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.
Formula
Copy the exact expression or work through it step by step below.
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
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Sources & References
- 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 (1971)
Related Content
Keep the topic connected
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Correlation Formula
The Pearson correlation formula: covariance of two return series over the product of their standard deviations. The key diversification input.
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.
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 that catches most people.