The Fama–French factor models

The lesson "CAPM, beta and the security market line" said that a single number, the market beta, should explain every asset's expected return, and "Factor models and systematic risk" showed how one common factor splits risk into a systematic and an idiosyncratic part. This lesson asks what happens when the market alone is not enough. It builds the Fama–French factors from sorted portfolios, runs the regressions that use them, and shows how to read the loadings, the alpha and the R² that come out.

Size and value anomalies under CAPM

Test CAPM on a stock or a portfolio by regressing its monthly excess return $R_i - R_f$ on the market's excess return $R_M - R_f$:

If CAPM holds, the intercept $\alpha_i$ is zero: the market beta accounts for the whole average return. In US data from the 1960s onwards two groups of portfolios break this. Portfolios of small companies earned more than their betas justify, and so did portfolios of value stocks, firms with a high ratio of book equity to market value. Growth stocks, with a low ratio, earned less. These positive and negative alphas are called the size and value anomalies. Fama and French (1993) responded by adding the returns that size and value themselves earn as extra factors.

Building factors from sorted portfolios

A factor here is the return of a long–short portfolio. Each June, sort all stocks by market capitalisation into small and big, and separately by book-to-market into growth (bottom 30%), neutral and value (top 30%). The intersections give six value-weighted portfolios. Two factors come out of them:

SMB (small minus big) buys small firms and sells big ones while holding the value mix fixed; HML (high minus low) buys value and sells growth while holding the size mix fixed. Because each is a difference of two portfolios, it costs nothing to hold in theory, and its average return is a premium per unit of exposure. The market factor MKT is $R_M - R_f$.

The five-factor model of 2015 adds two more built the same way. RMW (robust minus weak) is long firms with high operating profitability and short those with low profitability. CMA (conservative minus aggressive) is long firms that grow their assets slowly and short those that invest heavily. Carhart (1997) added momentum, UMD (up minus down), long last year's winners and short its losers; it is often appended to either model.

The three-factor and five-factor regressions

The Fama–French regressions are time-series regressions, one per fund or portfolio, with the factor returns as regressors. The three-factor model (FF3) is

and the five-factor model (FF5) adds $r_i\,\text{RMW}_t + c_i\,\text{CMA}_t$. Taking expectations and using $E[\varepsilon] = 0$ gives the model's prediction for the average excess return:

The model holds when $\alpha_i = 0$: then the average return is just the sum of each loading times its factor premium, exactly as CAPM's line had one beta times one premium.

Reading the loadings

Each loading is a sensitivity holding the other factors fixed. $b_i$ is close to the CAPM beta but usually not equal to it, because SMB and HML now absorb part of the co-movement. A positive $s_i$ means the fund behaves like small stocks, a negative one like large stocks; a positive $h_i$ means a value tilt and a negative one a growth tilt. A loading of 0.5 on SMB says that when small firms beat big ones by 1% in a month, the fund tends to gain 0.5% more than its market exposure alone would give. In the five-factor model, a positive $r_i$ points to profitable firms and a positive $c_i$ to firms that invest conservatively.

Alpha and its t-statistic

Alpha is the average return the factors do not explain. It is estimated from a finite sample, so it comes with a standard error, and the question is whether it differs from zero by more than noise. The usual test is the t-statistic

and $|t_\alpha| > 2$ is the rough threshold for a significant alpha at the 5% level. The standard error shrinks like $1/\sqrt{T}$ in the number of months $T$, so a given alpha becomes easier to detect in a longer sample.

Why does adding factors shrink alpha? Suppose the true model is FF3, but you fit CAPM. The omitted terms $s_i\,\text{SMB} + h_i\,\text{HML}$ then sit in the residual, and their mean ends up in the intercept:

A small-value fund therefore shows a positive CAPM alpha even if its manager has no skill: it is collecting the size and value premia. Once those factors enter the regression, that part of the alpha moves into the loadings. (The approximation ignores the correlation between the market and the omitted factors, which shifts the beta a little too.)

The figure simulates 240 months of factor returns and a fund whose true alpha is zero, with market loading 1, RMW loading 0.3, CMA loading 0, and size and value loadings set by the sliders. Start at the defaults, $s = 0.5$ and $h = 0.4$, with CAPM: the fitted alpha is 4.00% a year with a t-statistic of 2.08, apparently significant skill, and R² is 0.746. Switch to FF3: alpha falls to 1.28% (t = 1.04) and R² rises to 0.899, while the size and value bars land near their true marks. Switch to FF5: the remaining alpha, which was the profitability premium, drops to 0.29% (t = 0.26), R² reaches 0.919, and the RMW bar finds its loading of 0.3. Then set both tilts to zero and see the CAPM alpha shrink, or lengthen the sample and watch the bars settle.

Explained variance

The R² of the regression is the share of the fund's return variance the factors explain. With uncorrelated residuals, the variance splits as

where $\mathbf{b}_i$ collects the loadings and $\Sigma_f$ is the covariance matrix of the factors. With the market as the only factor this is the single-index split from the factor-models lesson, $\beta_i^2\sigma_M^2 + \sigma_\varepsilon^2$. Adding factors moves variance from the residual into the systematic part, so R² can only rise. A high R² says the fund is close to a mix of factor portfolios; it says nothing about whether the alpha is zero.

Testing many portfolios at once

A model is judged on many test portfolios together, for example 25 portfolios sorted by size and book-to-market. Run the time-series regression for each and look at all intercepts: a good model leaves them small and jointly insignificant. The Gibbons–Ross–Shanken (GRS) test turns the vector of alphas into one F-statistic, and the average absolute alpha gives an easy-to-read summary. FF3 cuts the mean absolute alpha of size–value portfolios far below CAPM's; FF5 does better still on portfolios sorted by profitability and investment, and makes HML nearly redundant, because value firms tend to be profitable and to invest conservatively.

Limits of factor models

Factors found by sorting historical data are at risk of data mining: test enough characteristics and some will show a premium by chance. Hundreds of candidate factors have been published, the so-called factor zoo, and many shrink or vanish out of sample or after costs. Whether a premium is a reward for risk or a mispricing is not settled by the regression either. In practice the models are most useful as a benchmark: they tell you which part of a fund's return could have been bought with cheap factor portfolios.

Alpha is always relative to a model. A CAPM alpha, an FF3 alpha and an FF5 alpha are different numbers for the same fund, and a manager's skill should be measured against the factors an investor could have held instead.

Summary

• SMB, HML, RMW, CMA and UMD are long–short portfolios built from sorts on size, book-to-market, profitability, investment and past returns.
• The time-series regression $R_i - R_f = \alpha + b\,\text{MKT} + s\,\text{SMB} + h\,\text{HML} (+ r\,\text{RMW} + c\,\text{CMA}) + \varepsilon$ gives loadings that say which styles a fund resembles.
• With zero alpha the expected excess return is the sum of loadings times factor premia; the CAPM alpha of a tilted fund is roughly its tilts times the omitted premia.
• Alpha is significant when $|t_\alpha| > 2$; R² measures how much of the variance is systematic, and rises as factors are added.
• Models are tested on many portfolios jointly (GRS, mean |α|), and the long list of published factors calls for caution about data mining.