FRM Exam Part I · Multivariate Random Variables

Skewness, Kurtosis, Coskewness and Cokurtosis for FRM Part I

Updated 11 October 2026 · Fact-checked

Skewness is the standardized third central moment and measures asymmetry. Kurtosis is the standardized fourth central moment and measures tail heaviness; a normal distribution has kurtosis 3. Coskewness and cokurtosis extend these to several variables, capturing joint asymmetry and joint tail behavior that covariance misses. Solve by computing deviations, raising them to the right power, and dividing by σ.

Understand Skewness, Kurtosis and Coskewness/Cokurtosis

Moments describe the shape of a distribution. The mean is the first moment. Variance is the second central moment. Skewness and kurtosis are the third and fourth moments, standardized so they have no units.

Skewness = E[(X − μ)³] ÷ σ³. It is zero for a symmetric distribution such as the normal. Positive skew means a long right tail: rare large gains. Negative skew means a long left tail: rare large losses. Cubing keeps the sign of each deviation, so the direction of the tail matters.

Kurtosis = E[(X − μ)⁴] ÷ σ⁴. The normal distribution has kurtosis 3. Excess kurtosis = kurtosis − 3. A value above 3 is leptokurtic: fat tails and more extreme outcomes than the normal predicts. Fourth powers are always positive, so kurtosis says nothing about direction. Fat tails matter for VaR, because a normal model understates the chance of large losses.

Covariance only captures how two variables move together in a linear way. Coskewness looks at joint asymmetry, for example E[(X − μX)²(Y − μY)] ÷ (σX² σY) or E[(X − μX)(Y − μY)²] ÷ (σX σY²). It tells you whether X tends to be extreme in one direction when Y is very high or very low. Cokurtosis uses fourth powers, such as E[(X − μX)²(Y − μY)²] ÷ (σX² σY²), and shows whether extreme moves occur together.

The total power on the deviations equals the moment order: 3 for coskewness, 4 for cokurtosis. Two variables give 2 coskewness measures (powers 2,1 and 1,2) and 3 cokurtosis measures (powers 3,1; 2,2; 1,3) in the standard presentation. Joint tail dependence is why diversification can fail in a crisis.

Key formulas to remember

Skewness: S = E[(X − μ)³] ÷ σ³; Zero for symmetric distributions. Negative means a longer left tail.
Kurtosis: K = E[(X − μ)⁴] ÷ σ⁴; Normal = 3. Always positive.
Excess kurtosis: Excess K = K − 3; Positive means fatter tails than the normal.
Coskewness: S(X,X,Y) = E[(X − μX)²(Y − μY)] ÷ (σX² σY); Swap the squared variable to get S(X,Y,Y). Powers sum to 3.
Cokurtosis: K(X,X,Y,Y) = E[(X − μX)²(Y − μY)²] ÷ (σX² σY²); Powers sum to 4. Other versions use powers 3,1 or 1,3.
Sample skewness (equal weights): Ŝ = [Σ(xi − x̄)³ ÷ n] ÷ σ̂³; Use the same σ̂ convention as the question states.

How to solve Skewness, Kurtosis and Coskewness/Cokurtosis questions

Use this method for any calculation or interpretation question on higher moments.

1Identify which moment is asked: third (skewness, coskewness) or fourth (kurtosis, cokurtosis).
2Compute the mean of each variable.
3Compute deviations from the mean and raise them to the required power, keeping signs for odd powers.
4Average the powered deviations using the probabilities or 1/n as the question gives.
5Divide by the standard deviation raised to the matching total power (σ³ or σ⁴, or the product of σ powers for joint moments).
6For kurtosis, compare with 3 (or subtract 3 for excess kurtosis).
7Interpret: sign of skew gives tail direction; kurtosis above 3 means fat tails.

Quickest way: Table of deviations

When to use it: Use when you are given a small data set or a probability table and must compute a moment.

Write mean, then deviations d in one column.
Square d for variance, then cube or fourth-power it in the next column.
Check the sign: if the largest absolute deviation is negative, expect negative skew.
Divide once at the end by σ³ or σ⁴.
Eliminate options with the wrong sign or with kurtosis below the plausible range before finishing the arithmetic.

Common mistakes in Skewness, Kurtosis and Coskewness/Cokurtosis

Saying kurtosis of the normal is 0.
Confusing kurtosis with excess kurtosis.
Fix: Normal kurtosis is 3. Excess kurtosis is 0. Read which one the question asks for.
Dividing by variance instead of σ³ or σ⁴.
Mixing up powers of σ and σ².
Fix: Divide by σ raised to the same power as the moment order.
Dropping the sign of deviations before cubing.
Treating deviations as absolute values.
Fix: Keep negative signs. A negative deviation cubed is negative.
Thinking zero skewness means a normal distribution.
Treating the normal as the only symmetric distribution.
Fix: Zero skew only shows symmetry. Check kurtosis too.
Assuming zero correlation means zero coskewness or cokurtosis.
Believing covariance captures all dependence.
Fix: Higher joint moments can be nonzero even when covariance is zero.
Thinking high kurtosis means more variance.
Equating fat tails with spread.
Fix: Kurtosis is standardized by σ. It describes tail shape, not the size of the spread.

Worked examples

Example 1

A return X takes values −4%, 0%, 2%, 2% with equal probability (each 1/4). Mean is 0%. Compute skewness. Use population moments.

Show the solution

Mean = (−4 + 0 + 2 + 2) ÷ 4 = 0.
Squares: 16, 0, 4, 4. Variance = 24 ÷ 4 = 6. σ = √6 ≈ 2.4495.
Cubes: −64, 0, 8, 8. Sum = −48. Third moment = −48 ÷ 4 = −12.
σ³ = 6 × 2.4495 ≈ 14.697.
Skewness = −12 ÷ 14.697 ≈ −0.816.

Answer: Skewness ≈ −0.82. The distribution has a longer left tail.

Example 2

A return series has kurtosis of 6. What is its excess kurtosis, and what does this mean for a normal-based VaR?

Show the solution

Excess kurtosis = K − 3 = 6 − 3 = 3.
A value above 0 (K above 3) means leptokurtic: fatter tails than the normal.
Extreme losses are more likely than a normal model predicts.

Answer: Excess kurtosis is 3. A normal-based VaR at high confidence is likely to understate tail losses.

Exam tips

Check whether the question asks for kurtosis or excess kurtosis. Many wrong options use the other one.
Questions often ask you to interpret, not calculate: sign of skew, kurtosis versus 3, and what zero covariance does not imply.
For coskewness and cokurtosis, count the powers. They must sum to 3 or 4.
Link fat tails to risk: normal-based VaR can understate losses when kurtosis is high or skew is negative.
Use the sign shortcut to remove options before doing heavy arithmetic.

Practice questions from Multivariate Random Variables

Skewness, Kurtosis and Coskewness/Cokurtosis: frequently asked questions

What is cokurtosis in finance?

Cokurtosis measures how the fourth powers of deviations of two or more variables move together. It shows whether extreme moves in assets tend to occur at the same time. It is a joint tail-risk measure.

What is the kurtosis of a normal distribution?

It is 3. Excess kurtosis is kurtosis minus 3, so the normal has excess kurtosis of 0. Values above 3 indicate fat tails.

Does negative skewness mean losses?

It means the left tail is longer, so large losses are more extreme than large gains. The mean can still be positive. It describes shape, not average return.

Why do coskewness and cokurtosis matter if I have covariance?

Covariance only measures linear co-movement. Two assets can have low covariance yet crash together, which coskewness and cokurtosis help reveal. They matter for portfolio tail risk.