Cost of capital — clear, compact explainer
Definition
– Cost of capital is the required return a company must earn on a new investment to maintain its market value and satisfy its providers of capital (creditors and shareholders). For investors, it represents the expected return from holding the company’s securities.
– Key idea: it is the minimum hurdle rate a project must clear to create value.
Why it matters
– Decision rule for investments: compare a project’s expected return (or internal rate of return) to the cost of capital; accept projects that exceed it.
– Capital-structure choice: helps management weigh debt versus equity financing because different sources carry different costs and tax effects.
– Investor perspective: signals the expected compensation for taking on a firm’s risk.
Main components
1. Cost of debt (rd)
– Definition: the effective interest rate a company pays on its borrowings.
– Tax effect: interest is usually tax-deductible, so the relevant cost is after-tax: rd_aftertax = rd_beforetax × (1 − T), where T is the company’s marginal tax rate.
– Practical estimate methods:
– Direct: interest expense ÷ total debt, then multiply by (1 − T).
– Market-based: risk-free rate + credit spread (then apply (1 − T)).
2. Cost of equity (re)
– Definition: the return equity investors demand for owning the company’s stock.
– Common estimator: Capital Asset Pricing Model (CAPM)
– CAPM formula: re = Rf + β × (Rm − Rf)
– Rf = risk-free rate (e.g., long-term government bond yield)
– β (beta) = sensitivity of the company’s returns to market returns (measure of systematic risk)
– Rm − Rf = market risk premium (expected excess return of the market)
– Notes: beta for private firms is often inferred from comparable public companies and adjusted for leverage.
Weighted Average Cost of Capital (WACC)
– Definition: the firm’s blended cost of capital, weighting each source by its share of total financing (typically market values).
– Formula (basic two-component form): WACC = we × re + wd × rd_aftertax
– we = weight of equity (market value of equity / total market value of capital)
– wd = weight of debt (market value of debt / total market value of capital)
– Include other instruments (preferred stock, convertible debt) as separate terms if present.
Checklist — steps to calculate a practical WACC
1. Decide the valuation scope (firm-wide or project-specific) and whether to use levered or unlevered rates.
2. Measure market values of capital components: market value of equity (market cap), market value of debt (book value approximations or market quotes).
3. Compute weights: we = E/(E + D), wd = D/(E + D) (plus other components if present).
4. Estimate cost of debt:
– From financial statements: interest expense ÷ total debt, or use current bond yields/credit spreads.
– Adjust for corporate tax: rd_aftertax = rd_beforetax × (1 − T).
5. Estimate cost of equity:
– Select Rf (e.g., 10‑year government bond yield).
– Choose market risk premium (historical or forward-looking).
– Get beta (public-company regression or industry average) and apply CAPM.
6. Compute WACC using the formula above.
7. Use WACC as the discount rate for projects with similar risk to the firm; adjust for project-specific risk if necessary.
8. Perform sensitivity checks (vary Rf, beta, weights, tax rate).
Worked numeric examples
Example A — compute after-tax cost of debt from financials
– Company pays $8 million in annual interest and has $200 million total debt. Corporate marginal tax rate = 25%.
– rd_beforetax = interest ÷ total debt = 8 / 200 = 0.04 = 4.0%
– rd_aftertax = rd_beforetax × (1 − T) = 4.0% × (1 − 0.25) = 3.0%
Example B — WACC for a simple capital structure
– Inputs:
– Market value
– Inputs (assumptions)
– Market value of equity, E = $600 million
– Market value of debt, D = $400 million
– Total firm value, V = E + D = $1,000 million
– Before‑tax cost of debt, rd_beforetax = 5.0%
– Corporate tax rate, T = 25%
– Risk‑free rate, Rf = 2.0%
– Market risk premium, (Rm − Rf) = 6.0%
– Equity beta, β = 1.20
Step 1 — compute after‑tax cost of debt
– rd_aftertax = rd_beforetax × (1 − T)
– rd_aftertax = 5.0% × (1 − 0.25) = 5.0% × 0.75 = 3.75%
Step 2 — compute cost of equity by CAPM (Capital Asset Pricing Model)
– re = Rf + β × (Rm − Rf)
– re = 2.0% + 1.20 × 6.0% = 2.0% + 7.2% = 9.2%
Step 3 — compute weights (market‑value weights)
– Weight of equity, We = E / V = 600 / 1,000 = 0.60
– Weight of debt, Wd = D / V = 400 / 1,000 = 0.40
Step 4 — compute WACC
– WACC = We × re + Wd × rd_aftertax
– WACC = 0.60 × 9.2% + 0.40 × 3.75%
– WACC = 5.52% + 1.50% = 7.02% (approx.)
