Covariance and correlation

In short

• Covariance is the probability-weighted average of the product of two variables' deviations from their expected values.
• Positive covariance: returns tend to be on the same side of their means at the same time. Negative: opposite sides. Zero: no linear relation.
• The covariance of a return with itself is its variance; and $\text{Cov}(R_i,R_j) = \text{Cov}(R_j,R_i)$.
• Sample covariance from historical data divides the sum of cross-products by $n - 1$.
• Correlation = covariance ÷ (product of the two standard deviations). It lies between −1 and +1; strength is read from its distance from 0.
• Both measure linear association only.