FRM Exam Part I · Sample Moments

Coskewness and Cokurtosis for FRM Part I

Updated 11 October 2026 · Fact-checked

Coskewness and cokurtosis are higher-order joint moments of two random variables. Coskewness measures whether one variable tends to be extreme when the other is extreme in a given direction. Cokurtosis measures joint tail behavior. You compute them as expected products of standardized deviations, such as E[(X−μX)²(Y−μY)] ÷ (σX²σY).

Understand Coskewness and Cokurtosis

Covariance tells you how two variables move together on average. It uses the product of two deviations. It cannot tell you whether they move together more in crashes than in rallies. For that you need higher moments of the joint distribution.

Coskewness uses three deviations. For X and Y there are two versions: S(X,X,Y) = E[(X−μX)²(Y−μY)] ÷ (σX²σY) and S(X,Y,Y) = E[(X−μX)(Y−μY)²] ÷ (σXσY²). The squared deviation is always positive. So the sign comes from the single deviation. A positive S(X,X,Y) means that when X is far from its mean (either way), Y tends to be above its mean. A negative value means Y tends to be below its mean. A value of zero says only that (X−μX)² and Y are uncorrelated, that is, Cov((X−μX)², Y) = 0. It does not mean large X moves have no link to Y.

Cokurtosis uses four deviations. There are three versions: K(X,X,X,Y), K(X,X,Y,Y) and K(X,Y,Y,Y). The symmetric one is K(X,X,Y,Y) = E[(X−μX)²(Y−μY)²] ÷ (σX²σY²). It captures the co-movement of squared deviations, regardless of sign. A value above 1 means E[(X−μX)²(Y−μY)²] exceeds σX²σY², so the squared deviations are positively correlated. Large moves in X and large moves in Y (in any direction) occur together more than they would under independence. This matters for portfolio risk, but it is a statement about squared deviations, not specifically about tail events. It is a signal, not proof of tail dependence.

If X and Y are independent (and the moments exist), the standardized coskewness values are 0. Under independence K(X,X,X,Y) = K(X,Y,Y,Y) = 0 and K(X,X,Y,Y) = 1. Note that K(X,X,Y,Y) = 1 is not enough to prove independence. If X = Y, coskewness becomes ordinary skewness and cokurtosis becomes ordinary kurtosis. Normal variables have skewness 0 and kurtosis 3.

Why it matters: variance and covariance alone underestimate risk when returns are skewed or fat-tailed. An asset with negative coskewness with the portfolio adds to crash risk even if its correlation looks modest. Investors who like positive skew prefer assets with positive coskewness.

Key formulas to remember

Coskewness S(X,X,Y): S(X,X,Y) = E[(X − μX)²(Y − μY)] ÷ (σX² σY); Standardized by σX² and σY. Sample version averages the products over observations.
Coskewness S(X,Y,Y): S(X,Y,Y) = E[(X − μX)(Y − μY)²] ÷ (σX σY²); The mirror version. Two coskewness measures exist for a pair.
Cokurtosis K(X,X,Y,Y): K(X,X,Y,Y) = E[(X − μX)²(Y − μY)²] ÷ (σX² σY²); Equals 1 if X and Y are independent. Other versions are K(X,X,X,Y) and K(X,Y,Y,Y).
Cokurtosis K(X,X,X,Y): K(X,X,X,Y) = E[(X − μX)³(Y − μY)] ÷ (σX³ σY); Reduces to the kurtosis of X when Y = X.
Special case X = Y: S(X,X,X) = skewness; K(X,X,X,X) = kurtosis; Normal distribution: skewness 0, kurtosis 3.
Counts of distinct comoments (two variables): Coskewness: 2; Cokurtosis: 3; Excluding the pure skewness and kurtosis of each variable.

How to solve Coskewness and Cokurtosis questions

Use this method whenever you are given paired data or a joint distribution and asked for a comoment.

1Identify which comoment is asked: S(X,X,Y), S(X,Y,Y), or one of the K versions. Note which variable is squared or cubed.
2Compute the means μX and μY.
3Compute each deviation (x − μX) and (y − μY) for every observation or outcome.
4Raise the deviations to the required powers and multiply them. Keep signs carefully for the odd powers.
5Take the expectation: sum of probability × product for a distribution, or average of the products for equal-weight data.
6Compute σX and σY using the same weighting (population divisor unless told otherwise).
7Divide by the standardizing term, for example σX²σY for S(X,X,Y).
8Interpret: sign for coskewness, size relative to 1 (independence benchmark) for K(X,X,Y,Y).

Quickest way: Table of deviations

When to use it: Use for small datasets or discrete joint distributions given in the question.

Build one table with columns: dx, dy, dx², dy², then the product needed.
Fill the sums in one pass and divide by n or by the probabilities.
Reuse dx² and dy² for the standard deviations, so you need no extra pass.
Check the sign first. If the answer options differ in sign, you may finish without full arithmetic.
If the question states variances or standard deviations, just plug them into the denominator.

