CAPM, APT, and Fama-French: From Beta to Multifactor Pricing

Asset-pricing models all ask the same question: which systematic risks deserve expected return? CAPM gives one answer, APT opens the door to many, and Fama-French turns that logic into an empirical benchmark family.

The Intuition

Latent factor methods assume the classical factor-model era is already on the table. That is efficient, but it leaves many readers without the bridge from familiar beta language to the modern factor zoo.

The bridge is:

• CAPM says one priced source of risk, the market portfolio, is enough
• APT says expected returns are linear in exposures to several systematic factors
• Fama-French provides a concrete empirical multifactor benchmark

These models are not just historical milestones. They define what later latent methods are trying to match, beat, or reinterpret.

CAPM: One Beta, One Premium

The Capital Asset Pricing Model says that the expected excess return on asset \(i\) is proportional to its market beta:

$$ \mathbb{E}[r_i^e] = \beta_i \, \mathbb{E}[r_m^e], $$

where

$$ \beta_i = \frac{\operatorname{Cov}(r_i, r_m)}{\operatorname{Var}(r_m)}. $$

The idea is elegant:

• idiosyncratic risk can be diversified away
• only covariance with the market should command a premium

In empirical work, CAPM is usually estimated through a time-series regression:

$$ r_{i,t}^e = \alpha_i + \beta_i r_{m,t}^e + \varepsilon_{i,t}, $$

where \(r_{i,t}^e = r_{i,t} - r_{f,t}\) is the excess return over the risk-free rate. Here \(\alpha_i\) is the pricing error. Under the model, expected alpha is zero.

This already introduces the key object that survives every later model: alpha as model-relative mispricing.

Why CAPM Was Not Enough

CAPM is powerful, but it struggles empirically. Portfolios sorted on size, value, momentum, and other characteristics often show average returns that market beta alone does not explain well.

This does not automatically prove CAPM false in a deep structural sense. It proves that as an empirical benchmark, one-factor pricing leaves systematic residual structure in the cross-section.

That motivates two extensions:

• add more factors
• rethink whether the factors should be pre-specified or estimated from data

APT: The Linear Pricing Logic

Arbitrage Pricing Theory drops the market-portfolio derivation and keeps a weaker conclusion: if returns are driven by a small number of common factors and arbitrage is limited, expected returns should be approximately linear in factor exposures. The approximation matters: the no-arbitrage logic is exact for well-diversified portfolios and only approximate at the individual-asset level.

Write returns as

$$ r_{i,t} = \alpha_i + \beta_i^\top f_t + \varepsilon_{i,t}, $$

where \(f_t \in \mathbb{R}^K\) are factor realizations and \(\beta_i\) are loadings. Then expected excess returns satisfy

$$ \mathbb{E}[r_i^e] \approx \beta_i^\top \lambda, $$

where \(\lambda\) is the vector of factor premia.

APT matters because it cleanly separates two concepts:

• factor realizations \(f_t\): shocks that move returns through time
• factor prices \(\lambda\): compensation investors demand for loading on those shocks

That distinction sits underneath later chapters too. A factor can explain variance without carrying much expected return. That is exactly the tension between attribution factors and priced factors in latent factor analysis.

Fama-French as an Empirical Multifactor Benchmark

Fama-French takes the multifactor idea and gives it concrete traded mimicking portfolios. The three-factor model writes

$$ r_{i,t}^e = \alpha_i + \beta_{i,M} MKT_t + \beta_{i,S} SMB_t + \beta_{i,H} HML_t + \varepsilon_{i,t}, $$

where:

• \(MKT\) is the market excess return
• \(SMB\) is small minus big
• \(HML\) is high book-to-market minus low

The five-factor extension adds profitability and investment factors, often denoted \(RMW\) and \(CMA\).

The conceptual change from CAPM is small but important:

• CAPM says one common source of priced risk
• Fama-French says several traded portfolios summarize recurring cross-sectional structure

These models are not purely statistical conveniences. They are empirical benchmark pricing models against which new anomalies and new factor estimators are judged.

