Cost of Capital

The cost of equity (Ke) is the return that equity investors require given the risk they bear. Two main methods:

1. Dividend Valuation Model (DVM) — without growth:

Where P₀ = current share price (ex-div), D = expected constant dividend per share.

Example: Share price £2.50, expected dividend 20p per year forever. Ke = 20 / 250 = 8%.

2. DVM with constant growth (Gordon Growth Model):

Where D₁ = expected next dividend (i.e., D₀ × (1+g)), g = constant growth rate (with g < Ke).

Example: Share price £3.00, just paid dividend 15p, expected growth 4% per year. D₁ = 15p × 1.04 = 15.6p. Ke = 15.6/300 + 0.04 = 0.052 + 0.04 = 9.2%.

Estimating growth (g):

• Historical method: g = (D_latest / D_earliest)^(1/n) − 1, where n = number of years between
• Earnings retention method (Gordon's): g = b × r, where b = retention ratio (1 − payout ratio), r = return on reinvested earnings

DVM advantages: Simple, directly uses observable data. Disadvantages: Only valid with dividends (can't use for non-dividend-paying companies); assumes constant growth forever; very sensitive to g estimate; ignores systematic risk and market conditions.

3. Capital Asset Pricing Model (CAPM):

Where:

• Rf = risk-free rate (usually yield on government bonds)
• Rm = expected return on the market portfolio
• Rm − Rf = equity risk premium (ERP) — historical average ~5-7% for UK/US
• β (beta) = sensitivity of the share's return to market movements

Example: Rf = 3%, ERP = 5%, β = 1.2. Ke = 3% + 1.2 × 5% = 3% + 6% = 9%.

CAPM assumptions (idealised):

• Investors are rational, risk-averse, and evaluate investments based on expected return and variance
• Single-period horizon
• All investors have the same expectations (homogeneous expectations)
• Frictionless market: no taxes, no transaction costs, unlimited borrowing/lending at Rf
• All investors hold diversified portfolios
• All investors can borrow and lend unlimited amounts at the risk-free rate

Systematic vs unsystematic risk:

• Systematic (market) risk: affects all shares — macroeconomic factors (interest rates, recession, war). Measured by beta. CANNOT be diversified away.
• Unsystematic (specific) risk: specific to a company or industry (management quality, product recall, strike). Can be eliminated by diversification (typically ~20-30 stocks suffice).

CAPM assumes investors hold diversified portfolios, so only systematic risk is compensated. Unsystematic risk earns no risk premium.

Beta (β):

• β = 1: the share moves with the market
• β > 1: more volatile than market (tech, cyclicals)
• β < 1: less volatile than market (utilities, defensive stocks)
• β = 0: return independent of market (theoretically — rare)
• β < 0: inversely correlated (very rare — some gold stocks sometimes)

Security Market Line (SML): A graphical representation of CAPM — plots expected return against beta. All fairly-priced shares lie on the SML. Shares above the SML are undervalued; below are overvalued (per CAPM theory).

CAPM advantages: Grounded in portfolio theory; applies to any share (not just dividend-payers); only systematic risk is compensated. Disadvantages: Strong assumptions; beta estimates are noisy and backward-looking; the "market portfolio" is unobservable in theory; single-period limitation.

The cost of debt (Kd) is the return lenders require. Because interest is tax-deductible, the company's effective cost is the after-tax cost: Kd × (1 − T), where T = corporation tax rate.

1. Irredeemable debt (never repaid — e.g., perpetual bonds, irredeemable preference shares):

Example: £100 nominal value bond paying 8% (£8 coupon), currently trading at £110. Tax rate 20%.
Kd (pre-tax) = £8 / £110 = 7.27%. Kd (after tax) = 7.27% × (1 − 0.20) = 5.82%.

2. Redeemable debt (repaid at a future date):

Requires calculating the yield to maturity (YTM) — the IRR that equates the bond's current market price with the PV of all future cash flows (coupons + redemption).

Cash flows:

• t=0: −Market value (price paid)
• t=1 to n: Interest (coupons × (1−T) if using after-tax cost)
• t=n: +Redemption value (typically nominal, but can be at a premium)

Calculate IRR by interpolation between two discount rates.

Example: £100 nominal bond with 6% coupon, 5 years to redemption at par, market value £95. Tax rate 20%.
Annual after-tax coupon = £6 × 0.80 = £4.80. Redemption £100 (no tax effect — capital).

Try 6% (higher than current yield):
PV of coupons: £4.80 × AF(5, 6%) = £4.80 × 4.212 = £20.22
PV of redemption: £100 × 0.747 = £74.73
Total: £94.95

Try 6% gives roughly £95, very close to market. So after-tax Kd ≈ 6%.

Note: Two conventions exist. Method 1: calculate IRR on after-tax cash flows (above). Method 2: calculate IRR on pre-tax cash flows, then × (1-T). Slightly different results due to differential tax timing. Method 1 is more precise; exam usually accepts either.

