CAPM vs Fama-French Three-Factor Model: Worked Examples for Equity Cost of Capital

CAPM (Capital Asset Pricing Model): E(R) = Rf + β × (Rm - Rf) Where: - E(R) = expected return on the asset (cost of equity) - Rf = risk-free rate (typically 10-year Treasury) - β = asset's beta relative to the market - Rm = expected return on the market portfolio - (Rm - Rf) = market risk premium (MRP), typically 5-7% Fama-French Three-Factor Model (1993): E(R) = Rf + βMKT × MKT + βSMB × SMB + βHML × HML Where: - Rf = risk-free rate - MKT = market risk premium (same as in CAPM) - SMB (Small Minus Big) = return of small stocks minus return of big stocks — premium for size - HML (High Minus Low) = return of high book-to-market (value) stocks minus return of low book-to-market (growth) stocks — premium for value - βMKT, βSMB, βHML = the asset's loading on each factor, obtained from time-series regression Critical difference: CAPM assumes only market risk is priced. Fama-French recognizes that size and value factors explain additional returns not captured by market beta alone. Historical factor premiums (1927-2025, US data, approximate annualized): - MKT: 6-8% (market risk premium) - SMB: 2-4% (size premium) - HML: 3-4% (value premium) When the premiums are all positive, a small-cap value stock would have higher expected return than a large-cap growth stock — reflecting more historical risk/return. This content is for educational purposes only and does not constitute financial advice.

CAPM was developed by Sharpe, Lintner, and Mossin in the 1960s. It's the foundational asset pricing model. Key assumptions: - Investors hold the market portfolio (diversified, includes all risky assets) - Only systematic (non-diversifiable) risk is priced - All idiosyncratic risk can be diversified away - Investors have homogeneous expectations - No transaction costs or taxes - Investors can borrow/lend at the risk-free rate Using CAPM in practice: Step 1 — Get the risk-free rate. Use the current 10-year Treasury yield for long-term investments. In 2026: approximately 4.3-4.7% depending on market conditions. Step 2 — Estimate the market risk premium. Use historical MRP of 5-7% (US equity market has produced roughly 6-7% above T-bills over long history). Some analysts use 5%, others 7%. Be consistent. Step 3 — Determine beta. Public stocks: use published beta from financial data providers (Bloomberg, Yahoo Finance). Regression-estimated beta from 5 years of monthly returns is standard. Private companies: use 'pure-play' beta from similar public companies, lever it for your company's capital structure. Step 4 — Apply the formula. Example: Beta = 1.2, Rf = 4.5%, MRP = 6.5% Cost of equity = 4.5% + 1.2 × 6.5% = 4.5% + 7.8% = 12.3% Strengths of CAPM: - Simple and intuitive - Uses single factor, easy to explain - Widely accepted in corporate finance practice - Clear theoretical foundation (modern portfolio theory) - Good starting point for most analysis Weaknesses of CAPM: - Empirical tests show mixed support — some premium for size (small caps outperform) and value (cheap stocks outperform) not captured by CAPM - Single-factor world is unrealistic for many stocks - Beta estimates can be unstable over time - Assumes efficient markets which may not hold in reality - May systematically underestimate required returns for some categories of stocks Where CAPM often produces reasonable results: - Large-cap, blue-chip stocks - Mid-cap stocks with stable operations - Diversified portfolios - Long-term investment horizons where short-term anomalies wash out Where CAPM may underestimate required returns: - Small-cap stocks (size effect) - Value stocks (high book-to-market) - Stocks with high financial distress risk - Highly leveraged companies - Emerging market exposure - Highly illiquid securities

