Covariance

A measure of how two variables move together: the average product of their deviations from their own means. A positive value means they tend to rise and fall together, a negative one that they move in opposite directions.

Its size depends on the units, which makes it hard to read on its own. Dividing it by the two standard deviations gives the correlation, which is scale-free: a covariance of 0.012 between two assets with return volatilities of 20% and 15% is a correlation of 0.4.

Portfolio risk is built from covariances. Portfolio variance is the weighted sum of every pair's covariance. So combining assets that don't move together lowers risk without lowering the expected return, and a covariance matrix estimated from a calm period understates risk in a crisis, when correlations rise together.

Diversification works through the covariances.