CFA Level I Exam · The Return and Risk of a Financial Portfolio

Covariance and Correlation of Asset Returns Explained

Updated 7 October 2026 · Fact-checked

Covariance measures whether two assets' returns move together, in squared-return units. Correlation is covariance divided by the product of the two standard deviations, so it is unit-free and lies between −1 and +1. From joint probabilities, covariance is the probability-weighted sum of the product of each asset's deviations from its expected return.

Understand Covariance and Correlation of Asset Returns

Variance tells you how much one asset's return varies around its mean. Covariance tells you how two assets' returns vary together. If asset A tends to be above its mean when asset B is above its mean, covariance is positive. If A tends to be above when B is below, covariance is negative. If there is no consistent linear pattern, covariance is near zero.

The problem with covariance is its size. Its units are the units of A times the units of B (for returns, squared percent). A covariance of 0.0048 is hard to judge as strong or weak. Covariance also changes if you change units, for example percent versus decimals.

Correlation fixes this. You divide covariance by the standard deviation of A times the standard deviation of B. The result has no units and always lies between −1 and +1. A value of +1 means a perfect positive linear relationship, −1 a perfect negative one, and 0 means no linear relationship.

The sign of covariance and the sign of correlation are always the same, because standard deviations are positive. Covariance of an asset with itself is its variance. Correlation measures only linear association. Two variables can have zero correlation and still be strongly related in a non-linear way.

This matters for portfolios. The lower the correlation between assets, the more portfolio risk falls through diversification. Portfolio variance uses covariance (or correlation times the standard deviations) in its cross-term.

Key formulas to remember

Covariance from joint probabilities: Cov(A,B) = Σ P(i) × [A(i) − E(A)] × [B(i) − E(B)]; Sum over all scenarios i. First find E(A) and E(B) as probability-weighted means.
Covariance, shortcut form: Cov(A,B) = E(AB) − E(A) × E(B); E(AB) = Σ P(i) × A(i) × B(i). Handy when deviations are messy.
Sample covariance from historical data: Cov(A,B) = Σ (Aᵢ − Ā)(Bᵢ − B̄) ÷ (n − 1); Use n − 1 for a sample. Use n only for a full population.
Correlation: Corr(A,B) = Cov(A,B) ÷ (σA × σB); Range is −1 to +1. Same sign as covariance.
Covariance from correlation: Cov(A,B) = Corr(A,B) × σA × σB; Rearranged form, used in portfolio variance questions.
Covariance with itself: Cov(A,A) = Var(A); Variance is a special case of covariance.

How to solve Covariance and Correlation of Asset Returns questions

Use this method for any covariance or correlation question, whether the data are joint probabilities, a list of returns or given statistics.

1Identify what is given: scenario probabilities and returns, historical return pairs, or covariance and standard deviations.
2If you need means, compute the expected return of each asset: E(A) = Σ P × A, or the simple average for historical data.
3Compute each asset's deviation from its mean in every scenario or period.
4Multiply the two deviations for each scenario, then weight by the probability (probability-weighted sum), or sum and divide by n − 1 for sample data.
5If correlation is asked, compute σA and σB (square roots of the variances) from the same data.
6Divide covariance by σA × σB. Check the result lies between −1 and +1.
7Check the sign makes sense against the data pattern, and read the answer: positive, negative or near zero, strong or weak.

Quickest way: Shortcut for joint-probability covariance and correlation

When to use it: Use when the question gives a small table of scenarios with probabilities and the answer options are far apart.

Estimate the sign first: if the two return columns rise and fall together across scenarios, covariance is positive. This often eliminates one option immediately.
Check the bounds: a correlation outside −1 to +1 is impossible, so eliminate it.
If standard deviations and correlation are given, just multiply: Cov = Corr × σA × σB. No table work needed.
If the table is needed, use the shortcut E(AB) − E(A)E(B) and enter values in the calculator in decimal form to avoid slips.
On the BA II Plus, the DATA/STAT worksheet (2nd DATA to enter the data, 2nd STAT to choose the mode) gives the correlation r and the standard deviations for historical data. Then covariance = r × sx × sy. For probability tables, compute directly with the memory keys.
Round only at the end, then pick the matching option from the three listed smallest to largest.