Fama and French (1993) observed that actual stock returns were systematically higher for: (1) small-cap vs large-cap stocks, and (2) value vs growth stocks. They added factors to CAPM to capture these effects. The SMB (Small Minus Big) factor: - Size premium: small-cap stocks historically earn 2-4% more per year than large-cap stocks - Economic rationale: small companies have higher business risk, less diversification, less access to capital, more bankruptcy risk - Beta on SMB measures the stock's exposure to the size factor - Small-cap stock: high positive βSMB (typically 0.5-1.5) - Large-cap stock: low or negative βSMB (typically -0.5 to 0.5) The HML (High Minus Low) factor: - Value premium: high book-to-market stocks (value) historically outperform low book-to-market stocks (growth) by 3-4% per year - Economic rationale: value stocks have higher distress risk, lower growth prospects perceived by market, more cyclical exposure - Beta on HML measures value vs growth exposure - Value stock: positive βHML (typically 0.5-1.2) - Growth stock: negative βHML (typically -0.3 to -1.2) How to obtain factor loadings: 1. Gather monthly returns for the stock over 5+ years 2. Obtain MKT, SMB, HML factor returns from Kenneth French's data library (accessible via university websites or direct from Fama-French research) 3. Run time-series regression: Stock_Return_t - Rf_t = α + βMKT×MKT_t + βSMB×SMB_t + βHML×HML_t + ε_t 4. Extract the three betas (βMKT, βSMB, βHML) Example calculations: Large-cap growth stock (e.g., Microsoft): βMKT = 1.0, βSMB = -0.3, βHML = -0.5 E(R) = 4.5% + 1.0 × 6.5% + (-0.3) × 2.5% + (-0.5) × 3.5% E(R) = 4.5% + 6.5% - 0.75% - 1.75% = 8.5% Compared to CAPM with beta 1.0: CAPM would estimate 11.0% (4.5% + 1.0 × 6.5%). Fama-French is 2.5% lower, reflecting the negative loadings on size and value. Small-cap value stock: βMKT = 1.1, βSMB = 1.0, βHML = 0.8 E(R) = 4.5% + 1.1 × 6.5% + 1.0 × 2.5% + 0.8 × 3.5% E(R) = 4.5% + 7.15% + 2.5% + 2.8% = 16.95% Compared to CAPM with beta 1.1: CAPM would estimate 11.65%. Fama-French is 5.3% higher, reflecting the positive loadings on size and value. Mid-cap core stock: βMKT = 1.0, βSMB = 0.3, βHML = 0.2 E(R) = 4.5% + 1.0 × 6.5% + 0.3 × 2.5% + 0.2 × 3.5% E(R) = 4.5% + 6.5% + 0.75% + 0.7% = 12.45% Compared to CAPM with beta 1.0: CAPM would estimate 11.0%. Fama-French is 1.45% higher — a small difference.

The difference between the two models is larger for stocks with specific characteristics: Large growth stocks (Microsoft, Apple, Alphabet): - Strong CAPM beta (1.0-1.5) - Negative SMB loading (large-cap) - Negative HML loading (growth) - Fama-French estimate is SIGNIFICANTLY LOWER than CAPM - Many investment banks use CAPM for these companies (simpler, and the difference isn't enormous) - In practice, analysts often apply company-specific adjustments Small value stocks (regional banks, cyclical industrials): - Beta typically 1.1-1.4 - Large positive SMB loading - Large positive HML loading - Fama-French estimate is SIGNIFICANTLY HIGHER than CAPM - CAPM materially underestimates required return - Recommended to use Fama-French for cost of capital Mid-cap blend (many S&P 400 stocks): - Beta typically 0.9-1.2 - Moderate SMB loading (small positive) - Moderate HML loading (near zero or small) - Fama-French and CAPM produce similar results (within 1%) - Either model works reasonably Real estate investment trusts (REITs): - Beta typically 0.6-0.9 (lower than market) - Positive HML loading (often high BV/MV) - Size varies - Fama-French often produces meaningfully different (usually higher) estimates due to HML loading Utility and consumer staples: - Low beta (0.4-0.7) - Large-cap usually (negative SMB) - Variable HML loading - Fama-French and CAPM often produce similar estimates because low beta dominates Tech stocks specifically: - Beta often >1.3 - Usually large-cap (negative SMB) - Growth-oriented (negative HML) - Net effect: CAPM often overestimates, Fama-French adjusts downward (sometimes materially) The 2015 five-factor model: Fama and French (2015) added two more factors: profitability (RMW) and investment (CMA). Five-factor model: E(R) = Rf + βMKT×MKT + βSMB×SMB + βHML×HML + βRMW×RMW + βCMA×CMA RMW (Robust Minus Weak profitability): profitability premium. CMA (Conservative Minus Aggressive investment): investment premium. The five-factor model often provides even better fit to actual returns, especially for companies with extreme profitability or investment patterns. Most investment practitioners still use three-factor because the additional factors' empirical support is more mixed.

