What is the number behind a beta of 1.2?

The performance course used beta and the factor course used loadings. Both are the slope of a regression, and this lesson is the regression itself — the slope, the fit, the residual, and what each one is worth — measured on AAPL.US against SPY.US.

The regression, measured 2026-09-04

Daily returns on adjusted closes, 2 September 2016 to 3 September 2026, 2,513 days. Regress Apple's return on the S&P 500 fund's:

Over the two years to 3 September 2026 alone: beta 1.08, correlation 0.62. The API's own Technicals.Beta for Apple read 1.086 on the same day, which matches the shorter window — a reminder that a published beta is a window's beta, and the window is rarely stated.

Slope, correlation, R²

They are three views of one relationship and they are routinely confused. The slope is the correlation times the ratio of the two standard deviations: Apple moves 1.2 times the market because it is 0.74 correlated and about 1.6 times as volatile. A share can have a beta of 1.2 with a correlation of 0.3 — a volatile share loosely tied to the market — or a beta of 0.5 with a correlation of 0.9. R² is the correlation squared: 0.74² is 0.55, and it is the only one of the three that says how much of the share's behaviour the regression describes. Beta says what the market does to the share; R² says how much of the share is the market.

The factor course's single-stock claim — beta explains often less than half — is this R². Apple, one of the most index-like shares in the world, is at 0.55. Most single shares are lower.

The residual is the point

What the regression leaves is the residual: the share's return minus what the market predicted. It is the raw material of alpha, of stock-specific risk, and of the factor loadings the factor course added — a second regression on the residual, against value and size and momentum, is the multi-factor model. A residual with structure — autocorrelated, clustered, fat-tailed — is a sign that something the regression did not include is doing work, and Apple's residual has all three properties.

Three things that break it

Non-stationary inputs: regress prices on prices and the previous lessons apply. Fat-tailed inputs: a single −10.9% market day carries enormous weight in a least-squares fit, and a beta estimated on a window containing March 2020 is partly a measurement of March 2020. Window choice: 1.20 over ten years, 1.08 over two, and neither is the true beta, because there is no such thing — only a beta on a window, like a volatility on a window, and the rolling-windows lesson makes the point with three at once.

In the data

Two /eod/ pulls joined on date, converted to returns, and one least-squares fit. /fundamentals/AAPL.US?filter=Technicals returns the provider's Beta; the lesson's own figure is the same quantity on a window you chose and can state.

Try it now

1. The slope of AAPL.US daily returns on SPY.US daily returns, both from /eod/ on adjusted closes, joined on date and fitted by least squares on 28 September 2026, for five windows ending 3 September 2026:

And here is the API's own figure, /fundamentals/AAPL.US?filter=Technicals:

Compare Beta with the table and write which window the API appears to use, and which windows it rules out. 2. The two series, so the slope has a picture:

3. Describe, a line each, what a beta of 1.2 with an R² of 0.2 would be, and what a beta of 0.6 with an R² of 0.8 would.