Interpretation
– With these inputs, the firm’s WACC ≈ 7.02%. This is the approximate discount rate for projects with the same systematic risk as the firm.
– Assumptions to note: market values used for weights, CAPM used to estimate re, debt treated as a tax‑deductible interest expense, and no preferred stock or minority interests included.
Sensitivity checks (quick examples)
– If beta rises to 1.50 (higher equity risk): re = 2% + 1.5×6% = 11.0%; WACC = 0.6×11.0% + 0.4×3.75% = 8.10%.
– If corporate tax rate rises to 30% (more tax shield): rd_aftertax = 5.0%×(1−0.30)=3.5%; WACC = 0.6×9.2% + 0.4×3.5% = 6.92%.
– These show WACC is sensitive to beta (cost of equity) and tax rate (debt tax shield).
Checklist — computing WACC correctly
1. Use market values (not book values) for equity and debt when possible.
2. Estimate cost
2. Estimate cost of equity (re)
– Primary method: CAPM (Capital Asset Pricing Model). Formula: re = rf + β × (RPM), where rf is the risk‑free rate, β (beta) measures the stock’s systematic risk relative to the market, and RPM is the market risk premium (expected market return minus rf). Note assumptions: CAPM is forward‑looking only if inputs are forward‑looking; betas may be estimated from historical regressions or adjusted/industry betas.
– Alternative method: Dividend Discount / Gordon Growth model for dividend‑paying firms: re = D1 / P0 + g (next dividend divided by current price plus long‑run growth rate). Use this only when dividends are stable and a reasonable g can be estimated.
– Practical guidance: state the source and date for rf, beta, and RPM; if using historical beta, consider adjusting toward 1 (Blume adjustment) or use an industry beta when company data are thin.
3. Estimate pre‑tax cost of debt (rd) and apply tax effect
– Use the market yield on outstanding debt when available (yield to maturity). If bonds are illiquid or firm is private, estimate rd using recent borrowing rates for comparable credits or the firm’s credit spread over the risk‑free curve.
– Convert to after‑tax cost: rd_aftertax = rd × (1 − Tc), where Tc is the corporate tax rate. This reflects the tax deductibility of interest.
– Document whether you used a marginal statutory tax rate, an effective tax rate, or an expected forward tax rate and why.
4. Include preferred stock (if applicable)
– If the firm has preferred shares, include their market value in weights and use cost of preferred = preferred dividend / preferred market price (no tax shield).
– Treat minority interests and other hybrid securities carefully—include only if they represent long‑term financing of operations.
5. Determine weights using market values
– Equity market value = shares outstanding × current share price.
– Debt market value = sum of market values of debt instruments; if market values are not available, use book value as an approximation but disclose the approximation and test sensitivity.
– Weights must sum to 100% and reflect the firm’s target or current capital structure as appropriate for the valuation purpose (target structure for long‑run valuation; current structure for short‑run).
6. Ensure consistency of cash flows and discount rate
– Match nominal vs. real: use nominal WACC with nominal cash flows (include inflation) and real WACC with real cash flows (exclude inflation).
– Match currency and country risk: WACC should be in the same currency and reflect the same sovereign risk as the cash flows being discounted.
– For project‑level analysis, adjust beta or use a project‑specific discount rate if the project’s risk profile differs from the firm’s (don’t blindly apply corporate WACC).
Quick worked example (step‑by‑step)
Assumptions: market cap = $150m, market value of debt = $50m, rf = 2.0%, RPM = 6.0%, beta = 1.1, rd = 6.0%, corporate tax rate Tc = 21%, no preferred stock.
1) Market weights: E = 150 / (150+50) = 0.75; D = 50 / 200 = 0.25.
2) Cost of equity (CAPM): re = 2.0% + 1.1×6.0% = 8.6%.
3) After‑tax cost of debt: rd_aftertax = 6.0% × (1 − 0.21) = 4.74%.