3. Convertible debt:

• Calculate YTM assuming the holder will CONVERT (typically if share price at conversion > conversion price) or REDEEM (if not)
• Use whichever gives the higher value to the holder
• Apply tax adjustment

4. Bank loans:

• Cost = actual interest rate charged by the bank × (1 − T)
• No market value calculation needed (not traded)

5. Preference shares:

• Dividends are paid out of post-tax profits — NOT tax-deductible — so NO tax adjustment
• Kp = Annual preference dividend / Market value of preference shares
• For redeemable preference shares: YTM-style calculation, no tax adjustment

Most companies use a mix of debt and equity. The WACC is the weighted average of the costs of each source, weighted by their market values.

Where:

• E = Market value of equity (share price × number of shares)
• D = Market value of debt
• Ke = cost of equity
• Kd = pre-tax cost of debt
• T = corporation tax rate

For more capital sources (e.g., preference shares, convertible bonds), add additional weighted terms.

Example: A company has equity market value £50m (Ke = 10%) and debt market value £20m (pre-tax Kd = 6%). Tax rate 25%.

• After-tax Kd = 6% × (1 − 0.25) = 4.5%
• WACC = (50/70) × 10% + (20/70) × 4.5%
• WACC = 0.714 × 10% + 0.286 × 4.5%
• WACC = 7.14% + 1.29% = 8.43%

Why market values (not book values)? The cost of capital is what investors CURRENTLY require on their investment — based on current market prices, not historical book values. Book values reflect past transactions; market values reflect current economic reality.

WACC assumptions:

1. The project is similar in risk to the existing business (so the business's cost of capital is appropriate)
2. The project is small relative to the company (doesn't significantly change the capital structure)
3. The financing mix will remain constant — if the project is financed with a disproportionate amount of debt, WACC may not apply
4. The company pays tax at a given rate (tax shield applies)

When WACC is NOT appropriate:

• Project is in a different industry (different business risk) — use a project-specific discount rate based on comparator betas
• Project significantly changes gearing (e.g., a large debt-financed acquisition) — use APV or regear the WACC
• Project uses specific financing (e.g., government-subsidised loan) — use APV to capture the financing side-effects
• Company is loss-making or has complex tax position (tax shield uncertain)

Modigliani-Miller (1958, 1963) published groundbreaking papers on how capital structure affects firm value.

M&M without tax (1958) — Propositions I and II:

Proposition I: The value of a firm is INDEPENDENT of its capital structure. V(levered) = V(unlevered).

• Rationale (arbitrage argument): If two firms have identical cash flows but different gearing, any value difference would be arbitraged away by "homemade leverage" (investors borrowing/lending personally)
• Implication: There's no "optimal" gearing in a tax-free world — all gearing levels give the same firm value

Proposition II: Cost of equity RISES linearly with gearing to compensate for financial risk. WACC is CONSTANT.

Where K₀ = cost of capital if ungeared (all equity); D/E = market value gearing.

Visually: As D/E rises, Ke rises linearly; WACC stays flat.

M&M with tax (1963) — the tax shield effect:

Interest is tax-deductible. This creates a tax shield — a benefit of debt financing. The analysis changes:

• V(levered) = V(unlevered) + PV of tax shield
• For irredeemable debt: PV of tax shield = D × T (where T = corporation tax rate)
• WACC FALLS as gearing increases
• Implication: Firms should maximise gearing (100% debt)! Obviously unrealistic — the theory doesn't account for bankruptcy costs.

Formulas (with tax):

Extensions to M&M:

Financial distress / bankruptcy costs: At high gearing, probability of bankruptcy rises. Costs include:

• Direct: legal and administrative costs of insolvency (small)
• Indirect: loss of customers/suppliers, higher costs (lenders worried about repayment demand higher rates), inability to retain key staff, missed investment opportunities (larger)

Trade-off theory: Optimal capital structure balances tax shield benefits against financial distress costs. At the optimum, marginal benefit of more debt = marginal cost of more debt.

Traditional view (pre-M&M):

• WACC is U-shaped with gearing: falls at first (low gearing benefits), reaches a minimum (optimal gearing), then rises (distress costs dominate)
• Cost of equity rises more slowly than M&M predicts
• There is an optimal gearing level that minimises WACC and maximises firm value
• More intuitive than M&M without tax; consistent with observed behaviour

Pecking order theory (Myers, 1984):

• Firms don't target an optimal capital structure; they follow a hierarchy: (1) internal finance, (2) debt, (3) equity
• Driven by information asymmetry — equity issues signal overvaluation
• Gearing is a RESIDUAL outcome of financing decisions, not a target
• Explains why profitable firms (abundant retained earnings) have lower gearing

In practice: Most firms have gearing between 20% and 60% debt/capital, suggesting a mixture of theories apply. Optimal capital structure is industry-specific and reflects risk appetite, growth opportunities, and tax position.

Equity beta vs asset beta:

• Equity beta (βₑ): observed from the market — reflects BOTH business risk AND financial risk (gearing)
• Asset beta (βₐ): also called "ungeared beta" or "unlevered beta" — reflects ONLY business risk (as if the firm were 100% equity financed)

Ungearing formula (removing financial risk from equity beta):

Assumes debt beta = 0 (debt is risk-free).

