Factor Models: CAPM, APT, and Fama-French

Setup

Context and Assumptions

Factor models decompose asset returns into systematic exposures to common risk factors and an idiosyncratic residual. They are used on every quant equity desk — for risk attribution, portfolio construction, and alpha signal design.

The central question factor models answer is: why do different assets earn different expected returns? The answer, in all factor frameworks, is compensation for bearing systematic risk that cannot be diversified away.

Notation throughout. Let:

• $R_{i,t}$ = excess return of asset $i$ at time $t$ (return minus risk-free rate $r_f$)
• $\mathbb{E}[R_i]$ = unconditional expected excess return of asset $i$
• $\beta_{i,k}$ = factor loading (sensitivity) of asset $i$ to factor $k$
• $\lambda_k$ = risk premium for factor $k$ (expected excess return per unit of factor exposure)
• $\varepsilon_{i,t}$ = idiosyncratic return; $\mathbb{E}[\varepsilon_{i,t}] = 0$, uncorrelated with factors

Key assumptions that vary by model are stated in each section.

Theory

1. CAPM: Capital Asset Pricing Model

Assumptions.

1. Investors are mean-variance optimisers (Markowitz 1952) with identical beliefs.
2. All assets are tradeable; no transaction costs, taxes, or short-selling constraints.
3. Returns are jointly normally distributed (or investors have quadratic utility).
4. A risk-free asset exists, lendable and borrowable at rate $r_f$.
5. All investors have the same investment horizon.

Under these assumptions, every investor holds the same risky portfolio — the market portfolio $M$, which in equilibrium is the value-weighted portfolio of all risky assets.

Derivation of the SML. Consider any asset $i$. Form a portfolio with weight $\alpha$ in asset $i$ and $(1-\alpha)$ in the market portfolio. Expected excess return and variance:

$\mathbb{E}[R_p] = \alpha \mathbb{E}[R_i] + (1-\alpha)\mathbb{E}[R_M]$ $\sigma_p^2 = \alpha^2 \sigma_i^2 + 2\alpha(1-\alpha)\sigma_{iM} + (1-\alpha)^2 \sigma_M^2$

In equilibrium, asset $i$ is already in the market portfolio, so the efficient portfolio locus through $(i, M)$ must be tangent to the Capital Market Line at $\alpha = 0$. Computing $d\mathbb{E}[R_p]/d\sigma_p$ at $\alpha = 0$ and equating to the Sharpe ratio of the market gives:

$\mathbb{E}[R_i] = \beta_i \, \mathbb{E}[R_M], \qquad \beta_i = \frac{\text{Cov}(R_i, R_M)}{\text{Var}(R_M)} = \frac{\sigma_{iM}}{\sigma_M^2}.$

This is the Security Market Line (SML). Assets plot on the SML in equilibrium; deviations from it are alphas — excess returns not explained by market beta.

Time-series regression form (Jensen 1968):

$R_{i,t} = \alpha_i + \beta_i R_{M,t} + \varepsilon_{i,t}.$

Under CAPM, $\alpha_i = 0$ for all assets in equilibrium.

2. APT: Arbitrage Pricing Theory

Assumptions (Ross 1976).

1. Returns are generated by a K-factor linear model: $R_{i,t} = \mathbb{E}[R_i] + \sum_{k=1}^K \beta_{i,k} F_{k,t} + \varepsilon_{i,t},$ where $F_{k,t}$ are zero-mean factor realisations and $\varepsilon_{i,t}$ are idiosyncratic, mutually uncorrelated, with bounded variance.
2. There are sufficiently many assets to form well-diversified portfolios.
3. No-arbitrage: no portfolio with zero cost, zero systematic risk, and positive expected return.

Result. In a no-arbitrage economy, expected returns satisfy (approximately):

$\mathbb{E}[R_i] = \lambda_0 + \sum_{k=1}^K \beta_{i,k} \lambda_k,$

where $\lambda_0 = r_f$ (zero-beta return) and $\lambda_k$ is the risk premium for factor $k$. The APT does not specify which factors matter — it only says that if factors explain covariance structure, their premia must exist to preclude arbitrage.

CAPM is a special case of APT with $K=1$ and $F_1 = R_M$, plus the equilibrium assumptions that fix $\lambda_1 = \mathbb{E}[R_M]$.

3. Fama-French Three-Factor Model

Motivation. Fama and French (1992, 1993) documented that CAPM beta does not fully explain the cross-section of expected returns. Two anomalies survive controlling for market beta:

• Size effect: small-cap stocks earn higher average returns than large-cap.
• Value effect: stocks with high book-to-market (B/M) ratio earn higher average returns than growth stocks.

Factor construction (Fama-French 1993).

Let $SMB_t$ (Small Minus Big) and $HML_t$ (High Minus Low) be zero-cost long-short factor portfolios constructed monthly:

• SMB: long bottom 50% of stocks by market cap, short top 50%.
• HML: long top 30% of stocks by B/M, short bottom 30%; sort within size buckets to control for size.

The three-factor model:

$R_{i,t} = \alpha_i + \beta_i^M R_{M,t} + \beta_i^{SMB} \, SMB_t + \beta_i^{HML} \, HML_t + \varepsilon_{i,t}.$

Under the model, $\alpha_i = 0$ in equilibrium. Empirically, $\alpha_i$ is close to zero for most equity portfolios but non-zero for momentum strategies — motivating Carhart (1997) to add a momentum factor $MOM$.