Lesson 8 of 25 · 6 min

Covariance and correlation, and which one the arithmetic uses

The formula runs on covariance, the number a person can interpret is correlation, and between them sit two volatilities.

Correlation is the number that gets quoted. Covariance is the number the variance formula actually needs. They are the same information with the scale left in or divided out, and which one you are looking at determines what a change in it means.

covariance = r x s1 x s2

covariance: The shared movement of two holdings, in squared-return units. What lesson 3's grid is really summing.
r: The correlation between them, from -1 to +1.
s1, s2: The two holdings' own volatilities, in the same units and over the same period as the covariance you want.

Two holdings correlated 0.41, with annualised volatilities of 28 and 19 percent. Covariance is 0.41 times 0.28 times 0.19 = 0.021812, or 218.12 if you prefer to work in percent-squared. Nothing in anyone's experience is measured in squared returns, which is the practical reason covariance is not the number people quote. It has no readable scale.

The difference between the two becomes concrete when one holding's volatility changes. Double the second holding's volatility from 19 to 38 percent and covariance doubles, from 0.021812 to 0.043624. The correlation stays exactly 0.41. So correlation is a measure of shape and covariance is a measure of shape and size together — and it is size and shape together that enters the variance. Halve both volatilities and covariance falls to a quarter of its previous value with the correlation unmoved.

Keep that asymmetry in view. A statement that correlations were stable is compatible with a covariance that quadrupled, and it is covariance that drives the number a lender computes. Lesson 22 works that case through.

Correlation is covariance with the volatilities divided out. Dividing them out is what makes it readable and also what makes it incomplete.

Key terms

Covariance: The average product of two holdings' deviations from their own means. Same information as correlation, with the two volatilities still in it.
Scale-free measure: A statistic that does not change when the underlying quantities are rescaled. Correlation is one; covariance is not.
Squared-return units: The units covariance and variance come in. They have no everyday interpretation, which is why volatility and correlation are quoted instead.

Worked example

One correlation, four covariances

A pair correlated 0.41. The correlation is held fixed while the volatilities are changed. Illustrative annualised figures unless stated.

Volatilities 28% and 19%: Covariance 0.021812
The same figure in percent-squared: 218.12
Second volatility doubled to 38%: Covariance 0.043624
Monthly volatilities of 8.08% and 5.49%: Covariance 18.19 per month
That monthly figure annualised: 218.25

One correlation produced four different covariances. The variance formula consumed the covariances, and only the correlation was quotable.

Check yourself

Two holdings correlated 0.36 have volatilities of 22% and 31%. What is their covariance, and what happens to it if both volatilities halve while the correlation holds?

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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.