, debt weight = 40%.
– WACC = 0.60×9.0% + 0.40×4.0% = 5.4% + 1.6% = 7.0%
Short checklist — how to calculate a company’s cost of equity
1. Decide which model to use: dividend model (requires stable dividends) or CAPM (works without dividends).
2. Gather inputs:
– Dividend model: D1, current share price, dividend growth rate.
– CAPM: current risk‑free rate, beta, expected market return (or market risk premium).
3. Compute CoE using the chosen formula.
4. Sensitivity check: vary growth, beta, and market premium to see impact.
5. If calculating WACC, collect after‑tax cost of debt and capital structure weights and calculate weighted average.
6. Document assumptions and sources for each input.
Special considerations and limitations
– Dividend model limitations: needs reliable dividends and a sustainable growth estimate. It ignores other shareholder returns (e.g., buybacks) unless you convert them into an equivalent dividend.
– CAPM limitations: relies on accurate beta estimation and a defensible market risk premium. Betas differ by sampling period and indexing choice; historical betas may not predict future risk precisely.
– Choosing the risk‑free rate: use a term consistent with the investment horizon (commonly long‑term
…Treasury yield — most practitioners use a government bond yield whose term roughly matches the cash‑flow horizon of the valuation (commonly the 10‑year U.S. Treasury for many corporations). For non‑USD or emerging‑market companies, use the sovereign yield in the company’s reporting currency or add a country risk premium (see checklist below).
Relevering/unlevering beta (practical steps)
– Why: Betas observed for comparable firms reflect their particular capital structures. To compare risk across firms or apply a beta to a target capital structure you must remove (unlever) debt effects and then re‑apply (relever) for your firm.
– Formulas (Hamada relationship):
– Unlevered beta (βu) = βL / [1 + (1 − Tc) · (D/E)]
– Relevered beta (βL,target) = βu · [1 + (1 − Tc) · (D/E)target]
– Where βL = observed (levered) beta, Tc = corporate tax rate, D/E = debt/equity ratio.
– Worked example:
– Three comparable firms’ levered betas: 1.10, 1.25, 0.95. Debt/equity ratios: 0.4, 0.6, 0.3. Use Tc = 25% (0.25).
– Compute each firm’s βu:
– Firm A: βu = 1.10 / [1 + 0.75·0.4] = 1.10 / 1.30 = 0.846
– Firm B: βu = 1.25 / [1 + 0.75·0.6] = 1.25 / 1.45 = 0.862
– Firm C: βu = 0.95 / [1 + 0.75·0.3] = 0.95 / 1.225 = 0.776
– Average βu = (0.846 + 0.862 + 0.776) / 3 = 0.828.
– If your target D/E = 0.5, relever: βL,target = 0.828 · [1 + 0.75·0.5] = 0.828 · 1.375 = 1.138.
– Use β = 1.138 in CAPM.
CAPM reminder and worked numeric example
– CAPM formula: Cost of Equity = Rf + β · (Rm − Rf)
– Rf = risk‑free rate, β = levered beta appropriate to the target capital structure, Rm − Rf = equity market risk premium (expected market return minus Rf).
– Example:
– Rf = 3.0% (10‑yr Treasury), β = 1.138 (from above), market risk premium = 5.0%.
– CoE = 3.0% + 1.138 × 5.0% = 3.0% + 5.69% = 8.69%.
Dividend Discount Model (Gordon growth) quick check
– Formula (for a stable‑growth dividend payer): CoE = D1 / P0 + g
– D1 = expected dividend next period, P0 = current share price, g = sustainable growth rate.
– Example:
– Expected dividend next year D1 = $2.00, price P0 = $50, g = 2.0%.
– CoE = 2 / 50 + 0.02 = 0.04 + 0.02 = 6.0%.
– Caveat: This model requires reliable dividend policy and a plausible long‑term growth rate. It doesn’t directly capture buybacks unless you convert total shareholder returns into a dividend equivalent.
Other model choices and adjustments
– Multifactor models (e.g., Fama‑French) add size and value (or profitability/momentum) premia; they can improve fit for portfolios but require factor returns and exposure estimates.
– Country and sovereign risk: add a country premium when valuing firms in markets with meaningful sovereign risk above the chosen risk‑free rate.
– Size premium: for small, illiquid firms consider a size premium added to CAPM, but document the source and methodology.
Using CoE in WACC (quick example)
– After‑tax cost of debt = Rd · (1
– After‑tax cost of debt = Rd · (1 − Tc), where Rd is the pre‑tax yield on debt and Tc is the corporate tax rate.
WACC formula and quick worked example
– WACC (weighted average cost of capital) = (E/V) · Re + (D/V) · Rd · (1 − Tc) + (P/V) · Rp
– E = market value of equity
– D = market value of debt
– P = market value of preferred stock (if any)
– V = E + D + P (total financing value; use market values)
– Re = cost of equity
– Rd = pre‑tax cost of debt
– Rp = cost of preferred stock (dividend / price)
– Tc = corporate tax rate
– Numeric example:
– Suppose a firm has market equity E = $600m, market debt D = $400m, no preferred stock (P = 0). So V = $1,000m.
– Re (from CAPM or other method) = 10.0%
– Rd (current yield on company debt) = 5.0%
– Tc = 21%
– After‑tax debt cost = 5.0% × (1 − 0.21) = 3.95%
– WACC = (600/1000) × 10.0% + (400/1000) × 3.95% = 6.00% + 1.58% = 7.58%
Practical notes on inputs
– Use market values, not book values, when possible. If only book values are available, disclose the limitation and consider sensitivity tests.