Common mistakes in Covariance and Correlation of Asset Returns

Treating covariance as a measure of strength of the relationship
Covariance is read like correlation, but its size depends on the units and on how volatile each asset is.
Fix: Covariance's sign shows direction, but its size is not comparable across pairs. Use correlation to judge strength.
Forgetting to weight by probability
Students multiply deviations and add them, as in a simple average.
Fix: With joint probabilities, always multiply each product of deviations by that scenario's probability.
Using n instead of n − 1 for sample data
The probability-table formula has no divisor, so the habit carries over.
Fix: For historical samples, divide by n − 1. Use n only if the question says it is the population.
Forgetting to square-root the variances before computing correlation
Variance is computed in the same sitting, and the number is used in the denominator by mistake.
Fix: Correlation divides by σA × σB, which are standard deviations, not variances.
Assuming zero correlation means the assets are unrelated
The word 'no relationship' is applied too broadly.
Fix: Zero correlation means no linear relationship only. A non-linear link can still exist.
Mixing percent and decimal units
Returns of 10% are entered as 10 in one place and 0.10 in another.
Fix: Pick one form. Covariance in percent-squared is 10,000 times the decimal value, so keep it consistent.

Worked examples

Example 1

Two assets have these joint scenarios. Bad: probability 0.30, A = −10%, B = −4%. Normal: probability 0.50, A = 8%, B = 6%. Good: probability 0.20, A = 20%, B = 12%. What is the covariance of A and B (in decimal form)? Options: (A) 0.0032 (B) 0.0063 (C) 0.0126

Show the solution

E(A) = 0.30 × (−10) + 0.50 × 8 + 0.20 × 20 = −3 + 4 + 4 = 5%.
E(B) = 0.30 × (−4) + 0.50 × 6 + 0.20 × 12 = −1.2 + 3 + 2.4 = 4.2%.
Deviations of A: −15, +3, +15. Deviations of B: −8.2, +1.8, +7.8.
Products: (−15)(−8.2) = 123; (3)(1.8) = 5.4; (15)(7.8) = 117.
Weighted: 0.30 × 123 = 36.9; 0.50 × 5.4 = 2.7; 0.20 × 117 = 23.4. Sum = 63.0 in percent-squared.
Convert to decimals: 63.0 ÷ 10,000 = 0.0063.
Match 0.0063 to option B. The other options are about half and double this value.

Answer: (B) 0.0063

Example 2

Asset X has a standard deviation of 12% and asset Y has a standard deviation of 20%. Their covariance is 0.0096. What is the correlation? Options: (A) 0.40 (B) 0.80 (C) 4.00

Show the solution

Convert standard deviations to decimals: σX = 0.12, σY = 0.20.
Product: 0.12 × 0.20 = 0.024.
Correlation = 0.0096 ÷ 0.024 = 0.40.
Check bounds: 0.40 is within −1 to +1. Option C (4.00) is impossible. Option B would need covariance 0.0192.

Answer: (A) 0.40

Exam tips

Eliminate impossible correlations first. Any option above +1 or below −1 is wrong, and this often removes one of the three choices.
Check the sign before calculating. If the scenario returns move in the same direction, the answer must be positive.
Watch units. If standard deviations are in percent and covariance is in decimals, convert before dividing.
Expect portfolio links: a question may give correlation and standard deviations and ask for covariance, then use it in portfolio variance.
For definition questions, remember: covariance gives direction, correlation gives direction and strength on a −1 to +1 scale, and both capture linear association only.

Practice questions from The Return and Risk of a Financial Portfolio

Covariance and Correlation of Asset Returns in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Covariance and Correlation of Asset Returns: frequently asked questions

What is the difference between covariance and correlation?

Covariance shows the direction of co-movement and is measured in squared-return units, so its size is hard to interpret. Correlation is covariance scaled by both standard deviations, giving a unit-free value from −1 to +1. Both always have the same sign.

How do you calculate covariance from joint probabilities?

Find each asset's expected return. For each scenario, multiply the two deviations from the means and then by the scenario probability. Add the results. The shortcut E(AB) − E(A)E(B) gives the same answer.

Can correlation be greater than 1?

No. Correlation always lies between −1 and +1. If your calculation gives a value outside this range, you have made an error, often by using variances instead of standard deviations.

Does zero correlation mean two assets are independent?

Not necessarily. Zero correlation means no linear relationship. Assets can still be related in a non-linear way, so zero correlation does not prove independence.