The Regression View

Most empirical factor work boils down to two linked regression steps.

Time-Series Step

Estimate each asset's exposures:

$$ r_{i,t}^e = \alpha_i + \beta_i^\top f_t + \varepsilon_{i,t}. $$

Cross-Sectional Step

Check whether estimated betas explain average returns:

$$ \bar{r}_i^e = \gamma_0 + \beta_i^\top \lambda + u_i. $$

This is the generic two-pass pricing idea: estimate exposures first, then ask whether those estimated exposures explain average returns. The specific Fama-MacBeth procedure runs repeated cross-sectional regressions over time and averages the resulting risk-premium estimates for inference.

The notation varies across papers, but the point is stable:

• betas tell you what risk an asset loads on
• lambdas tell you what that risk is worth

A Worked Example

Stylized example with illustrative parameters.

Suppose a stock has:

• market beta \(1.1\)
• size loading \(0.4\)
• value loading \(0.6\)

and suppose the annualized factor premia (in decimal) are

$$ \lambda = \begin{bmatrix} 0.05 \\ 0.02 \\ 0.03 \end{bmatrix}, $$

corresponding to a 5% market premium, 2% size premium, and 3% value premium.

Then the model-implied expected annualized excess return is

$$ 1.1(0.05) + 0.4(0.02) + 0.6(0.03) = 0.055 + 0.008 + 0.018 = 0.081, $$

or 8.1%.

If the stock's realized average excess return is materially above 8.1%, the difference is alpha relative to the model. In actual empirical work, however, those betas are themselves estimated, so the second-stage pricing test inherits measurement error rather than operating on known exposures.

That one calculation is the empirical heart of decades of asset-pricing research.

Where Anomalies Enter

An anomaly is usually a return spread not well explained by the benchmark model currently in use.

• Under CAPM, value and momentum look anomalous.
• Under Fama-French 3F, some profitability and investment patterns still look anomalous.
• Under richer models, the residual menu gets smaller but never disappears completely.

This is why benchmark choice matters so much. A strategy's alpha is never absolute. It is alpha relative to a pricing model.

That is also why test assets and evaluation design matter so much. If you change the benchmark, you change what counts as unexplained.

Why This Matters for Latent Factors

Latent methods such as PCA, IPCA, RP-PCA, and adversarial SDF estimation do not replace the old factor models by decree. They challenge them along three dimensions:

• maybe the true factors are not the named portfolios we hand-constructed
• maybe exposures are time-varying and characteristic-driven
• maybe variance maximization is the wrong objective if we care about pricing

But the old models remain the benchmark language.

If you do not understand CAPM, APT, and Fama-French, you cannot clearly state what latent methods improve:

• do they explain more covariance?
• do they reduce pricing errors?
• do they produce economically cleaner risk dimensions?

In Practice

Keep this ladder in mind:

1. CAPM gives the one-factor baseline and the alpha/beta vocabulary.
2. APT gives the linear multifactor pricing logic.
3. Fama-French gives traded benchmark factors used in actual empirical work.
4. Latent methods (PCA, IPCA, RP-PCA) ask whether factors should be estimated rather than fully specified in advance.

That sequence is not obsolete history. It is the map the modern methods are built on top of.

Common Mistakes

• Treating beta as a universal measure of "risk" rather than market exposure under one model.
• Confusing factor realizations with factor premia.
• Reading alpha as a model-free truth instead of a benchmark-relative residual.
• Assuming Fama-French factors are the final word rather than a strong empirical benchmark family.
• Forgetting that a factor can explain covariance well and still be weakly priced.

Connections

This primer supports Chapter 14's shift from named factors to latent factor estimation. It connects directly to the factor-zoo primer, RP-PCA, IPCA, SDF estimation, and the later portfolio and risk chapters where factor exposures become operational objects rather than only empirical tests.