Factor Model (Fama-French)

Definition

A factor model explains a portfolio’s returns through exposure to systematic factors rather than the market alone. The widely used Fama-French three-factor model extends CAPM with size and value factors.

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Core Ideas

The three factors

• Market excess return — the portfolio’s return less the risk-free rate.
• SMB (Small Minus Big) — the size factor: small-market-cap firms have historically generated higher returns than large-caps.
• HML (High Minus Low) — the value factor: value stocks with high book-to-market ratios have historically outperformed the market.

Together these three factors explain a large share of cross-sectional return variation that a single market-beta model misses.

Arbitrage Pricing Theory (APT)

Factor models generalize as Arbitrage Pricing Theory. Excess returns R of N stocks decompose as

$R=Xb+u$

• X — N × F matrix of factor exposures (a.k.a. factor loadings): each stock’s regression sensitivity to a factor (its market beta, its SMB sensitivity, its HML sensitivity). Often normalized to mean 0, std 1 across the universe.
• b — F-vector of factor returns.
• u — N-vector of stock-specific (idiosyncratic) returns.

Statistical factors (PCA)

One class of factor model needs nothing but historical returns: statistical factors extracted via Principal Component Analysis (PCA), rather than named fundamental or macro factors.

Why factor models work — and their drawback

Fundamental/macro factor models depend on investors persisting in valuing companies by the same metric — i.e. factor returns must have momentum (see Mean Reversion and Momentum). This breaks when preferences rotate: the value (HML) factor is usually positive, but growth led during the late-1990s internet bubble, 2007, and 2017–2020. Consequently factor models carry relatively long holding periods and long drawdowns through regime switches.

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Relationships

• Quantitative Trading — factor exposures as strategy signals
• Mean Reversion and Momentum — factor returns need momentum to persist
• Portfolio Risk Management — decomposing portfolio risk by factor
• Sharpe Ratio — risk-adjusted evaluation of factor returns
• Trading & Finance — parent topic