When to use CAPM: - Introductory or educational contexts - Large-cap stocks with diversified exposure - Portfolio-level applications where individual stock errors cancel out - Simple relative valuation (not trying to generate absolute price targets) - Quick back-of-envelope estimates - When the stakeholder values simplicity over precision When to use Fama-French: - Small-cap or mid-cap stocks - Value stocks specifically - Emerging markets with different factor structures - Private equity or VC valuation of cheap-multiple companies - Investment-grade analysis where precision matters - Academic or research contexts where factor sensitivities are the point of analysis When to use five-factor or more: - Highly specialized stocks (high profitability companies) - Factor-specific analysis - Academic portfolio attribution - Empirical asset pricing tests Hybrid approaches in practice: 1. CAPM baseline + sector adjustments: many practitioners start with CAPM and add sector-specific premiums or discounts based on their judgment. 2. Build-up method: risk-free rate + market premium + size premium (from Duff & Phelps or Morningstar) + specific company risk premium. Essentially a simplified multi-factor approach. 3. Multiple model triangulation: calculate cost of equity using CAPM, Fama-French, and build-up method. Present a range rather than a single number. This acknowledges the uncertainty in estimation. Calculation examples using Finance Professional's data: Scenario: Estimating cost of equity for a small-cap value financial services company with: - Historical CAPM beta: 1.2 - Fama-French βMKT = 1.15, βSMB = 0.85, βHML = 0.92 - Rf = 4.5%, MRP = 6.5%, SMB premium = 2.5%, HML premium = 3.5% CAPM estimate: 4.5% + 1.2 × 6.5% = 12.30% Fama-French estimate: 4.5% + 1.15 × 6.5% + 0.85 × 2.5% + 0.92 × 3.5% = 4.5% + 7.48% + 2.13% + 3.22% = 17.33% The 5% gap is material for valuation. If you discount this company's future cash flows at 12.3%, you'll produce a valuation roughly 25-30% higher than discounting at 17.3%. In a transaction context, this is a meaningful divergence. Sources of factor data: - Kenneth French data library (ken.french.dartmouth.edu): free factor time-series for US and international markets - Ibbotson/Duff & Phelps data: commercial factor premiums with specific size buckets - WRDS (Wharton Research Data Services): academic access with detailed factor data - Morningstar Stocks: retail access to factor analysis This content is for educational purposes only and does not constitute financial advice.

Both models have critics. Understanding the limitations helps you interpret results appropriately. Criticisms of CAPM: 1. Empirical tests show the CAPM relationship (beta vs return) is weak or non-existent for many time periods. 2. Single-factor model ignores observable systematic return drivers (size, value, profitability). 3. Beta estimates are unstable — a stock's beta can change by 30-50% over a 5-year rolling window. 4. Assumes stock returns follow normal distribution — they don't (fat tails, skewness). 5. Market portfolio definition is imprecise (S&P 500? Global? Includes bonds?). 6. Domestic CAPM may underestimate required returns in global portfolios due to currency and country risk. 7. Long-term MRP estimates vary significantly between sources (3% to 8%). Criticisms of Fama-French: 1. Factor definitions are specific to the data set used (e.g., US only; international factors differ). 2. Size premium has been weaker in recent decades — some argue it may be disappearing. 3. Value premium has also been less pronounced in recent decades. 4. Factor loadings must be estimated, adding another source of error. 5. Three factors may be insufficient (hence five-factor and newer models). 6. Explanatory power varies widely by stock — some stocks fit the model well, others don't. 7. Assumes the factors are priced across all time periods — empirical evidence is mixed. Both models have limitations: - Both assume stock returns are driven primarily by systematic risk factors. - Neither captures company-specific idiosyncratic risk that drives most of the actual return variation for individual stocks. - Both require estimation of betas and factor premiums, introducing uncertainty. - Both ignore taxes and transaction costs. Practicing analysts often: 1. Use CAPM or Fama-French as a starting point, not a final answer. 2. Apply sector-specific adjustments based on judgment. 3. Consider multiple valuation methods (DCF with CAPM, multiples-based, precedent transactions) and triangulate. 4. Sensitivity test the cost of capital assumption (±1-2% typically). 5. Present the range of valuations rather than a single point estimate. 6. Explicitly note the cost-of-capital assumptions in their analysis. The 'right' cost of capital depends on purpose. For internal decision-making (capital budgeting), a single estimate works. For valuation (transactions, fairness opinions), a range is more defensible. This content is for educational purposes only and does not constitute financial advice. Cost-of-capital decisions for investment should involve consultation with a licensed financial professional.