1985issue C021-3
Beta and alpha as chosen regression parameters
Beta and the alpha intercept are the slope and intercept of a linear regression of a security's period return on the contemporaneous market change. A usable beta names the market or group index, the historical window, and whether the pairs are daily, weekly, or monthly.
- Total risk is framed as a firm-specific part that can be reduced by adding holdings and a market-wide part that remains after diversification.
- Beta is the slope of the security-versus-market return fit and is defined as expected sensitivity to economy-wide events, not as the firm-specific residual.
- Alpha is the vertical intercept of that same fitted line and is described as per-period price appreciation or depreciation not attributed to the market term.
- A usable beta names the market or group index, the historical window, and whether observations are daily, weekly, or monthly.
Two parts of total risk
A security's total risk is framed as a firm-specific part that can be reduced by adding holdings and a market-wide part that remains after diversification. The firm-specific residual is the diversifiable risk. The market-wide component is the nondiversifiable risk that remains after diversification and that beta is meant to measure.
Beta is defined as the expected sensitivity of a security or portfolio to economy-wide events, not as the firm-specific residual. The reference market series is assigned a beta of 1.0. A security beta of 1.5 is treated as 50 percent more volatile, so a 10 percent index move is mapped to a 15 percent expected security move.
The fitted line and its intercept
The estimation model is a linear regression of a security's period return on the contemporaneous market change. The intercept is set so the residual has expected value zero, and the slope is the return sensitivity to the market. Linear regression is the line of best fit through paired security and market changes over a chosen sample and interval.
On the security-versus-market scatter, alpha is the vertical intercept described as per-period price appreciation or depreciation, and beta is the slope of that line. Alpha is the per-period price change not attributed to the market term. The market index is the reference series against which sensitivity is measured. Theoretically that series is every economic asset. Practically it is the broadest available proxy or a chosen industry group.
Three parameters that define a usable beta
A usable beta requires three explicit parameters: which market or group index to use, which historical window to include, and whether observations are daily, weekly, or monthly. The observation interval is that sampling bar, used to form the return pairs.
More observations generally improve the estimate, but events such as mergers, divestitures, or large debt-to-equity shifts can change a firm's beta, so the window must stay relevant to the current firm.
Across 195 common stocks, weekly and monthly intervals produced significantly different issue-level betas, yet that interval gap lost significance when the same names were compiled as a large portfolio. Daily bars emphasize short-term fluctuations and monthly bars emphasize longer-range moves. No interval is inherently more accurate, and each parameter set yields its own beta whose quality depends on index relevance, observation count, and interval.