Beta

Beta is the slope of a line fitted through an asset's returns plotted against the market's returns. A beta of 1.3 says that, over the window measured, a 10 percent move in the index came alongside a move of roughly 13 percent in the asset. Written out, it is $\mathrm{Cov}(r_a, r_m) / \mathrm{Var}(r_m)$: how the two move together, scaled by how much the market moves on its own.

It is the standard measure of exposure to systematic risk, the part of risk that spreading holdings cannot remove. That makes it the input to the capital asset pricing model, where an asset's expected return is the risk-free rate plus its beta times the market risk premium. It also sums up a whole portfolio neatly, since portfolio beta is the weighted average of the betas inside it. A company's equity beta tends to rise with its financial leverage, because the same swing in operating results lands on a thinner slice of equity.

The mistake is reading beta as a fixed property of a company. It is an estimate from a sample, and it moves with every choice behind it: which index counts as the market, how long the window runs, and whether returns are measured daily, weekly or monthly. The same stock can be published at 0.8 by one data provider and 1.2 by another with neither of them doing the arithmetic wrong. A low beta is also not the same as low risk, only low co-movement with the index. A company can carry plenty of its own unsystematic risk while barely tracking the market at all.

The denominator of beta is a plain statistical variance, the average squared distance of the market's returns from their own mean: variance.