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Covariance and Correlation

A single asset's risk is its variance. A portfolio's risk depends on how its holdings move together, and that is what covariance measures.

Covariance is the expected product of the two variables' deviations from their means:

Covariance
Cov(X,Y)=E[(X−μX)(Y−μY)]=E[XY]−E[X]E[Y]\text{Cov}(X, Y) = E\left[(X - \mu_X)(Y - \mu_Y)\right] = E[XY] - E[X]E[Y]

How two variables move together. The second form is the one you compute with.

The sign is what it tells you clearly. Positive means the variables tend to sit on the same side of their means together; negative means one is above when the other is below. Zero means no linear relationship, a caveat we return to below.

The magnitude, however, is close to uninterpretable: covariance carries the units of both variables multiplied together, so re-quoting a price in cents rather than dollars changes it by a factor of 100 without anything real having changed.

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