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Factor Models

"How many factors would you use, and why does this premium pay?" is a common question at systematic equity funds. Behind it sit three others: what a factor model is, how its factors are built, and how to tell a real factor from one found by searching. A factor model is also the risk model under nearly every equity portfolio, so the ideas matter well beyond the interview.

The model

A factor model explains each asset's return with a few shared factor returns and an asset-specific remainder:

ri=αi+∑k=1Kβikfk+εir_i = \alpha_i + \sum_{k=1}^{K} \beta_{ik} f_k + \varepsilon_i

The βik\beta_{ik} are the asset's exposures, the fkf_k are the factor returns, and εi\varepsilon_i is the specific return, assumed uncorrelated across assets. With exposures in an N×KN \times K matrix BB, factor covariance Ω\Omega and a diagonal matrix DD of specific variances, the covariance matrix of all the assets is

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