Estimating CAPM beta with a simple regression

In short

• CAPM: $E(r_i) = r_f + \beta_i[E(r_m) - r_f]$. It is a one-factor model: only systematic (market) risk, measured by beta, earns a premium; idiosyncratic risk can be diversified away.
• Expected returns cannot be observed, so we use realised returns and estimate $Z_{i,t} = b_0 + b_1 Z_{m,t} + \varepsilon_{i,t}$, where $Z = r - r_f$ are excess returns. $\hat b_1$ is the beta estimate.
• Steps: gather asset, index and risk-free returns (e.g., monthly for 3–5 years or weekly for 1–3 years) → compute excess returns → run OLS → plug $\hat b_1$ into CAPM for the expected return.
• Read the output as usual: slope = beta; t = $\hat b_1 / s_{\hat b_1}$; confidence interval $\hat b_1 \pm t_c s_{\hat b_1}$; $R^2$ = share of the asset's excess-return variance explained by the market (systematic part).
• Beta > 1 amplifies market moves, beta < 1 dampens them, negative beta moves against the market (a hedge). A change in the market risk premium changes the expected excess return by beta × the change.
• Evidence for CAPM is mixed, yet it remains widely used for portfolio management, performance evaluation, cost of capital and project valuation.