– Use a target capital structure (management guidance or industry median) when valuing a going concern; if valuing a liquidation, use current capital structure.
– Exclude non‑operating cash from the enterprise value used to compute E and D; treat cash separately.
– For preferred stock, use Rp = preferred dividend / current market price of the preferred issue.
– If debt is quoted at different maturities or coupons, estimate a weighted average pre‑tax Rd or use the market yield on outstanding debt issues.
Estimating and applying beta (re‑levering/unlevering)
– Unlevered (asset) beta removes financial leverage and reflects business risk alone:
– Beta_unlevered = Beta_levered / [1 + (1 − Tc) · (D/E)]
– Re‑lever to a target capital structure:
– Beta_target = Beta_unlevered × [1 + (1 − Tc) · (D_target/E_target)]
– Worked example:
– Comparable company levered beta = 1.20
– Comparable D/E = 0.50, Tc = 21%
– Beta_unlevered = 1.20 / [1 + (1 − 0.21) × 0.50] = 1.20 / [1 + 0.79 × 0.50] = 1.20 / [1 + 0.395] = 1.20 / 1.395 ≈ 0.86
– Target D/E = 0.80 → Beta_target = 0.86 × [1 + 0.79 × 0.80] = 0.86 × [1 + 0.632] = 0.86 × 1.632 ≈ 1.40
– Use Beta_target in CAPM: Re = Rf + Beta_target × Equity Risk Premium
Common adjustments and caveats
– Country/sovereign risk: add a country premium to Re (or increase equity risk premium) for firms operating in countries with meaningful sovereign/default risk.
– Size/liquidity premium: for small, illiquid firms add a size or illiquidity premium; document the source and magnitude.
– Buybacks and total shareholder return: if dividends are low but buybacks are substantial, convert expected total shareholder payouts into a dividend equivalent when using dividend‑based models.
– Choice of risk‑free rate: match the tenor of the risk‑free rate to the cash‑flow horizon (e.g., use long‑term Treasury for long-duration cash flows). Using mismatched tenors biases Re.
– Implied vs historical equity risk premium: implied (from market prices) and historical averages can differ materially; show sensitivity to both.
Step‑by‑step checklist to estimate WACC
1. Decide scope: enterprise or equity valuation; determine cash‑flow horizon.
2. Collect market
2. Collect market inputs: gather current market prices and observable rates tied to your chosen cash‑flow horizon. Key items:
– Stock price and shares outstanding → market equity value (Market cap = price × shares).
– Debt market value: use bond prices/YTM where available; if not, approximate by book value adjusted for recent trades or credit spreads.
– Preferred stock market value and stated dividend, if any.
– Risk‑free rate matched to cash‑flow tenor (e.g., 10‑year Treasury for long‑duration free cash flows).
– Market equity risk premium (ERP) or historical premium, and comparable firms’ betas (levered β). Date‑stamp every input.
3. Compute capital structure weights (market‑value weights):
– We = Market value of equity / Total market value of capital
– Wd = Market value of debt / Total market value of capital
– Wp = Market value of preferred / Total market value of capital
Total market value of capital = equity + debt + preferred (use market values). Avoid using book values unless a market value is unavailable and you document the reason.
4. Estimate the cost of equity (Re). Common methods:
– CAPM (capital asset pricing model): Re = Rf + β × ERP
– Rf = matched risk‑free rate; β = levered beta (sensitivity of stock returns to market returns); ERP = equity risk premium.
– Dividend Discount Model (DDM): Re = (D1 / P0) + g where D1 = next dividend, P0 = price, g = growth rate.
– Implied ERP: back out the market’s expected ERP from aggregate prices and cash‑flow forecasts.
Adjustments: add sovereign/default risk premium for exposures to high‑risk countries; add size/illiquidity premium for small, thinly traded firms. Document and justify any adjustments.
5. Estimate the cost of debt (Rd). Use observable yields on the firm’s debt or model as Rf + credit spread. For new financing, use expected forward rates or current market spreads for the firm’s rating. Convert nominal yields to effective rates if cash‑flow timing differs. Use after‑tax cost of debt in WACC: Rd_after = Rd × (1 − Tc), where Tc is the corporate tax rate.
6. Cost of preferred (Rp). If preferred shares exist: Rp = preferred dividend / market price of preferred. Preferred is included in WACC as a separate weighted component (Wp × Rp) and is not tax‑adjusted.
7. WACC formula (market‑value weights):
WACC = We × Re + Wd × Rd × (1 − Tc) + Wp × Rp
Assumptions: formulas assume cash flows to the firm are financed by the same long‑run mix of capital; weights reflect market values.
Worked numeric example
– Market equity = $500 million
– Market debt = $200 million
– Market preferred = $50 million
Total capital = $750 million
Weights: We = 500/750 = 0.6667; Wd = 200/750 = 0.2667; Wp = 50/750 = 0.0667
Inputs:
– Risk‑free rate (10‑yr Treasury) = 3.00%
– Beta = 1.10
– ERP = 5.00% → Re (CAPM) = 3.00% + 1.10×5.00% = 8.50%
– Rd (market YTM) = 5.00%
– Corporate tax rate Tc = 21% → Rd_after = 5.00% × (1 − 0.21) = 3.95%
– Preferred dividend yield Rp = 6.00%
WACC = 0.6667×8.50% + 0.2667×3.95% + 0.