What is a Factor: From CAPM to Fama–French

Layer 1: The Intuition

1.1 Where does the taste of a soup come from?

Is a soup very salty because of the salt and soy sauce added, or because of the chef's special skill? If you know how much salt and soy sauce were put into the soup, you can estimate how much the "skill" contributed.

A fund earns 20%:

• If the market rose 18% that year, and the stocks in this fund are inherently more volatile than the market, then most of that 20% is "salt" (market exposure);
• If it primarily buys small-cap stocks, and small-cap stocks generally outperformed that year, that is another kind of "soy sauce" (size factor exposure);
• After deducting these, what remains is the "skill" (alpha).

A factor model is a tool to decompose returns into "ingredients" and "skill."

1.2 Why this matters

Layer 2: Evolution of Factor Models

2.1 CAPM: Only one factor

The Capital Asset Pricing Model (Sharpe, Lintner, etc., 1960s) posits that:

$$E[R_i] - R_f = \beta_i \big(E[R_m] - R_f\big),$$

The excess return of an asset depends only on its exposure $\beta_i$ to the market. High beta stocks have higher risk, and thus higher expected returns.

The problem: In empirical studies, many phenomena cannot be explained by CAPM. For example, small-cap stocks and low price-to-book ("cheap") stocks have long-term average returns that are higher than what CAPM predicts.

2.2 Fama–French Three-Factor Model (1993)

$$R_i - R_f = \alpha_i + \beta_i\,\text{MKT} + s_i\,\text{SMB} + h_i\,\text{HML} + \varepsilon_i$$

2.3 More factors

2.4 Where does the factor premium come from?

This is one of the most debated issues in academia:

For example, the value factor performed poorly for a long time around 2010, triggering intense discussions on whether "the value factor is dead."

Layer 3: Perform a Factor Regression

3.1 Performing a factor regression with simulated data

We will first "create" a portfolio: we know its true exposures are Market 1.1, Size 0.5, and Value −0.3, with a monthly alpha of 0.1%, plus random noise. Then, we will use regression to see if we can recover these numbers.

import numpy as np, pandas as pd, statsmodels.api as sm
rng = np.random.default_rng(42)
n = 120  # 10 years of monthly data
f = pd.DataFrame({
    "MKT": rng.normal(0.006, 0.045, n),
    "SMB": rng.normal(0.002, 0.030, n),
    "HML": rng.normal(0.003, 0.030, n),
})
true_alpha, true_beta = 0.001, {"MKT": 1.1, "SMB": 0.5, "HML": -0.3}
excess = true_alpha + sum(true_beta[k] * f[k] for k in f) + rng.normal(0, 0.02, n)

X = sm.add_constant(f)
res = sm.OLS(excess, X).fit()
print(res.params.round(3).to_dict())
print(res.tvalues.round(2).to_dict())
print("R2", round(res.rsquared, 3))

Output:

{'const': 0.002, 'MKT': 0.999, 'SMB': 0.459, 'HML': -0.375}
{'const': 0.8, 'MKT': 18.5, 'SMB': 7.39, 'HML': -6.1}
R2 0.803

3.2 How to read these results

The last line is the most important lesson: alpha is hard to prove. Factor exposures (beta) are usually estimated accurately because the factors themselves have high volatility; whereas alpha is a very small constant buried in a large amount of noise. Roughly speaking, the t-value of alpha grows with the square root of the sample length; a true but tiny alpha might require decades of data to confirm.

3.3 Engineering details for practical use

• Factor Data: Professor Kenneth French's website provides free factor return data for US and global markets, which is the standard source for academic research;
• Details in factor construction are important: How "small-cap" is defined, how often rebalancing occurs, and whether financial stocks are excluded will all affect the results;
• Time-varying exposure: A fund manager's style changes; use rolling window regression to observe changes in beta;
• Standard Errors: When return series exhibit autocorrelation and heteroscedasticity, use robust standard errors such as Newey–West.

res_robust = sm.OLS(excess, X).fit(cov_type="HAC", cov_kwds={"maxlags": 3})

Summary