Fama-French Factor Model

Expands the Capital Asset Pricing Model (CAPM) by adding two additional factors to explain the cross-section of stock returns. The model posits that small-cap stocks and high book-to-market (value) stocks tend to outperform the market over long periods.

• Market Risk (Rm - Rf): The traditional CAPM beta, capturing the excess return of the market portfolio over the risk-free rate.
• Size (SMB - Small Minus Big): The return spread between small-capitalization and large-capitalization stocks, capturing the size premium.
• Value (HML - High Minus Low): The return spread between stocks with high book-to-market ratios (value) and low book-to-market ratios (growth), capturing the value premium.

In 2015, Fama and French augmented the model with two additional factors to capture anomalies related to profitability and investment patterns, improving explanatory power.

• Profitability (RMW - Robust Minus Weak): The return spread between companies with high operating profitability and those with weak profitability.
• Investment (CMA - Conservative Minus Aggressive): The return spread between companies that invest conservatively (low asset growth) and those that invest aggressively (high asset growth).
• Dividend Discount Model Logic: This extension is theoretically grounded in the dividend discount model, where higher expected profitability and lower required investment imply higher valuations.

The model is expressed as a linear regression where the excess return of a portfolio or asset is a function of factor exposures (betas) and factor premiums.

Equation: R_it - R_ft = α_i + β_1(R_mt - R_ft) + β_2(SMB_t) + β_3(HML_t) + ε_it

• R_it: Total return of asset i at time t.
• R_ft: Risk-free rate of return.
• α_i (Alpha): The intercept representing abnormal return unexplained by the factors. A statistically significant positive alpha indicates manager skill.
• β_1, β_2, β_3: Factor loadings or sensitivities.
• ε_it: The idiosyncratic error term.

The factors are constructed using a rigorous sorting mechanism to isolate the specific risk premiums. Portfolios are rebalanced periodically to maintain factor purity.

• Double Sorting: Stocks are first sorted by size (market cap) and then independently by the target characteristic (e.g., book-to-market ratio).
• Breakpoints: The universe is typically split using median market cap for size and 30th/70th percentiles for value, profitability, and investment.
• Intersectional Portfolios: This creates six (2x3) or more intersectional portfolios (e.g., Small-Value, Big-Growth) from which the long-short factor returns (SMB, HML) are calculated as weighted averages.

The primary application is decomposing a portfolio's historical returns to distinguish between returns derived from known risk factors and true managerial skill.

• Factor Replication: If a portfolio's returns are fully explained by the factors, its alpha is zero, suggesting it is a factor-closet rather than a source of unique alpha.
• Style Analysis: Investors use factor loadings to monitor style drift. A sudden increase in the HML loading indicates a shift toward value stocks.
• Risk Management: The model identifies unintended factor bets. A portfolio manager might hedge out SMB exposure if they have no directional view on the size premium.

While robust, the model does not fully explain all return variations. Specific anomalies persist, challenging the efficient market hypothesis.

• Momentum (WML): The tendency for stocks that have performed well in the past to continue performing well in the short term is a significant anomaly not captured by the five-factor model.
• Low Volatility Anomaly: The empirical observation that low-beta or low-volatility stocks often generate higher risk-adjusted returns than high-beta stocks contradicts the core CAPM assumption.
• Factor Decay: Critics argue that factor premiums diminish after publication due to crowding and arbitrage, though Fama and French maintain the premiums are compensation for systematic risk.

The single-factor predecessor to Fama-French that explains expected returns solely through market beta. CAPM posits that the market portfolio is mean-variance efficient and that an asset's risk premium is proportional to its covariance with the market.

• Single Factor: Expected return = risk-free rate + β(Market Risk Premium)
• Key Limitation: Fails to explain the size effect and value premium that Fama-French captures
• Historical Context: Developed by Sharpe (1964) and Lintner (1965), it remains the foundational benchmark against which multi-factor models are tested

An extension of the Fama-French three-factor model that adds a momentum factor (WML) —winners minus losers. Carhart demonstrated that stocks with strong past performance tend to continue outperforming in the near term.

• Fourth Factor: Captures the momentum anomaly that Fama-French cannot explain
• Practical Application: Widely used in mutual fund performance evaluation to control for momentum exposure
• Critique: Momentum is transaction-cost intensive and prone to crashes during market reversals

Fama and French's own 2015 expansion that adds profitability (RMW) and investment (CMA) factors to the original three. The model captures the empirical observation that profitable firms and conservative investors outperform.

• RMW (Robust Minus Weak): High operating profitability firms minus low profitability firms
• CMA (Conservative Minus Aggressive): Low asset growth firms minus high asset growth firms
• Key Insight: With these additions, the value factor (HML) becomes redundant for describing average returns

The theoretical foundation for multi-factor models, developed by Stephen Ross in 1976. APT posits that asset returns are driven by multiple unspecified systematic risk factors, each with its own risk premium.

• No-Arbitrage Principle: Mispricing is eliminated by arbitrageurs, enforcing a linear relationship between factor loadings and expected returns
• Flexibility: Unlike CAPM, APT does not require identification of the market portfolio
• Relationship to Fama-French: The three-factor model is an empirical implementation of APT using size and value as the systematic factors

A term describing the proliferation of hundreds of published factors claiming to predict returns. Research by Harvey, Liu, and Zhu (2016) suggests most factors fail to replicate after accounting for data mining and publication bias.

• Multiple Testing Problem: With enough backtesting, spurious factors emerge by chance
• Survivorship: Only the Fama-French factors and momentum survive rigorous out-of-sample testing
• Implication: Portfolio managers must apply higher statistical thresholds (t-stat > 3.0) when evaluating new factors

The unifying mathematical framework underlying all asset pricing models, including Fama-French. The SDF is a random variable that discounts future payoffs to present value while incorporating risk adjustments.

• Fundamental Equation: Price = E[SDF × Future Payoff]
• Factor Model Connection: In linear factor models, the SDF is expressed as a linear combination of factors: SDF = a + b₁(SMB) + b₂(HML) + b₃(Mkt-Rf)
• No-Arbitrage Condition: A positive SDF ensures the absence of arbitrage opportunities in the pricing kernel