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FRM Exam Part I · Multivariate Random Variables

Covariance and Correlation for FRM Part I

Updated 11 October 2026 · Fact-checked

Covariance measures how two random variables move together: Cov(X,Y) = E[XY] − E[X]E[Y]. Correlation rescales it to a value between −1 and +1: ρ = Cov(X,Y) ÷ (σX σY). To solve questions, find the means, compute E[XY], subtract the product of means, then divide by both standard deviations.

Understand Covariance and Correlation

Covariance tells you whether two variables tend to be above or below their means at the same time. A positive value means they usually move in the same direction. A negative value means they move in opposite directions. Zero means no linear co-movement.

The problem with covariance is its units. It is measured in units of X times units of Y, so its size is hard to read. A covariance of 40 could be strong or weak depending on the scale of the data.

Correlation fixes this. You divide covariance by the product of the two standard deviations. The result has no units and always lies between −1 and +1. A value of +1 means a perfect increasing linear relationship. A value of −1 means a perfect decreasing linear relationship.

Correlation measures only linear dependence. Two variables can be strongly dependent and still have zero correlation. For example, if X is symmetric around zero and Y = X², then Cov(X,Y) = 0 even though Y is fully determined by X. So independence implies zero covariance (when the moments exist), but zero covariance does not imply independence.

In risk work, correlation drives diversification. Portfolio variance depends on covariances. Correlations also tend to rise in a crisis and are unstable over time, which is a key limitation. Sample correlation is also sensitive to outliers.

Key formulas to remember

Covariance (definition)
Cov(X,Y) = E[(X − μX)(Y − μY)]
Average product of deviations from the means.
Covariance (shortcut)
Cov(X,Y) = E[XY] − E[X]E[Y]
Fastest form when you have a joint probability table.
Correlation
ρXY = Cov(X,Y) ÷ (σX × σY)
Always between −1 and +1. Needs both standard deviations to be non-zero.
Sample covariance
s_XY = Σ(Xi − X̄)(Yi − Ȳ) ÷ (n − 1)
Divide by n − 1 for the unbiased sample estimate. Divide by n for a population of n equally likely points.
Covariance with itself
Cov(X,X) = Var(X)
Variance is a special case of covariance.
Scaling and shifting
Cov(aX + b, cY + d) = ac × Cov(X,Y)
Constants b and d drop out. Correlation is unchanged if a and c have the same sign and flips sign if they differ.
Variance of a sum
Var(X + Y) = Var(X) + Var(Y) + 2Cov(X,Y)
For a difference, the covariance term is subtracted: Var(X − Y) = Var(X) + Var(Y) − 2Cov(X,Y).
Independence
X, Y independent ⇒ Cov(X,Y) = 0
The reverse is not true in general.

How to solve Covariance and Correlation questions

Use this order for any covariance or correlation question, whether the data come as a list, a joint table or a statement of variances.

  1. 1Identify what is given: raw data, a joint probability table, or summary figures such as variances and a correlation.
  2. 2Compute E[X] and E[Y]. For a joint table, sum each variable times its probability, using the marginal probabilities or the joint cells.
  3. 3Compute E[XY] by multiplying x × y × probability for each cell and summing. For raw data, compute the average of the cross-products or the sum of deviation products.
  4. 4Apply Cov = E[XY] − E[X]E[Y]. For a sample, use Σ(deviation products) ÷ (n − 1).
  5. 5If correlation is needed, compute Var(X) = E[X²] − (E[X])² and the same for Y, take square roots, then divide the covariance by σX σY.
  6. 6Check the result. Correlation must lie in [−1, +1], and its sign must match the sign of the covariance.
  7. 7If asked about a linear combination, use the variance-of-a-sum formula with the covariance term and the weights squared or multiplied as required.

Quickest way: Joint table shortcut with E[XY] − E[X]E[Y]

When to use it: Use when the question gives a small joint probability table or a few data points and asks for covariance or correlation.

  1. Skip the deviation form. Compute only E[X], E[Y] and E[XY].
  2. Ignore cells where x = 0 or y = 0 when finding E[XY], since they add nothing.
  3. Subtract E[X] × E[Y] from E[XY] to get covariance.
  4. For correlation, find E[X²] and E[Y²] only if standard deviations are not given.
  5. Sanity check the sign: if high X goes with high Y in the table, covariance should be positive.
  6. On the calculator, use the statistics mode for raw data to get sample standard deviations and correlation directly, but confirm whether the question wants a sample or population figure.

