Covariance and Correlation (Calculations for CFA® and FRM® Exams)

Understand the relationship between covariance and correlation for CFA and FRM exams. We start with a quick variance recap, then compute covariance step by step, explain interpretation (positive, negative, zero), show why variance is a special case of covariance, and finish with correlation as standardized covariance, including the key insight that zero correlation ≠ independence when relationships are non-linear.

AnalystPrep's Concept Capsules for CFA® and FRM® Exams
This series of video lessons is intended to review the main calculations required in your CFA and FRM exams.

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Covariance
The covariance is a measure of the degree of co-movement between two random variables. For instance, we could be interested in the degree of co-movement between the rate of interest and the rate of inflation.
X = interest rate
Y =inflation

The covariance between two random variables can be positive, negative, or zero. A positive number indicates co-movement (i.e. the variables tend to move in the same direction); a value of zero indicates no relationship, and a negative value shows that the variables move in opposite directions.

Correlation
Correlation is the ratio of the covariance between two random variables and the product of their two standard deviations, i.e., it measures the strength of the linear relationship between two variables. While the covariance can take on any value between negative infinity and positive infinity, the correlation is always a value between -1 and +1.

A value of -1 indicates a perfect inverse relationship (i.e. a unit change in one means that the other will have a unit change in the opposite direction). A value of +1 indicates a perfect linear relationship (i.e. the two variables move in the same direction with the unit changes being equal). If there is no linear relationship at all, then the correlation will be zero.

How does correlation impact portfolio risk?
Correlation ranges from -1 to +1
⮚ +1 = returns are perfectly positively correlated.
⮚ 0 = returns of two assets are not correlated.
⮚ -1 = returns are perfectly negatively correlated.

What happens to portfolio risk (in a portfolio of two risky assets) when the two assets are perfectly correlated?
⮚Risk is unaffected; no diversification benefit.

What happens to portfolio risk (in a portfolio of two risky assets) when the two assets are not perfectly correlated?
⮚Overall portfolio risk is reduced; there is a diversification benefit.

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