A factor model is a framework that decomposes a stock's return into exposures to a small number of systematic drivers (factors) plus an idiosyncratic component. The CAPM has one factor (market beta); Fama-French has three (market, size, value); modern multi-factor models like APEX cover quality, momentum, accruals, sentiment, and others.
Mathematically: r_i = α_i + Σ β_ij · F_j + ε_i, where r_i is stock i's return, F_j is factor j's return, β_ij is stock i's exposure to factor j, and ε_i is idiosyncratic noise. Factor models let us rank stocks cross-sectionally, decompose performance into compensated and uncompensated risk, and identify whether observed alpha is genuinely uncorrelated or just an unmeasured factor exposure.