Cost of Equity and CAPM

Definition of required return on equity
Cost of Equity — the return required by shareholders for investing in a company. This is the opportunity cost of capital and a critical input for WACC and valuation. CAPM is the dominant model for estimating the cost of equity.

Concept of Cost of Equity

Required return: shareholders require a return compensating for the risk of investment. Higher risk — higher required return. Cost of equity is the minimum return a company must earn to satisfy equity investors.

Opportunity cost: investors can invest elsewhere. Cost of equity reflects the return of alternative investments of comparable risk.

Not directly observable: Unlike the cost of debt (coupon rate, yield), the cost of equity is not directly observable. Estimation is required.

CAPM: Capital Asset Pricing Model

CAPM formula:
$Re = Rf + \beta \times (Rm - Rf)$.

Where $Re$ is the cost of equity, $Rf$ is the risk-free rate, $\beta$ is beta (systematic risk), $(Rm - Rf)$ is the equity risk premium.

Intuition: expected return = risk-free return + premium for bearing systematic risk. The premium is proportional to the amount of systematic risk (beta).

Risk-free rate ($Rf$)

Theoretical: return on a zero-risk investment.

In practice: government bonds of the issuer’s currency (US Treasuries for USD, Bunds for EUR).

Maturity matching: use $Rf$ matching the project/valuation horizon. For long-term valuation, 10-year or 20-year government bonds.

Current rate vs historical average: use the current rate for forward-looking analysis. The historical average may smooth short-term distortions.

Beta ($\beta$)

Systematic risk measure: beta indicates the sensitivity of stock returns to market returns.
$\beta = \text{Covariance}(\text{stock}, \text{market}) / \text{Variance}(\text{market})$.

Interpretation:
$\beta = 1$: stock moves with the market;
$\beta > 1$: more volatile than the market (amplifies movements);
$\beta < 1$: less volatile than the market (dampens movements).

Estimation: regress historical stock returns against market index returns. The slope of the regression = beta. Bloomberg, Reuters provide estimated betas.

Issues: beta is unstable over time, depends on estimation period, frequency, index choice.

Adjusted beta: $(2/3 \times \text{raw beta}) + (1/3 \times 1)$ — mean-reversion adjustment.

Equity Risk Premium $(Rm - Rf)$

ERP: excess return investors expect from equities over the risk-free rate. Compensation for bearing equity market risk.

Historical approach: average historical excess return of stocks over bonds. US long-term: ~5-7% (depends on period, methodology).

Forward-looking approach: implied ERP from current market prices using DDM. If market P/E is high, implied ERP is lower.

Current practice: practitioners use 4-6% ERP for developed markets. Emerging markets add a country risk premium.

CAPM Example

$Rf = 3%$, $\beta = 1.2$, $ERP = 5%$.

Cost of equity $= 3% + 1.2 \times 5% = 9%$.

Interpretation: investors require 9% return to invest in this stock. 3% for time value (risk-free), 6% for systematic risk.

CAPM Limitations

Assumptions: investors are rational, markets are frictionless, single-period horizon, investors hold the market portfolio. Reality differs.

Single factor: CAPM uses only market risk. Research shows other factors matter (size, value, momentum). Multi-factor models (Fama-French) address this.

Beta estimation: historical beta may not predict the future. Industry average or comparable company beta is often used.

Alternative Approaches

Build-up method: $Rf + ERP +$ Size premium $+$ Company-specific risk. For private companies without tradable beta.
Dividend Growth Model (DDM): $P = \dfrac{D_1}{Re-g}$, solve for $Re = \dfrac{D_1}{P} + g$. Works if dividends are stable, growth is predictable.
Bond yield plus risk premium: $Re = \text{company's bond yield} +$ equity premium (3-5%). Simple approximation.

Unlevered and Relevered Beta

Observed beta includes the effect of financial leverage. To isolate business risk:
unlevered beta $= \dfrac{\beta}{1 + (1-T) \times D/E}$.

Applying to a different capital structure: relever using target $D/E$.
$\beta_{levered} = \beta_{unlevered} \times (1 + (1-T) \times D/E)$.

Industry beta: unlever peer betas, average, relever for the target company. Useful for private companies or changing capital structure.

Practical Considerations

Sensitivity: small changes in assumptions ($Rf$, ERP, beta) significantly impact the cost of equity and valuation.
Cross-check: compare estimated cost of equity to industry norms, company’s historical returns, alternative methods.
Document assumptions: clearly state $Rf$, ERP, beta source/methodology. Enables reviewers to assess and challenge.