Beta is one number that says how much a stock moves with the market, and the Beta Calculator computes it two ways: paste paired return series in the Return Series mode, or slide a correlation and two volatilities in the Quick Estimate mode. The first mode divides the covariance of the two series by the variance of the market series, the second multiplies the correlation by the stock volatility over the market volatility, and both print the answer to four decimals.
Covariance over variance, one pass
The Return Series mode takes two comma-separated lists, one return per entry, and the tool splits each list on commas, drops any token that is not a number, and refuses to compute unless at least three pairs remain and the two lists are exactly the same length. It then averages both lists, adds up every (stock deviation) times (market deviation) product for the covariance, adds up every squared market deviation for the market variance, and divides the first sum by the second: beta equals covariance over variance, and when the market variance is exactly zero the tool prints 0 instead of dividing by zero. The same population style divides by n, not n minus 1, so a hand recomputation from the same paste matches the panel. Volatility is the square root of those variances, and the Sharpe ratio pairs a return with its volatility in the Sharpe ratio calculator.
Four monthly returns, walked pair by pair
Paste 0.02, -0.02, 0.04, -0.04 for the market and 0.03, -0.03, 0.05, -0.03 for the stock. The market average is 0 and the stock average is 0.005. The covariance adds 0.0005, 0.0007, 0.0018 and 0.0014, divides 0.0044 by 4, and lands at 0.0011; the market variance adds 0.0004, 0.0004, 0.0016 and 0.0016, divides 0.004 by 4, and lands at 0.001. Beta is 0.0011 over 0.001, which the panel prints as 1.1000: for every one percent the market moves, this stock tends to move about 1.1 percent in the same direction.
What the result panel shows
Under the four-decimal beta the panel lists the alpha, the regression fit, and the two volatilities. The alpha is the stock average minus the risk-free rate minus beta times the market premium, and with the risk-free slider at its default 3.0 percent the example reads 0.80 percent: the stock out-earns the CAPM prediction by a hair. The R2 line prints 0.9490, the two volatilities print 3.57 percent for the stock and 3.16 percent for the market, and because beta 1.1000 is above 1 the caption under the panel says the stock is more volatile than the market. Beta at exactly 1 would say it moves with the market, and below 1 it says less volatile.
The R2 number and the line it measures
The R2 is not measured against the CAPM line. The tool runs an ordinary least squares fit of stock return on market return with the line through the two averages, the point (0, 0.005) in this example, and R2 is the share of the stock variance that line explains, 0.9490 here. A line pinned at the risk-free point (0.03, 0.03) would be a different fit and would give a different, even negative, R2, so the number you read is a fit statistic for the regression, not a verdict on how well CAPM holds. When the stock series has zero variance the panel prints 0 for the fit.
Quick estimate: correlation times the volatility ratio
The Quick Estimate mode skips the paste entirely: slide the correlation, set the stock volatility and the market volatility as percentages, and the beta is the correlation times the stock volatility over the market volatility. With the default sliders the estimate is 0.8 times 25 over 15, which the panel prints as 1.3333. The correlation slider runs from -1 to 1 in steps of 0.05 and the volatility sliders run from 1 to 100, and the correlation input is the same coefficient that the correlation matrix tool computes across a whole set of assets.
Where beta sits in a portfolio workflow
Once a beta number exists, it feeds the risk side of a portfolio: the value at risk calculator sizes the loss you expect at a confidence level, and beta tells you which holding drives that loss. On the position side the DCA calculator spreads fixed-interval buying across the holdings a beta ranking would order, and on the article side how to calculate Monte Carlo stock returns simulates the path beta assumes, how to calculate value at risk unpacks the loss statistic the beta feeds, and how to calculate CAGR converts the return the beta accompanies into one comparable annual rate.