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CFA Level I Exam · Statistical Characteristics of Asset Returns

Covariance vs Correlation of Returns for CFA Level I

Updated 7 October 2026 · Fact-checked

Covariance measures whether two return series move together, but its size depends on the units. Correlation divides covariance by the product of the two standard deviations, giving a unit-free value from -1 to +1. To solve questions, compute covariance, then divide by both standard deviations, then check for outliers or spurious links.

Understand Correlation and Covariance of Returns

Covariance tells you the direction of the linear relationship between two variables. A positive value means they tend to be above or below their means at the same time. A negative value means one tends to be above its mean when the other is below. A value near zero means no linear pattern.

The problem is size. Covariance is measured in the product of the two units (for returns, percent squared), so a value of 40 or 400 tells you little about strength. You cannot compare covariances across different pairs of assets.

The correlation coefficient fixes this. You divide covariance by the standard deviation of each variable. The result has no units and always lies between -1 and +1. A value of +1 is a perfect positive linear relationship, -1 is a perfect negative one, and 0 means no linear relationship.

A scatter plot shows the pairs of observations as dots. An upward-sloping tight cloud means high positive correlation. A downward-sloping tight cloud means high negative correlation. A shapeless cloud means correlation near zero. A curved pattern warns you that correlation, which measures only linear association, may understate the true link.

Correlation has limits. Outliers can inflate or deflate it sharply. Spurious correlation is a high correlation with no real economic link, caused by chance, a hidden third variable, or using the same denominator in both variables. Nonlinear relationships can have a correlation near zero even when the variables are strongly related. Correlation also says nothing about cause and effect or about the size of the move in one variable for a move in the other.

Key formulas to remember

Sample covariance
Cov(X,Y) = Σ (Xi − X̄)(Yi − Ȳ) ÷ (n − 1)
Use n − 1 for a sample. Divide by n only if the question says it is a population.
Correlation coefficient
r = Cov(X,Y) ÷ (sX × sY)
Unit-free. Always between −1 and +1.
Covariance from correlation
Cov(X,Y) = r × sX × sY
Rearranged form. Common when a question gives correlation and standard deviations.
Covariance of a variable with itself
Cov(X,X) = Var(X)
Useful check: the diagonal of a covariance matrix holds variances.
Sample correlation in terms of deviations
r = Σ (Xi − X̄)(Yi − Ȳ) ÷ [√Σ(Xi − X̄)² × √Σ(Yi − Ȳ)²]
The n − 1 terms cancel, so you can skip them when you compute r directly.

How to solve Correlation and Covariance of Returns questions

Use this order for any covariance or correlation question.

  1. 1Read what is asked: covariance, correlation, or an interpretation of a scatter plot or limitation.
  2. 2Check whether data is a sample or a population. Samples use n − 1.
  3. 3Compute the mean of each series.
  4. 4Compute each deviation from the mean, multiply the pairs, and sum them.
  5. 5Divide the sum by n − 1 to get covariance.
  6. 6Divide covariance by the product of the two standard deviations to get correlation. Confirm it lies between −1 and +1.
  7. 7Interpret the sign and size, then check for outliers, nonlinearity or a spurious link before choosing an answer.
  8. 8Eliminate options that break a rule, such as a correlation above 1 or a covariance given as unit-free.

Quickest way: Shortcut with the calculator and rule checks

When to use it: When you have a few data pairs, or when a question gives two of covariance, correlation and standard deviations.

  1. If given r and both standard deviations, multiply to get covariance. If given covariance and standard deviations, divide to get r.
  2. Check the sign first. If the covariance is negative, correlation must be negative, so remove any positive option.
  3. Check range. Any correlation option outside −1 to +1 is wrong.
  4. For raw data on the TI BA II Plus: press 2ND DATA, then clear old data with 2ND CLR WORK. Enter X01, press ENTER, press the down arrow, enter Y01, press ENTER, press the down arrow, and repeat for each pair (X and Y go in separate rows).
  5. Then press 2ND STAT, press 2ND SET (2ND ENTER) until the method shows LIN, and scroll down to read n, X̄, Sx, Ȳ, Sy, a, b and r (Sx and Sy are the sample standard deviations).
  6. Covariance = r × Sx × Sy.
  7. For concept questions, scan for the keywords: outlier, spurious, nonlinear, causation.

