Beta is the slope coefficient of the regression of a stock's returns against market returns. Mathematically: β = Covariance(stock returns, market returns) / Variance(market returns). A positive beta above 1.0 means the stock amplifies market moves; between 0 and 1.0 means it participates but at lower amplitude; negative beta means it tends to move opposite to the market (rare outside of specific hedge strategies like inverse ETFs or gold in certain regimes). The regression is typically run on 36-60 months of monthly returns against a benchmark like the S&P 500.

Beta interpretation by sector: utilities, consumer staples, and healthcare tend to have betas of 0.4-0.8 — defensive sectors less correlated with economic cycles. Technology and discretionary companies tend toward 1.2-1.8 — more volatile and more economically sensitive. Financial stocks historically have betas above 1.0 and exhibit left-tail risk (their losses in crises tend to be larger than their beta predicts because of embedded leverage). Biotech and speculative growth stocks can have betas of 2.0+ and also exhibit high idiosyncratic volatility not captured by beta.

CAPM uses beta to define the expected return of an asset: Expected Return = Risk-Free Rate + β × (Market Return - Risk-Free Rate). The term (Market Return - Risk-Free Rate) is the equity risk premium (ERP) — historically around 4-6% for the US market. A stock with beta 1.5 and ERP of 5% has an expected return of 1.5 × 5% = 7.5% above the risk-free rate. CAPM provides a benchmark: if a stock's actual expected return exceeds its CAPM-predicted return, it offers positive alpha potential; below CAPM prediction implies negative alpha.

Alpha in CAPM is the portion of actual return not explained by beta: Alpha = Actual Return - (Risk-Free Rate + β × ERP). Persistent positive alpha — outperforming what beta exposure would explain — is what distinguishes skilled active management. But CAPM's single-factor model underpredicts returns for high-book-to-market (value) stocks and small-cap stocks because it misses those factor premiums — which is why Fama and French extended to a three-factor and later five-factor model that adds size and value (and later profitability and investment) as additional return factors.

Beta has significant limitations. It is backward-looking: calculated from historical returns that may not reflect future sensitivity. Beta is also time-varying — a stock's beta during calm markets may differ substantially from its beta during crises. Research documents 'beta expansion' in crashes: stocks that appeared to have moderate betas under normal conditions suddenly exhibit much higher market correlation when all correlations spike in risk-off episodes. High-leverage companies and banks are particularly prone to beta expansion in crises.

For practical portfolio risk management, beta is most useful as a relative comparison tool within an asset class rather than an absolute prediction of future volatility. Portfolio beta (weighted average of individual position betas) measures the overall directional market exposure of the portfolio. A portfolio manager targeting market-neutral exposure aims for portfolio beta near zero. A levered long portfolio might maintain beta of 1.2-1.5. Monitoring portfolio beta over time ensures that position changes aren't unintentionally adding or removing market exposure beyond the intended level.