What Is the Fama-French Three-Factor Model? Formula and How It Works

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The Fama-French three-factor model is an asset pricing model that explains a stock’s or portfolio’s return in excess of the risk-free rate using three sources of risk: the market, company size and value. Eugene Fama and Kenneth French built it in the early 1990s as an extension of the capital asset pricing model (CAPM). CAPM relies on market beta alone. The three-factor model adds a size factor (small minus big, or SMB) and a value factor (high minus low, or HML) because those two characteristics explained differences in average returns that beta did not.

Where the model came from

CAPM says a stock’s expected excess return depends only on its beta to the market. Research in the 1980s documented returns that beta could not explain: small companies and companies with high book value relative to market value earned higher average returns than CAPM predicted. Fama and French’s 1992 paper, “The Cross-Section of Expected Stock Returns,” showed that size and book-to-market explained the cross-section of U.S. stock returns far better than beta. Their 1993 paper turned those characteristics into tradable factor portfolios and set out the three-factor regression used today.

The Fama-French formula

The model is usually written as a time-series regression:

R(i,t) − R(f,t) = α(i) + β(1) × [R(M,t) − R(f,t)] + β(2) × SMB(t) + β(3) × HML(t) + ε(i,t)

Where:

• R(i,t) = return on stock or portfolio i in period t
• R(f,t) = risk-free rate (typically a short-term Treasury bill rate)
• R(M,t) − R(f,t) = excess return of the market portfolio
• SMB(t) = size premium, the return of small stocks minus the return of big stocks
• HML(t) = value premium, the return of high book-to-market stocks minus low book-to-market stocks
• β(1), β(2), β(3) = factor loadings (often labeled b, s and h)
• α(i) = intercept, the average return not explained by the three factors
• ε(i,t) = error term

The left side is the portfolio’s excess return. Each term on the right is a factor premium multiplied by the portfolio’s sensitivity to it.

How SMB and HML are built

The factors are returns on long-short portfolios, not company characteristics. In the standard construction, stocks are split into two size groups (small and big) at the median market capitalization. They are also split into three book-to-market groups: the top 30% (value), the middle 40% (neutral) and the bottom 30% (growth). That produces six portfolios.

• SMB is the average return of the three small portfolios minus the average return of the three big portfolios.
• HML is the average return of the two value portfolios minus the average return of the two growth portfolios.

Averaging across the other dimension keeps SMB from picking up value effects, and HML from picking up size effects. Kenneth French publishes monthly and daily factor returns built this way, and most academic and practitioner analyses use that series rather than rebuilding the factors.

Interpreting the coefficients

Running the regression on a fund’s history gives four numbers that describe it:

• Market beta (β1): sensitivity to the overall market. A value near 1 means the portfolio moves roughly with the market.
• SMB loading (β2): a positive value indicates a tilt toward small caps. A negative value indicates a tilt toward large caps. A large-cap fund tracking the S&P 500 typically shows an SMB loading near zero or slightly negative.
• HML loading (β3): a positive value indicates a value tilt. A negative value indicates a growth tilt.
• Alpha: what remains after the factor exposures are accounted for. A positive, statistically significant alpha suggests returns beyond what the portfolio’s factor tilts would predict.

R-squared shows how much of the return variation the three factors explain. Fama and French reported that market, size and value together explain up to about 95% of the variation in returns of diversified portfolios, a much larger share than CAPM explains. A high R-squared with an alpha near zero means the portfolio’s returns are largely accounted for by its exposures to market, size and value.

Example of the Fama-French three-factor model

Hypothetical example. A small-cap value fund returned 13% in a year. Assume the following figures:

• Risk-free rate: 3%
• Market excess return: 6%
• SMB: 2%
• HML: 3%
• Estimated loadings: market beta 1.1, SMB 0.4, HML 0.6

Return explained by the factors:

• Market: 1.1 × 6% = 6.6%
• Size: 0.4 × 2% = 0.8%
• Value: 0.6 × 3% = 1.8%
• Total factor-explained excess return: 9.2%

Adding the 3% risk-free rate gives a predicted return of 12.2%. The fund’s actual 13% implies a three-factor alpha of 0.8%.

Under CAPM, the same fund with a beta of 1.1 would have a predicted return of 3% + 6.6% = 9.6%, for an alpha of 3.4%. Most of the apparent outperformance came from holding small and value stocks, which the three-factor model treats as compensated exposure rather than skill. In a real analysis, the loadings come from regressing many periods of fund returns on factor returns, and alpha is judged by its statistical significance, not a single year.

How investors and analysts use it

• Performance evaluation: separating a manager’s factor tilts from stock-picking ability. This is a more demanding test than raw returns or a single-number measure like the Sharpe ratio.
• Portfolio construction: deliberately tilting toward size or value to target exposures an investor is willing to bear. Factor-tilted index funds and ETFs are built around this idea.
• Cost of equity: estimating an expected return for a company or project using its factor loadings instead of beta alone.
• Event studies and research: measuring abnormal returns after controlling for known risk factors.

Why the size and value premiums exist: the debate

The model describes returns. It does not settle why small and value stocks earned more. Supporters of efficient markets, including Fama, argue that the premiums are compensation for risk. Small firms and high book-to-market firms tend to be more vulnerable in bad economic times, and many value stocks are financially distressed companies with a higher cost of capital, since a depressed share price pushes book-to-market up. The behavioral view holds that investors misprice these stocks, for example by extrapolating past poor performance too far, and that the premiums reflect correctable errors.

Five-factor model and momentum

In 2014 Fama and French proposed a five-factor model that adds:

• RMW (robust minus weak): firms with high operating profitability minus firms with low profitability.
• CMA (conservative minus aggressive): firms that invest conservatively minus firms that grow assets aggressively. Aggressive investors have tended to earn lower average returns.

In their tests, HML became largely redundant once profitability and investment were included.

Momentum is not part of either Fama-French model. Mark Carhart added a momentum factor in 1997, often labeled WML (winners minus losers) or UMD, to create a four-factor model. It is commonly used alongside the Fama-French factors in fund evaluation.

Limitations

• Empirical, not theoretical: the factors were chosen because they fit the data, not derived from a pricing theory, which raises concerns about data mining.
• Unstable premiums: the size and value premiums vary widely across periods and markets and can disappear for years.
• Estimation sensitivity: loadings and alpha depend on the sample period, return frequency and factor data used.
• Book value issues: book equity can understate the value of companies whose main assets are intangible, which affects how HML classifies them.

Three-factor model vs. CAPM

CAPM uses one factor and rests on an equilibrium theory: only market risk should be priced. The three-factor model adds size and value because they explain returns empirically. Because both use the same regression logic, CAPM is a special case of the three-factor model with the SMB and HML loadings set to zero. Which benchmark is appropriate depends on whether size and value exposure is considered risk or a source of manager value.

The Bottom Line

Investors and analysts who judge fund managers, build factor-tilted portfolios or estimate a cost of equity can use the model to separate size and value exposure from stock-picking skill. A fund that looks strong under CAPM may show little alpha once its small and value tilts are counted. Results depend on the sample period and factor data, and the size and value premiums can fade for years.