Conditional Expectation and Variance Revisited

A factor observation updates a return forecast from $\mathbb{E}[X]$ to $\mathbb{E}[X\mid Y]$. Before the observation arrives, the updated forecast is itself random. This lesson derives total expectation and total variance, then explains the orthogonality of the forecast error. These results support factor forecasting and the decomposition of portfolio risk into explained and residual components.

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

Setup. Assume $\mathbb{E}[|X|]<\infty$ for conditional expectations. Assume $\mathbb{E}[X^2]<\infty$ for every variance and orthogonality statement below. Equalities between conditional expectations hold up to events of probability zero.