Common mistakes in Correlation and Covariance of Returns

  • Dividing by n instead of n − 1 for sample covariance

    It feels like averaging, so students divide by the number of observations.

    Fix: Unless the data is stated to be a population, use n − 1. Look for the word sample.

  • Interpreting covariance size as strength of relationship

    A large covariance looks like a strong link.

    Fix: Covariance depends on units and volatility. Use correlation to judge strength.

  • Assuming zero correlation means no relationship

    Students forget that correlation captures only linear association.

    Fix: Two variables can be strongly related in a curve and still have correlation near zero. Say no linear relationship.

  • Reading high correlation as causation

    A tight scatter plot looks convincing.

    Fix: Correlation shows co-movement only. A spurious link can come from chance, a third variable, or shared denominators.

  • Ignoring outliers

    Students trust the computed r without looking at the data.

    Fix: One extreme point can create a high correlation or hide a real one. Check the scatter plot, and consider whether the outlier is an error or genuine data.

  • Forgetting to take the square root of the variance

    Questions give variances, and students plug them in directly as standard deviations.

    Fix: Take √variance first, then divide covariance by the product of the standard deviations.

Worked examples

Example 1

Asset X and Asset Y have a sample covariance of 54 (%²). The standard deviation of X is 6% and of Y is 12%. What is the correlation? A) 0.45 B) 0.75 C) 1.50

Show the solution
  1. r = Cov ÷ (sX × sY).
  2. sX × sY = 6 × 12 = 72.
  3. r = 54 ÷ 72 = 0.75.
  4. Option C is above 1, so it is impossible. Option A would need covariance of 32.4.

Answer: B) 0.75

Example 2

Returns (%) for two funds over three periods: Fund A: 2, 4, 6. Fund B: 10, 6, 8. What is the sample covariance? A) −2 B) 1 C) 2

Show the solution
  1. Mean of A = (2 + 4 + 6) ÷ 3 = 4. Mean of B = (10 + 6 + 8) ÷ 3 = 8.
  2. Deviations of A: −2, 0, 2. Deviations of B: 2, −2, 0.
  3. Products: (−2)(2) = −4; (0)(−2) = 0; (2)(0) = 0. Sum = −4.
  4. Sample covariance = −4 ÷ (3 − 1) = −2 (%²).

Answer: A) −2

Exam tips

  • Questions often give correlation and standard deviations and ask for covariance. Memorize both directions of the formula.
  • Use the sign and the −1 to +1 range to eliminate two options quickly, since there is no penalty for guessing.
  • Concept questions test limitations: outliers, spurious correlation and nonlinearity. Match the scenario to the right limitation.
  • Do not confuse correlation with causation. Any option claiming one variable causes the other based only on correlation is usually wrong.
  • Read whether data is a sample or population before dividing.

Practice questions from Statistical Characteristics of Asset Returns

Correlation and Covariance of Returns in other exams

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

Correlation and Covariance of Returns: frequently asked questions

What is the difference between covariance and correlation?

Covariance shows the direction of the linear relationship but depends on the units of the variables. Correlation is covariance scaled by both standard deviations, so it is unit-free and lies between −1 and +1. That makes correlation the measure for comparing strength.

How do I calculate the correlation coefficient on the calculator?

On the TI BA II Plus, press 2ND DATA and clear old data with 2ND CLR WORK. Enter each pair in separate rows: X01, ENTER, down arrow, Y01, ENTER, down arrow, and so on. Then press 2ND STAT, press 2ND SET (2ND ENTER) until the method shows LIN, and scroll down through n, X̄, Sx, Ȳ, Sy, a and b to reach r. The same list gives the sample standard deviations Sx and Sy.

What is spurious correlation?

It is a correlation that looks meaningful but has no real economic link. It can arise by chance, from a hidden third variable that drives both series, or from using the same denominator in both. The exam expects you to question it before accepting a relationship.

Can correlation be zero when two variables are related?

Yes. Correlation measures only linear association. A curved relationship can have a correlation near zero even though one variable clearly determines the other. A scatter plot helps you spot this.