FRM Exam Part I · Fundamentals of Probability

Covariance, Correlation and Moments for FRM Part I

Updated 11 October 2026 · Fact-checked

Covariance measures how two variables move together: Cov(X,Y) = E[XY] − E[X]E[Y]. Correlation rescales it to between −1 and +1 by dividing by both standard deviations. Moments describe a distribution's shape: mean, variance, skewness (asymmetry) and kurtosis (tail heaviness). Solve questions by applying these formulas step by step.

Understand Covariance, Correlation and Moments

Covariance tells you whether two random variables tend to move in the same direction. A positive value means they tend to be above or below their means together. A negative value means one tends to be high when the other is low. Zero means no linear co-movement.

Covariance has a problem: its size depends on the units. Cov of returns in percent differs from Cov in decimals. Correlation fixes this. It divides covariance by the product of the two standard deviations, so it always lies between −1 and +1. Correlation measures only linear dependence. Two variables can be strongly dependent and still have zero correlation.

The moments of a distribution summarise its shape. A raw moment is E[X^k], taken about zero. A central moment is E[(X − μ)^k], taken about the mean. The first raw moment is the mean. The second central moment is the variance. The third and fourth central moments, once standardised, give skewness and kurtosis.

Skewness is the third central moment divided by σ³. It is zero for a symmetric distribution such as the normal. Positive skew means a long right tail. Negative skew means a long left tail, which matters in risk because large losses are more likely. Kurtosis is the fourth central moment divided by σ⁴. The normal distribution has kurtosis of 3. Excess kurtosis is kurtosis minus 3. Values above 3 mean fat tails and more extreme outcomes than a normal model implies.

Key formulas to remember

Covariance: Cov(X,Y) = E[(X − μX)(Y − μY)] = E[XY] − E[X]E[Y]; Cov(X,X) = Var(X).
Correlation: ρ = Cov(X,Y) ÷ (σX σY); Always between −1 and +1. Unitless.
Variance of a sum: Var(aX + bY) = a²σX² + b²σY² + 2ab·Cov(X,Y); Use −2ab·Cov for a difference X − Y with a = b = 1: Var(X − Y) = σX² + σY² − 2Cov.
Covariance scaling: Cov(aX + c, bY + d) = ab·Cov(X,Y); Constants added do not change covariance.
Raw moment: k-th raw moment = E[X^k]; Taken about zero.
Central moment: k-th central moment = E[(X − μ)^k]; Taken about the mean. Second central moment is variance.
Skewness: Skew = E[(X − μ)³] ÷ σ³; Zero for symmetric distributions.
Kurtosis: Kurt = E[(X − μ)⁴] ÷ σ⁴; Excess kurtosis = Kurt − 3; Normal has kurtosis 3.
Independence and covariance: Independent ⇒ Cov = 0, but Cov = 0 does not imply independent; Zero covariance only rules out linear dependence.

How to solve Covariance, Correlation and Moments questions

Use this routine for any question on covariance, correlation or moments.

1Identify what is asked: covariance, correlation, variance of a combination, or a moment measure.
2List the given inputs: variances or standard deviations, covariance or correlation, weights, and any joint probabilities.
3If only correlation is given, convert to covariance with Cov = ρσXσY before using the variance of a sum.
4For a combination, apply Var(aX + bY) = a²σX² + b²σY² + 2ab·Cov(X,Y). Mind the sign of b.
5For a joint distribution, compute E[X], E[Y] and E[XY] from the probabilities, then Cov = E[XY] − E[X]E[Y].
6For moments, find the mean first, then average the powers of deviations, and divide by σ³ or σ⁴ for standardised measures.
7Take a square root only at the end if you need a standard deviation.
8Check sense: correlation within ±1, variance not negative, normal kurtosis equals 3.

Quickest way: Shortcut for portfolio variance and joint tables

When to use it: Use when a question gives standard deviations and a correlation, or a small joint probability table.

Convert everything to variances and covariance in decimals first.
Compute the three pieces of the variance formula separately, then add them.
For a table, compute E[XY] = Σ p·x·y, subtract E[X]E[Y].
Eliminate options with a negative variance or a correlation beyond ±1.
If both variables have equal weights and equal volatility, variance of the sum is 2σ²(1 + ρ).

