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Beta and Correlation vs SPY: Formulas and What They Mean

What beta and correlation measure, how they're calculated from daily log returns vs SPY over 63 and 252 sessions, and what they don't tell you.

Reviewed · Sources at the end · How Vyreon measures
Vyreon's per-security reports are not live yet; this page describes what they will measure. Examples are illustrative.

Beta measures how much a security's daily returns have moved, on average, for each 1% move in a benchmark; correlation measures how consistently the two moved together. Vyreon measures both against SPY, the S&P 500 ETF, over the last 63 sessions (about three months) and 252 sessions (about one year).

Beta Formula

Beta = covariance(stock returns, SPY returns) ÷ variance(SPY returns)

A beta of 1 means the security's moves were, on average, the same size as SPY's in the same direction. Above 1 means larger moves; between 0 and 1, smaller; below 0, a tendency to move the opposite way.

Correlation Formula

Correlation = covariance(stock returns, SPY returns) ÷ (standard deviation of stock returns × standard deviation of SPY returns)

Correlation runs from −1 to 1. Near 1, their daily moves lined up closely in the same direction; near 0, there was little consistent linear relationship.

How Beta And Correlation Fit Together

Beta = correlation × (stock volatility ÷ SPY volatility)

That is why a volatile stock can have a high beta without being closely tied to the market.

Illustrative example: a stock whose daily moves are twice as large as SPY's, with a correlation of 0.6, has a beta of 0.6 × 2 = 1.2. A stock with a correlation of 0.95 and the same volatility as SPY has a beta of 0.95.

How Vyreon Calculates It

Vyreon uses daily log returns, ln(close ÷ previous close), from closes adjusted for splits and dividends, for both the security and SPY. It matches the two on the same session dates and uses the last 63 or 252 returns. The covariance and variances divide by the number of returns. If the security does not have enough matched history, the value is shown as unavailable rather than computed on a shorter window.

The 63-session figures react faster to recent changes; the 252-session figures are steadier.

What Beta And Correlation Do Not Tell You

  • They describe past co-movement. Both change over time, sometimes sharply around earnings, sector shifts or market stress.
  • Beta is not a complete measure of risk: a stock with low correlation can have a low beta and still be very volatile.
  • Neither says anything about future returns.

Related: relative performance · realized volatility · max drawdown

Sources

  • William F. Sharpe, "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk", Journal of Finance (1964): beta as a measure of a security's sensitivity to the market.
  • John Lintner, "The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets", Review of Economics and Statistics (1965).
  • Karl Pearson, "Note on Regression and Inheritance in the Case of Two Parents", Proceedings of the Royal Society of London (1895): the correlation coefficient.