Common mistakes in Covariance and Correlation

  • Treating zero correlation as independence.

    Students remember that independent variables have zero covariance and reverse the statement.

    Fix: Remember that correlation captures only linear dependence. Y = X² with symmetric X has zero correlation but full dependence.

  • Forgetting to subtract E[X]E[Y] when computing covariance.

    E[XY] is the last number computed and gets reported as the answer.

    Fix: Always write Cov = E[XY] − E[X]E[Y] first and fill in all three terms.

  • Reading covariance size as strength of relationship.

    Covariance looks like a measure of strength, but it depends on units.

    Fix: Use correlation to judge strength. Covariance only gives direction and a unit-dependent magnitude.

  • Dropping the 2Cov term in a variance of a sum.

    Students recall that variances add and forget this holds only for uncorrelated variables.

    Fix: Write Var(X + Y) = Var(X) + Var(Y) + 2Cov(X,Y) every time, and use a minus sign for a difference.

  • Using σ² where σ is needed, or the reverse, in the correlation denominator.

    Questions give variances in some cases and standard deviations in others.

    Fix: Convert to standard deviations first. The denominator is σX × σY, the square root of the product of variances.

  • Using n instead of n − 1 for sample covariance.

    Population and sample formulas look alike.

    Fix: If the data are described as a sample, divide by n − 1. Note that the sample correlation is the same either way if both covariance and standard deviations use the same divisor.

Worked examples

Example 1

X and Y have the joint distribution: (X=0, Y=0) with probability 0.30; (X=0, Y=2) with 0.20; (X=1, Y=0) with 0.10; (X=1, Y=2) with 0.40. Calculate Cov(X,Y).

Show the solution
  1. E[X] = 0 × (0.30 + 0.20) + 1 × (0.10 + 0.40) = 0.50.
  2. E[Y] = 0 × (0.30 + 0.10) + 2 × (0.20 + 0.40) = 2 × 0.60 = 1.20.
  3. E[XY]: only cells with both non-zero count. (1, 2) gives 1 × 2 × 0.40 = 0.80.
  4. Cov = E[XY] − E[X]E[Y] = 0.80 − 0.50 × 1.20 = 0.80 − 0.60 = 0.20.

Answer: Cov(X,Y) = 0.20 (positive, so X and Y tend to move together).

Example 2

Using the same distribution as above, calculate the correlation between X and Y.

Show the solution
  1. Cov(X,Y) = 0.20 from the previous example.
  2. X takes values 0 and 1, so E[X²] = 0.50. Var(X) = 0.50 − 0.50² = 0.25. σX = 0.50.
  3. Y takes values 0 and 2 with P(Y=2) = 0.60, so E[Y²] = 4 × 0.60 = 2.40. Var(Y) = 2.40 − 1.20² = 2.40 − 1.44 = 0.96. σY = √0.96 = 0.9798.
  4. ρ = 0.20 ÷ (0.50 × 0.9798) = 0.20 ÷ 0.4899 = 0.4082.

Answer: ρ ≈ 0.41

Exam tips

  • Joint table questions reward the shortcut E[XY] − E[X]E[Y]. Practise it until it takes under two minutes.
  • Expect conceptual items on correlation versus dependence. Zero correlation never proves independence, except in special cases such as the bivariate normal.
  • Watch for questions that change units or scale a variable. Covariance scales with the constants, while correlation does not change in magnitude.
  • Variance of a portfolio or of X − Y is a common use. Check the sign of the covariance term before you compute.
  • If the question mentions a crisis, think of correlations rising and diversification benefit shrinking.

Practice questions from Multivariate Random Variables

Covariance and Correlation 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: frequently asked questions

What is the difference between covariance and correlation?

Covariance shows the direction of the linear co-movement and is measured in the product of the two variables' units. Correlation divides covariance by both standard deviations, so it has no units and lies between −1 and +1. This makes correlation comparable across pairs of variables.

How do I calculate covariance from a joint probability table?

Find E[X] and E[Y] from the marginal probabilities. Then compute E[XY] by summing x × y × probability over all cells. Subtract E[X] × E[Y] from E[XY] to get covariance.

Does zero correlation mean the variables are independent?

No. Zero correlation means there is no linear relationship, but a nonlinear relationship can still exist. Independence implies zero correlation, but not the other way round, apart from special cases such as jointly normal variables.

Do I divide by n or n − 1 for sample covariance?

Use n − 1 for the unbiased sample estimate when the data are a sample. Use n only when the data are the full population or equally likely outcomes. Read the wording of the question to decide.