Risk Models: APT and Factor Structure

Every optimiser in Note 4 consumed a covariance matrix $\Sigma$; this note explains where it comes from. The raw problem is hopeless — a 2,000-stock universe has over two million covariances to estimate — and the industry's answer is structure: assume returns are driven by a modest number of common factors plus independent stock-specific noise, and the covariance matrix collapses to $\Sigma = B\Sigma_F B^\top + S$. The derivation is two lines of covariance algebra; the substance is in the three rival calibration philosophies (macroeconomic, fundamental, statistical) — which quantity you fix and which you estimate — and in their trade-offs, which explain why the fundamental approach (Barra, Axioma) dominates production risk systems.

Learning objectives. By the end of this note you should be able to: