Modeling Equities Returns: The Linear Case

This post details the mathematical formulation of linear factor models for equity returns, including the equation \mathbf r_t = \mathbf B_t\mathbf f_t + \mathbf \epsilon_t. It explains how factor models simplify covariance matrix estimation from O(n^2) to O(mn + m^2) space and computation, and how they are used for portfolio optimization by maximizing the Sharpe Ratio. The post also introduces the concept of decomposing expected returns into 'alpha parallel' (driven by factors) and 'alpha orthogonal' (idiosyncratic alpha), and provides a formula for the Sharpe Ratio based on these components.