Common mistakes in Coskewness and Cokurtosis

Using the wrong standardizing term, such as σXσY instead of σX²σY.
Students copy the covariance pattern.
Fix: Match the powers: the denominator powers equal the exponents in the numerator.
Believing zero correlation means zero coskewness or cokurtosis.
Correlation feels like a full dependence measure.
Fix: Correlation captures only linear co-movement. Higher comoments can be non-zero even when covariance is zero.
Treating K(X,X,Y,Y) = 1 as proof of independence.
It is the independence value, so it feels like an iff.
Fix: Independence implies 1, but 1 does not imply independence.
Losing the sign of an odd-power deviation.
Squaring is done mentally and the single deviation is dropped.
Fix: Write each deviation with its sign in the table before multiplying.
Confusing the benchmark: using 3 for cokurtosis of two different variables.
3 is the normal kurtosis of a single variable.
Fix: 3 applies to K(X,X,X,X) for a normal. For independent X,Y the K(X,X,Y,Y) benchmark is 1.
Thinking positive coskewness is bad for an investor.
The word skew is linked to risk in the mind.
Fix: Investors generally prefer positive skewness. Negative coskewness with the portfolio adds downside tail risk.

Worked examples

Example 1

Two returns X and Y have equally likely outcomes (X, Y): (−1, −2), (0, 0), (1, 2). Compute S(X,X,Y) = E[(X−μX)²(Y−μY)] ÷ (σX²σY).

Show the solution

μX = (−1 + 0 + 1) ÷ 3 = 0. μY = (−2 + 0 + 2) ÷ 3 = 0.
Products (x²)(y): (1)(−2) = −2; (0)(0) = 0; (1)(2) = 2.
E of products = (−2 + 0 + 2) ÷ 3 = 0.
The numerator is 0, so the standardized coskewness is 0 ÷ (σX²σY) = 0 (σX² = 2/3 and σY = √(8/3) are positive).

Answer: S(X,X,Y) = 0. This means X² and Y are uncorrelated, even though X and Y are perfectly linearly dependent (Y = 2X). The zero comes from the symmetric data.

Example 2

Equally likely outcomes for (X, Y): (2, 2), (−1, 0), (−1, −2). Compute K(X,X,Y,Y) = E[(X−μX)²(Y−μY)²] ÷ (σX²σY²) and compare with 1.

Show the solution

μX = (2 − 1 − 1) ÷ 3 = 0. μY = (2 + 0 − 2) ÷ 3 = 0.
x² values: 4, 1, 1. y² values: 4, 0, 4.
Products x²y²: 16, 0, 4. Mean = 20 ÷ 3 = 6.6667.
σX² = (4 + 1 + 1) ÷ 3 = 2. σY² = (4 + 0 + 4) ÷ 3 = 8/3 = 2.6667.
σX²σY² = 2 × 2.6667 = 5.3333.
K = 6.6667 ÷ 5.3333 = 1.25.

Answer: K(X,X,Y,Y) = 1.25, above the independence value of 1. In this small sample, the squared deviations of X and Y are positively related. Treat this as illustrative only: with just three data points, a K above 1 is not evidence of tail co-movement.

Exam tips

Memorize the pattern: the exponents in the numerator are repeated as powers of σ in the denominator.
Know the counts: two coskewness and three cokurtosis measures for a pair of variables.
Remember benchmarks: independence gives coskewness 0 and K(X,X,Y,Y) = 1; single normal variable gives skewness 0 and kurtosis 3.
Conceptual questions often ask what a negative coskewness means for a portfolio. Answer: more downside tail risk and less appeal to investors.
When asked why covariance is not enough, say it ignores asymmetry and tail dependence.

Practice questions from Sample Moments

Coskewness and Cokurtosis: frequently asked questions

What is coskewness in finance?

Coskewness measures how one asset's deviation from its mean, squared, relates to another asset's deviation. It shows whether two assets share asymmetric extreme moves. Negative coskewness with a portfolio signals added crash risk.

How many coskewness and cokurtosis measures exist for two variables?

There are two coskewness measures, S(X,X,Y) and S(X,Y,Y). There are three cokurtosis measures, K(X,X,X,Y), K(X,X,Y,Y) and K(X,Y,Y,Y). Each is standardized by the matching powers of the standard deviations.

Is cokurtosis of independent variables equal to 3?

No. For independent X and Y, K(X,X,Y,Y) equals 1. The value 3 is the kurtosis of a single normal variable, K(X,X,X,X).

Why do higher comoments matter for portfolio risk?

Variance and covariance miss skewness and tail dependence. Assets can look diversified on correlation but still fall together in extreme events. Higher comoments capture that joint tail risk.