4) WACC = E×re + D×rd_aftertax = 0.75×8.6% + 0.25×4.74% = 6.45% + 1.185% = 7.635% ≈ 7.64%.
Common pitfalls and checks
– Don’t mix book values for equity with market values for debt; be consistent and explain approximations.
– Don’t use historical average returns for rf and RPM without checking whether they’re appropriate for forward valuation.
– Watch beta adjustments: failing to relever/unlever beta correctly when changing capital structure will bias re.
– Forgetting taxes: using pre‑tax rd without the (1 − Tc) factor underestimates the tax shield.
– Using corporate WACC on a new project with different risk; instead, compute a project‑specific discount rate.
– Currency and inflation mismatches: discounting foreign cash flows with a domestic WACC gives
biased results — match currencies and inflation assumptions (e.g., real cash flows with a real WACC; nominal cash flows with a nominal WACC).
Practical additions, formulas, and worked examples
1) Relevering and unlevering beta (step‑by‑step)
– Purpose: adjust a comparables’ observed beta for differences in capital structure. Beta measures systematic risk (sensitivity of returns to the market).
– Unlever beta (remove financial leverage):
beta_unlevered = beta_levered / [1 + (1 − Tc) × (D / E)]
where Tc = corporate tax rate, D/E = debt-to-equity ratio (market values).
– Relever to a new capital structure:
beta_relevered = beta_unlevered × [1 + (1 − Tc) × (D_new / E_new)]
Worked numeric example
– Observed (levered) beta = 1.20; observed D/E = 0.50; Tc = 25% (0.25).
– beta_unlevered = 1.20 / [1 + (1 − 0.25) × 0.50] = 1.20 / [1 + 0.375] = 1.20 / 1.375 ≈ 0.873.
– Relever for target D/E = 0.25:
beta_relevered = 0.873 × [1 + 0.75 × 0.25] = 0.873 × 1.1875 ≈ 1.035.
– If rf (risk‑free rate) = 2.5% and ERP (equity risk premium) = 5.0%, then CAPM cost of equity:
re = rf + beta_relevered × ERP = 2.5% + 1.035 × 5% ≈ 7.675% ≈ 7.68%.
2) After‑tax cost of debt
– Formula: rd_aftertax = rd × (1 − Tc). It reflects interest tax shields (interest is tax‑deductible).
– Example consistent with earlier WACC step: if rd = 6.32% and Tc = 25% then rd_aftertax = 6.32% × 0.75 = 4.74%.
3) Which discount rate for what cash flow?
– Equity cash flows (dividends or levered free cash flow to
equity) should be discounted at the cost of equity (re). Firm cash flows (unlevered free cash flow to the firm, FCFF) should be discounted at the weighted average cost of capital (WACC) — or by the unlevered cost of capital when using APV. Do not mix: discounting FCFF with re or discounting levered cash flows with WACC produces inconsistent values.
Below are practical rules, formulas, a worked numeric example that continues the numbers from your context, and checklists for common valuation situations.
Quick rules (one-line)
– Use cost of equity (re) to discount equity cash flows (dividends, levered free cash flow to equity — LFCFE).
– Use WACC to discount FCFF if the company (or project) will maintain the stated target capital structure.
– Use APV (adjusted present value) or a project-specific unlevered discount rate if the leverage will change materially or if debt tax shields have different risk.
– When using CAPM, apply the relevered beta appropriate to the target D/E you assumed.
Formulas (reminder)
– re (CAPM) = rf + beta_levered × ERP.
– rd_aftertax = rd × (1 − Tc).
– WACC = we × re + wd × rd_aftertax, where we and wd are market-value weights of equity and debt (we + wd = 1).
– Firm value (FCFF perpetuity) = FCFF1 × (1 + g) / (WACC − g) when stable growth applies and WACC > g.
– Equity value = Firm value − Market value of debt (for non-financial firms, if debt is priced at market value).
Worked numeric example (continuing your numbers)
Assumptions carried forward:
– Target D/E = 0.25 (debt = 0.25 × equity).
– Re (CAPM, relevered) ≈ 7.68% (you computed ≈7.675%).
– rd = 6.32% and corporate tax Tc = 25% → rd_aftertax = 6.32% × 0.75 = 4.74%.
Step 1 — compute weights from D/E
– Let E = 1 (normalized). Then D = 0.25.
– Total capital = E + D = 1.25