Regearing formula (adding different financial risk back):

When to ungear and regear: When calculating a project-specific cost of capital. The key insight: if a project is in a DIFFERENT industry from your existing business, your company's beta doesn't reflect the project's business risk.

Approach — Pure-play method:

1. Find a comparator company (or several) operating in the project's industry (a "pure play")
2. Obtain the comparator's equity beta (observable from market data)
3. Ungear the comparator's beta using the comparator's D/E ratio → gives the asset beta for the industry
4. Regear the asset beta using YOUR company's D/E ratio → gives the equity beta appropriate to the project risk, adjusted for your own financing
5. Use this beta in CAPM to calculate the project-specific Ke
6. Calculate a project-specific WACC (using your company's financing mix)

Worked example (approach): Your company (industry A) is considering a project in industry B. A pure-play company in B has an equity beta of 1.3 and D/E of 50%. Your company has D/E of 25%. Tax rate 20%.

• Step 1 — Ungear comparator's beta: βₐ = 1.3 × [1 / (1 + 0.5 × 0.8)] = 1.3 × [1/1.4] = 0.929
• Step 2 — Regear using your D/E: βₑ = 0.929 × [(1 + 0.25 × 0.8) / 1] = 0.929 × 1.2 = 1.114
• Step 3 — Project Ke = Rf + 1.114 × (Rm − Rf)
• Step 4 — Project WACC = weighted average using this Ke and your cost of debt

Key insight: Ungearing isolates pure BUSINESS risk; regearing adds YOUR financial risk. Result is a discount rate reflecting the industry's business risk as experienced by your (differently-geared) company.

Multiple comparators: In practice, use several comparators and take the AVERAGE asset beta. This reduces noise from idiosyncratic variation.

APV is an alternative to WACC that separately values the project assuming it is all-equity-financed, and then adds financing side-effects.

Steps:

1. Calculate the "base-case" NPV — project cash flows discounted at the UNGEARED cost of capital (K₀), as if the project were financed entirely by equity
2. Add the PV of financing side-effects:
   • Tax shield on debt used: For irredeemable debt = D × T. For redeemable debt = PV of interest × T (at pre-tax Kd)
   • Issue costs: Deduct any floatation or debt issuance costs (net of tax saving, if any)
   • Subsidies: PV of any below-market-rate financing (e.g., government loans, grants)
   • Financial distress costs: Deduct expected costs if gearing is high (in theory; difficult in practice)

When to use APV instead of WACC:

• Project financing is SIGNIFICANTLY DIFFERENT from company's normal financing mix (WACC assumption violated)
• Project uses SPECIFIC financing with identifiable cash flows (e.g., government-subsidised loan)
• Capital structure will CHANGE over the project's life (e.g., debt gradually repaid)
• Analysing a leveraged buyout (LBO)

APV advantages:

• Transparently separates operating decision (base-case NPV) from financing decisions (tax shields, etc.)
• Flexible — easy to incorporate specific financing features
• Works when capital structure changes over time

APV disadvantages:

• Requires an estimate of the UNGEARED cost (K₀), which must come from ungeared betas of comparable firms — introduces its own measurement issues
• More complex than WACC approach

K₀ (ungeared cost of equity / cost of equity if ungeared):

• Also called "cost of equity in an all-equity firm" or "asset return"
• From CAPM: K₀ = Rf + βₐ × (Rm − Rf) where βₐ = ungeared (asset) beta
• Represents the return required by investors on the BUSINESS risk alone, before adding financial risk

Worked example overview: A project has a 5-year life with annual cash flows of £200k. Base-case NPV (discounted at K₀ = 10%) = £758k − £600k initial investment = £158k. If financed with £300k of debt at 6% (pre-tax), tax rate 20%:

• Annual tax shield = £300k × 6% × 20% = £3,600
• PV of tax shield (5 years at 6% — pre-tax cost of debt) = £3,600 × AF(5, 6%) = £3,600 × 4.212 = £15,163
• APV = £158k + £15k = £173k

The project adds £173k of value — higher than the base-case NPV because of the tax shield benefit of using debt.

Decision framework for choosing the discount rate:

Common exam scenarios:

Scenario 1 — Normal capital investment: Company considering a standard capital investment within its existing business. Use the company's current WACC.

Scenario 2 — Diversification: Company expanding into a new industry. Ungear a comparator company's beta; regear using your own gearing; calculate project-specific Ke and WACC.

Scenario 3 — Debt-financed acquisition: Large acquisition changing the company's capital structure. APV approach: calculate base-case NPV at K₀, add tax shield from new debt, deduct issue costs.

Scenario 4 — Government-subsidised loan: Project qualifies for below-market-rate loan. APV: calculate base-case NPV at K₀; add PV of tax shield on the actual debt used; add PV of the subsidy (difference between market-rate interest and subsidised rate × (1−T) × loan amount, discounted).

Practical considerations:

• WACC is easier for standard projects and widely understood; executives can relate to it
• APV is more flexible but requires more data (K₀, all financing side-effects)
• For exam purposes: if the question specifies a specific financing arrangement with clear cash flows, APV is often the expected method
• Always EXPLAIN your choice of discount rate in your answer