Arbitrage Pricing Theory and Multi-Factor Models › The Fama-French Factor Models

Eugene Fama and Kenneth French revolutionized empirical asset pricing with their 1993 three-factor model, later extended to five factors in 2015. These models have become the standard benchmark for evaluating investment performance and understanding return patterns.

The Fama-French three-factor model augments CAPM with two additional factors:

$$R_i - r_f = \alpha_i + \beta_i^{MKT} \cdot MKT + \beta_i^{SMB} \cdot SMB + \beta_i^{HML} \cdot HML + \epsilon_i$$

where:

• $R_i$: return on asset $i$
• $r_f$: risk-free rate
• $\alpha_i$: intercept (abnormal return)
• $\beta_i^{MKT}, \beta_i^{SMB}, \beta_i^{HML}$: sensitivities to the respective factors
• $MKT$: market factor ($R_m - r_f$)
• $SMB$: size factor (Small Minus Big)
• $HML$: value factor (High Minus Low)
• $\epsilon_i$: idiosyncratic error term

The model's structure reveals its purpose. The left-hand side measures the asset's excess return over the risk-free rate, which represents the premium investors earn for bearing risk. The right-hand side decomposes this premium into components: compensation for market risk, for size-related risk, for value-related risk, and any residual alpha that the factors cannot explain.

The factors are defined as follows:

• MKT (Market): The excess return on a broad market portfolio, identical to the CAPM market factor. $MKT = R_m - r_f$

• SMB (Small Minus Big): The return on a portfolio of small-cap stocks minus the return on a portfolio of large-cap stocks. SMB captures the size premium: small stocks' tendency to outperform large stocks

• HML (High Minus Low): The return on a portfolio of high book-to-market (value) stocks minus the return on a portfolio of low book-to-market (growth) stocks. HML captures the value premium

The construction of SMB and HML as long-short portfolios is deliberate. By going long small stocks and short large stocks, SMB isolates the pure effect of size, controlling for other characteristics. Similarly, by going long value stocks and short growth stocks, HML isolates the pure effect of valuation. This long-short construction ensures that the factors are approximately uncorrelated with the market factor, making them useful for explaining return variation beyond what CAPM captures.

The construction methodology matters for understanding what these factors capture. Each year at the end of June, Fama and French sort stocks as follows:

Size sort: Stocks are ranked by market capitalization and divided at the median into Small and Big groups.

Book-to-market sort: Stocks are independently ranked by book-to-market ratio and divided into three groups: Low (bottom 30%), Medium (middle 40%), and High (top 30%).

This creates six portfolios from the intersection of two size groups and three book-to-market groups. The factors are then computed as:

$$\begin{aligned} SMB &= \frac{1}{3}(\text{Small/Low} + \text{Small/Medium} + \text{Small/High}) - \frac{1}{3}(\text{Big/Low} + \text{Big/Medium} + \text{Big/High}) \\ HML &= \frac{1}{2}(\text{Small/High} + \text{Big/High}) - \frac{1}{2}(\text{Small/Low} + \text{Big/Low}) \end{aligned}$$

where:

• $\text{Small/High}$: portfolio of small-cap, high book-to-market stocks
• $\text{Big/High}$: portfolio of large-cap, high book-to-market stocks
• $\text{Small/Medium}$: portfolio of small-cap, medium book-to-market stocks
• $\text{Big/Medium}$: portfolio of large-cap, medium book-to-market stocks
• $\text{Small/Low}$: portfolio of small-cap, low book-to-market stocks
• $\text{Big/Low}$: portfolio of large-cap, low book-to-market stocks

The averaging process in these formulas serves an important purpose. The SMB factor averages across book-to-market groups, isolating the size effect. By including small value, small medium, and small growth stocks on the long side, and big value, big medium, and big growth stocks on the short side, the factor captures the pure size effect without being contaminated by any particular book-to-market exposure.

The HML factor averages across size groups, isolating the value effect. By including both small and big value stocks on the long side, and both small and big growth stocks on the short side, the factor captures the pure value effect without being contaminated by size effects.

In 2015, Fama and French extended their model with two additional factors based on profitability and investment:

$$R_i - r_f = \alpha_i + \beta_i^{MKT} \cdot MKT + \beta_i^{SMB} \cdot SMB + \beta_i^{HML} \cdot HML + \beta_i^{RMW} \cdot RMW + \beta_i^{CMA} \cdot CMA + \epsilon_i$$

where:

• $R_i, r_f, \alpha_i, \epsilon_i$: defined as in the three-factor model
• $\beta_i^{j}$: sensitivity to factor $j$
• $MKT, SMB, HML$: market, size, and value factors
• $RMW$: profitability factor (Robust Minus Weak)
• $CMA$: investment factor (Conservative Minus Aggressive)

Empirical observation and theoretical reasoning motivated adding these factors. Empirically, researchers found that profitability and investment patterns explained return variation not captured by the original three factors. Theoretically, these factors connect to fundamental valuation principles.

The new factors are:

• RMW (Robust Minus Weak): The return on stocks with robust (high) operating profitability minus stocks with weak (low) profitability. Companies with higher profit margins tend to earn higher returns

• CMA (Conservative Minus Aggressive): The return on stocks of companies with conservative (low) investment minus aggressive (high) investment. Firms that invest less tend to earn higher returns

The profitability factor has intuitive appeal. All else equal, a more profitable company should be worth more. If two companies have similar market prices but different profitabilities, the more profitable one offers better value and should earn higher subsequent returns. The investment factor relates to the rate at which companies are expanding their asset base. Firms investing heavily may be pursuing growth at the expense of current profitability, or they may be making poor capital allocation decisions. Either interpretation suggests that conservative investors, those who invest less aggressively, may earn higher returns.

While not part of the original Fama-French framework, momentum has become a standard factor in many models. The momentum factor (often called UMD for "Up Minus Down" or WML for "Winners Minus Losers") captures the tendency of recent winners to continue outperforming:

$$MOM = R_{winners} - R_{losers}$$

where:

• $R_{winners}$: return on the portfolio of recent top-performing stocks
• $R_{losers}$: return on the portfolio of recent bottom-performing stocks

Stocks are sorted based on their past 12-month returns (excluding the most recent month to avoid microstructure effects), and the factor is the return difference between the top and bottom deciles.

The exclusion of the most recent month is a subtle but important detail. Stock prices exhibit short-term reversal at very short horizons due to microstructure effects like bid-ask bounce. By skipping the most recent month, the momentum factor captures medium-term continuation rather than short-term noise.

Mark Carhart's 1997 four-factor model combined the Fama-French three factors with momentum:

$$R_i - r_f = \alpha_i + \beta_i^{MKT} \cdot MKT + \beta_i^{SMB} \cdot SMB + \beta_i^{HML} \cdot HML + \beta_i^{MOM} \cdot MOM + \epsilon_i$$

where:

• $R_i$: return on asset $i$
• $r_f$: risk-free rate
• $\alpha_i$: intercept (abnormal return)
• $\epsilon_i$: idiosyncratic error term
• $MKT, SMB, HML$: Fama-French three factors
• $MOM$: Momentum factor
• $\beta_i^{j}$: sensitivity to factor $j$

This model has become particularly popular for evaluating mutual fund performance. By including momentum, the model can distinguish between managers who generate alpha through stock selection and those who simply ride momentum trends.