Common mistakes in Covariance, Correlation and Moments

Forgetting the covariance term, or using the wrong sign, in Var(X − Y).
Students remember that variances add and apply it blindly.
Fix: Write the full formula every time. For a difference the covariance term is subtracted.
Using correlation where covariance is needed in the variance formula.
The question gives ρ and the formula asks for Cov.
Fix: Convert first: Cov = ρσXσY.
Believing zero correlation means independence.
Correlation is taught as a measure of dependence.
Fix: Remember it captures only linear dependence. For example Y = X² can have zero correlation with X yet depend on it fully.
Confusing kurtosis with excess kurtosis.
Some questions quote one and some the other.
Fix: Normal kurtosis is 3 and excess kurtosis is 0. Read which one the question states.
Mixing raw and central moments.
Both use E[X^k], so formulas look similar.
Fix: Raw is about zero. Central is about the mean. Variance = E[X²] − (E[X])² links them.
Squaring a weight incorrectly or dropping the factor 2 on the covariance term.
Rushing through the expansion.
Fix: Expand (aX + bY)² explicitly: the cross term is 2ab.

Worked examples

Example 1

X and Y have standard deviations of 10% and 20% and a correlation of 0.30. A portfolio holds 40% in X and 60% in Y. Find the portfolio standard deviation.

Show the solution

σX = 0.10, σY = 0.20, ρ = 0.30, a = 0.4, b = 0.6.
Cov = 0.30 × 0.10 × 0.20 = 0.006.
a²σX² = 0.16 × 0.01 = 0.0016.
b²σY² = 0.36 × 0.04 = 0.0144.
2ab·Cov = 2 × 0.4 × 0.6 × 0.006 = 0.00288.
Variance = 0.0016 + 0.0144 + 0.00288 = 0.01888.
Standard deviation = √0.01888 = 0.13740.

Answer: About 13.74%

Example 2

X and Y have the joint distribution: (X=1, Y=2) with probability 0.5; (X=3, Y=6) with probability 0.3; (X=5, Y=4) with probability 0.2. Find Cov(X,Y).

Show the solution

E[X] = 0.5×1 + 0.3×3 + 0.2×5 = 0.5 + 0.9 + 1.0 = 2.4.
E[Y] = 0.5×2 + 0.3×6 + 0.2×4 = 1.0 + 1.8 + 0.8 = 3.6.
E[XY] = 0.5×2 + 0.3×18 + 0.2×20 = 1.0 + 5.4 + 4.0 = 10.4.
E[X]E[Y] = 2.4 × 3.6 = 8.64.
Cov = 10.4 − 8.64 = 1.76.

Answer: 1.76

Exam tips

Questions often hide the covariance inside a stated correlation. Convert it before you start.
Know that kurtosis above 3 means fat tails, and negative skew means larger left-tail losses.
Expect conceptual items on zero correlation versus independence.
Check whether the question asks for variance or standard deviation before you pick an option.
Watch for scaling: Cov(aX, bY) = ab·Cov(X,Y), and adding constants changes nothing.

Practice questions from Fundamentals of Probability

Covariance, Correlation and Moments in other exams

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

Covariance, Correlation and Moments: frequently asked questions

What is the difference between covariance and correlation?

Covariance shows the direction of joint movement but its size depends on units. Correlation divides by both standard deviations, giving a unitless value between −1 and +1. Use correlation to compare strength across pairs.

What is the difference between central and raw moments?

Raw moments are E[X^k], measured about zero. Central moments are E[(X − μ)^k], measured about the mean. Variance, skewness and kurtosis are built from central moments.

What kurtosis value does a normal distribution have?

A normal distribution has kurtosis of 3, so its excess kurtosis is 0. A higher value indicates fatter tails and more extreme outcomes than the normal.

How do I find the variance of the sum of two random variables?

Use Var(X + Y) = σX² + σY² + 2Cov(X,Y). For a difference, subtract 2Cov(X,Y). If the variables are uncorrelated, the covariance term drops out.