Return-generating models, the market model and beta

In short

• A return-generating model estimates expected return from given parameters. The general form is a multi-factor model; factors can be macroeconomic, fundamental or statistical.
• Fama-French add size and book-to-market to the market factor; Carhart adds momentum.
• The single-index model: $E(R_i) - R_f = \beta_i[E(R_m) - R_f]$. It splits total variance into $\beta_i^2\sigma_m^2$ (systematic) + $\sigma_{e_i}^2$ (nonsystematic).
• The market model $R_i = \alpha_i + \beta_iR_m + e_i$ is estimated by regression; it is used to estimate beta and abnormal returns.
• $\beta_i = \text{Cov}(R_i, R_m)/\sigma_m^2 = \rho_{i,m}\sigma_i/\sigma_m$. The market's beta is 1, the risk-free asset's is 0, and the average stock's beta is 1.
• Short estimation windows are more current but noisier; three to five years is more accurate but may be out of date.