Definition
A factor model expresses the return of an asset as a linear combination of exposures to a small set of common risk factors plus an asset-specific residual. The canonical form is r_i = a_i + b_1 F_1 + b_2 F_2 + ... + b_k F_k + e_i, where each F is a systematic factor (the market, a size premium, a value premium, momentum, quality, low volatility), each b is the asset's loading on that factor, and e is the idiosyncratic component that cannot be explained by the common factors.
The promise of the framework is that thousands of correlated asset returns can be summarised by a handful of factor returns and a sparse set of loadings. That compression turns an unmanageable covariance matrix into something portfolio construction, risk attribution, and performance evaluation can act on.
Why it matters
How it works
Building a factor model has three steps. First, choose the factors — either statistically (principal components of historical returns) or economically (the Fama-French market, size, and value factors; Carhart momentum; Asness quality-minus-junk). Second, estimate each asset's loadings by regressing its excess returns on the factor returns over a rolling window. Third, use the estimated loadings to forecast covariance, attribute past performance, or build a portfolio with a target factor exposure.
The output is two things at once. The fitted part — the b_i F terms — describes the systematic, replicable component of return: this is what an index fund or factor ETF can deliver cheaply. The residual e_i is the unexplained part, the candidate for true skill-driven alpha. Distinguishing the two is the entire reason institutional investors run factor models: paying active fees for what is actually a tilt toward small-cap value is the canonical mistake the framework was built to prevent.