Fama–French three-factor model

Which three factors? Start with CAPM

The three Fama–French factors are ① market excess return Rₘ − Rᶠ, ② size SMB (Small Minus Big: small-stock minus big-stock returns), and ③ value HML (High Minus Low: high minus low book-to-market returns). The risk-free rate Rᶠ is the return benchmark, not a fourth factor. Alpha is not one of these three factors either. The CAPM calculator isolates that market contribution.

CAPM starts by relating returns to exposure to broad market risk. The three-factor model adds size and value to investigate return differences between portfolios with similar market exposure. It is an explanatory and estimation framework, not three categories guaranteed to make money.

Market factor: add only the excess over the risk-free rate

Assume a market return of 8% and a risk-free rate of 2%. Market excess return is 6%. With β = 1, the market contributes 1 × 6% = 6 percentage points; adding the 2% benchmark gives the CAPM estimate of 8%. Adding the full 8% market return to 2% would double-count its risk-free component.

Beta measures exposure to co-movement with the market factor, not an ownership percentage. Beta 1.2 gives 2% + 1.2 × 6% = 9.2% under these assumptions. It does not mean holding 120% of a particular stock or guarantee that realized returns will equal this estimate.

SMB: the small-minus-big return spread

The size factor compares returns on portfolios of smaller and larger companies, with official series using specified portfolio-construction rules. If illustrative small-stock and big-stock portfolios return 6% and 4%, their teaching spread is SMB = +2%. It is not a difference between two share prices, and company assets or market capitalization are not substituted directly into the equation.

The loading s measures exposure to SMB. With s = 0.5 and SMB = 2%, the contribution is one percentage point. A negative loading gives opposite exposure. Small-company risks do not imply positive SMB in every period or that every small company beats every large one.

HML: high-minus-low book-to-market returns

The value factor sorts portfolios by book-to-market B/M: book equity relative to equity market value. Illustrative returns of 5% for high B/M and 2% for low B/M give a simplified HML spread of 3%. An official factor still follows its provider's sorting and weighting methodology.

With h = 0.4, HML of 3% contributes 1.2 percentage points. B/M is not a price-to-earnings ratio, nor does a low share price necessarily imply high B/M. High B/M is not proof of undervaluation. The loading h is estimated return exposure, not a company's B/M ratio inserted directly.

Add the three contributions to the common benchmark

The complete teaching assumptions are Rᶠ = 2%, market excess return 6%, β = 1, SMB = 2%, s = 0.5, HML = 3% and h = 0.4. Contributions are the 2% benchmark, 6% market, 1% size and 1.2% value, totaling 10.2%. Market-only CAPM gives 8%; the 2.2-percentage-point difference is the contribution of the two additional factors.

That difference is neither alpha nor risk-free outperformance. It is a model difference under these inputs. Multiply each return by its loading before adding, and align periods and currencies: monthly SMB cannot simply be added to an annual market premium.

Change SMB to observe exposure and sign

The interaction holds other assumptions fixed and changes SMB. Moving it from +2% to −2% changes the s = 0.5 contribution from +1% to −1%, lowering the total estimate from 10.2% to 8.2%. Size exposure remains present but contributes negatively when the factor is unfavorable.

Ask what both the loading and premium represent and how adverse scenarios affect them, rather than only why the estimate exceeds CAPM. The sign and magnitude come from their product. With s = 0, changing SMB alone contributes nothing. This is factor attribution, not an automatic improvement to a portfolio.

Historical explanation, forward estimates and limits

Empirical research commonly regresses contemporaneous excess returns on the three factors to estimate β, s and h, potentially including an alpha intercept and residual. A forward required-return estimate also needs assumptions about future premiums; one realized factor return is not a permanent expected value. The interaction demonstrates substitution, not statistical estimation.

Sample period, region, investment universe, currency and frequency affect estimates, and loadings can change. The model does not capture every risk, and more factors do not guarantee a better explanation for every asset. Causality, forecasting ability and suitability require further evidence; the equation alone is not a trading recommendation.

Scope and limitations

Factor loadings are estimates, premiums may be negative, and relationships can change by sample, region and period. Match units and frequencies. This is not an investment recommendation or a prediction.