Single-Factor Model

Single-Factor Model (OLS Regression)

The single-factor model explains an asset's excess return as a linear function of a single factor's excess return:

$R_i - r_f = \alpha + \beta (R_f^{\text{factor}} - r_f) + \varepsilon$

This is estimated via Ordinary Least Squares (OLS) regression.

OLS Formulas

Given two series of observations (x, y), the OLS estimates are:

$\beta = \frac{\sum_t (x_t - \bar{x})(y_t - \bar{y})}{\sum_t (x_t - \bar{x})^2}$

$\alpha = \bar{y} - \beta \bar{x}$

where we regress excess asset returns (y) on excess factor returns (x).

Application

The factor exposure (beta) tells us how much the asset co-moves with the factor per unit of factor return. The alpha is the return unexplained by the factor.

Your Task

Implement:

• factor_exposure(asset_returns, factor_returns) — OLS beta of asset returns on factor returns
• fama_french_alpha(asset_returns, factor_returns, rf) — alpha from regression of excess asset returns on excess factor returns