Formula, in Risk & Portfolio Construction

Beta Formula

Beta is the covariance between an asset's returns and the market's returns divided by the variance of the market's returns. It is the slope coefficient when asset returns are regressed on market returns, and it quantifies how much the asset moves, on average, for each one-percent move in the market.

4 variables. Published May 26, 2026. By AI Fin Hub Research.

Formula

beta = Cov(R_i, R_m) / Var(R_m)

Equivalently:  beta = rho(i,m) x (sigma_i / sigma_m)

Variables

Cov(R_i, R_m)

Covariance of asset and market

The average product of the asset's and market's deviations from their means. Positive covariance means the asset tends to rise when the market rises.

Var(R_m)

Market variance

The variance of market returns. Dividing by it normalizes the covariance into a slope, so a beta of 1.0 means the asset moves one-for-one with the market on average.

rho(i,m)

Correlation

Correlation between asset and market returns. The equivalent form shows beta as correlation scaled by the ratio of volatilities, separating co-movement direction from relative magnitude.

sigma_i, sigma_m

Asset and market volatility

Standard deviations of the asset and the market. A volatile asset that is only weakly correlated with the market can still have a modest beta.

Step by step

1

Collect paired periodic returns for the asset and the market over the same window.

Use 36 monthly returns for a stock and its index.
2

Compute the covariance between the asset and market return series.

Covariance of the monthly returns is 0.00072.
3

Compute the variance of the market return series.

Market monthly variance is 0.0016 (standard deviation 4%).
4

Divide the covariance by the market variance.

0.00072 / 0.0016 = 0.45.

Worked example

Estimating a defensive stock's beta from monthly data

Cov(asset, market)

0.00108

Market variance

0.0016

Equivalent: correlation

0.60

beta = 0.00108 / 0.0016 = 0.675. Cross-check with the correlation form: asset volatility implied by Cov / (rho x sigma_m) = 0.00108 / (0.60 x 0.04) = 0.045, so beta = 0.60 x (0.045 / 0.04) = 0.60 x 1.125 = 0.675.

Beta of about 0.68. The stock moves roughly two-thirds as much as the market: in a 10% market rally it would be expected to rise about 6.8%, and to fall about 6.8% in a 10% selloff, before any idiosyncratic moves.

Common variations

Adjusted (Blume) beta: shrinks the raw estimate toward 1.0, since betas tend to revert over time, using roughly two-thirds raw plus one-third 1.0.

Levered and unlevered beta: adjusts for a firm's capital structure to separate business risk from financial leverage.

Downside beta: estimates the slope only over periods when the market fell, capturing asymmetric exposure.

Run the numbers next

Calculators Risk-Adjusted Returns Calculator
Paste a returns CSV. Sharpe, Sortino, Calmar, Omega, alpha, beta, tracking error, information ratio, max drawdown, and tail moments — plus.
Calculators Efficient Frontier Calculator
Paste a multi-asset returns CSV. See the Markowitz mean-variance frontier, the minimum-variance portfolio, the max-Sharpe (tangency) portfolio.

Sources

Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk , William F. Sharpe, Journal of Finance (1964)
On the Assessment of Risk , Marshall E. Blume, Journal of Finance 26(1), 1-10 (1971)

Keep the topic connected

Formula CAPM Formula
The CAPM formula: expected return equals the risk-free rate plus beta times the market risk premium. The basis for pricing systematic risk.
Formula Correlation Formula
The Pearson correlation formula: covariance of two return series over the product of their standard deviations. The key diversification input.
Formula Jensen's Alpha Formula
Jensen's alpha formula: realized return minus the CAPM-expected return given beta. The excess return a manager earns beyond market risk.
Learn Beta
Beta as factor sensitivity: what it measures, why a beta of 1 doesn't mean 'tracks the market', and the rolling-vs-static distinction to watch.