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FRM Exam Part I · Random Variables

Moments, Skewness and Kurtosis for FRM Part I

Updated 11 October 2026 · Fact-checked

Moments summarise a distribution. The first raw moment is the mean; central moments measure spread around the mean. Skewness is the third central moment ÷ σ³ and shows asymmetry. Kurtosis is the fourth central moment ÷ σ⁴ and shows tail heaviness. A normal distribution has skewness 0 and kurtosis 3.

Understand Moments, Skewness and Kurtosis

A moment is the expected value of a power of a random variable. The k-th raw moment is E(Xᵏ). The first raw moment is the mean. Raw moments are measured around zero.

A central moment is measured around the mean: the k-th central moment is E[(X − μ)ᵏ]. The first central moment is always 0. The second is the variance. The third and fourth feed into skewness and kurtosis.

Raw moments depend on where the data sits. Central moments do not change if you shift every value by a constant. That is why risk work uses them. Standardised moments divide by σᵏ, so they have no units and do not change if you rescale the data.

Skewness = E[(X − μ)³] ÷ σ³. It measures asymmetry. Positive skew means a longer right tail, with the mean usually above the median. Negative skew means a longer left tail. Negative skew matters in risk because it signals occasional large losses. For a symmetric distribution, skewness is 0.

Kurtosis = E[(X − μ)⁴] ÷ σ⁴. It measures tail heaviness and peakedness. The normal distribution has kurtosis 3. Excess kurtosis = kurtosis − 3. A distribution with excess kurtosis above 0 is leptokurtic: fat tails and more extreme outcomes than the normal. Below 0 is platykurtic: thin tails. Equal to 0 is mesokurtic. Financial returns are usually leptokurtic, so normal-based VaR can understate tail risk.

Key formulas to remember

Raw moment
k-th raw moment = E(Xᵏ)
First raw moment is the mean μ.
Central moment
k-th central moment = E[(X − μ)ᵏ]
First is 0; second is variance σ².
Variance from raw moments
σ² = E(X²) − [E(X)]²
Handy when raw moments are given.
Skewness
Skew = E[(X − μ)³] ÷ σ³
Unitless. 0 for symmetric distributions. Positive means right tail.
Kurtosis
Kurt = E[(X − μ)⁴] ÷ σ⁴
Normal distribution = 3.
Excess kurtosis
Excess kurtosis = Kurt − 3
Above 0 is leptokurtic (fat tails); below 0 is platykurtic.

How to solve Moments, Skewness and Kurtosis questions

Use this order for any question on moments, skewness or kurtosis.

  1. 1Identify what is asked: raw moment, central moment, skewness, kurtosis or excess kurtosis.
  2. 2List the outcomes and probabilities. Check the probabilities sum to 1.
  3. 3Compute the mean μ = ΣxP(x).
  4. 4Compute the deviations (x − μ) and the powers needed: squared for variance, cubed for skewness, fourth power for kurtosis.
  5. 5Weight each power by its probability and sum to get the central moment.
  6. 6Divide by σ³ or σ⁴ to standardise. Use σ = √variance, not variance itself.
  7. 7For excess kurtosis subtract 3. Then interpret the sign: skew direction or fat versus thin tails.

Quickest way: Shortcut using symmetry and the normal benchmark

When to use it: Use when the question asks you to interpret a distribution or compare values, not compute them in full.

  1. If the distribution is symmetric, skewness is 0 without calculation.
  2. Compare kurtosis with 3, or excess kurtosis with 0, to classify the tails.
  3. Negative skew means the left tail is longer, so the mean is usually below the median.
  4. If the central moments are given, just divide: skew = m3 ÷ σ³, kurt = m4 ÷ σ⁴, with σ = √m2.
  5. Eliminate options that confuse kurtosis with excess kurtosis.

Common mistakes in Moments, Skewness and Kurtosis

  • Treating kurtosis of 3 as fat-tailed.

    Students forget that 3 is the normal benchmark.

    Fix: Compare with 3, or use excess kurtosis and compare with 0.

  • Dividing by variance instead of σ³ or σ⁴ powers of the standard deviation.

    Mixing up σ and σ².

    Fix: Take the square root of variance first, then cube or raise to the fourth.

  • Using raw moments where central moments are needed.

    The names sound alike.

    Fix: Central moments subtract the mean before raising to a power.

  • Reading positive skew as 'more losses'.

    Positive sounds good or bad depending on view.

    Fix: Positive skew means a long right tail; negative skew means a long left tail, which is the loss side for returns.

  • Saying a zero third moment proves normality.

    Normal has skew 0, so students reverse the logic.

    Fix: Skew 0 only shows no asymmetry. Many non-normal distributions are symmetric.

Worked examples

Example 1

A variable takes values 0, 2 and 4 with probabilities 0.5, 0.25 and 0.25. Find its skewness.

Show the solution
  1. Mean: μ = 0(0.5) + 2(0.25) + 4(0.25) = 1.5.
  2. Deviations: −1.5, 0.5, 2.5.
  3. Variance: 2.25(0.5) + 0.25(0.25) + 6.25(0.25) = 1.125 + 0.0625 + 1.5625 = 2.75.
  4. σ = √2.75 = 1.6583; σ³ = 2.75 × 1.6583 = 4.5604.
  5. Third central moment: (−3.375)(0.5) + (0.125)(0.25) + (15.625)(0.25) = −1.6875 + 0.03125 + 3.90625 = 2.25.
  6. Skewness = 2.25 ÷ 4.5604 = 0.4934.

Answer: Skewness ≈ 0.49 (positive, right-skewed).

Example 2

A return distribution has variance 4 and a fourth central moment of 80. Find the kurtosis and excess kurtosis, and classify the tails.

Show the solution
  1. σ² = 4, so σ⁴ = 16.
  2. Kurtosis = 80 ÷ 16 = 5.
  3. Excess kurtosis = 5 − 3 = 2.
  4. Excess kurtosis above 0 means fatter tails than the normal.

Answer: Kurtosis = 5, excess kurtosis = 2, leptokurtic.

Exam tips

  • Know the normal benchmarks cold: skewness 0, kurtosis 3, excess kurtosis 0.
  • Read carefully whether the question gives kurtosis or excess kurtosis.
  • Questions often ask for the risk implication: fat tails and negative skew mean normal VaR understates losses.
  • Calculation questions usually use a small table. Compute the mean first, then the powers, and keep four decimals.
  • Use your calculator's memory to store σ and avoid rounding errors in σ³ and σ⁴.

Practice questions from Random Variables

Moments, Skewness and Kurtosis in other exams

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

Moments, Skewness and Kurtosis: frequently asked questions

What is the difference between raw and central moments?

Raw moments are E(Xᵏ), measured around zero. Central moments are E[(X − μ)ᵏ], measured around the mean. Central moments describe shape and do not change when you shift the data.

What does excess kurtosis tell me?

It is kurtosis minus 3, the normal value. A positive value means leptokurtic, with fatter tails and more extreme outcomes than the normal. A negative value means thinner tails.

Why is negative skewness a concern in risk management?

It means the left tail is longer, so large losses are more likely than a symmetric model suggests. Models that assume normality can understate this risk.

Is the first central moment useful?

No. It is always 0 because positive and negative deviations from the mean cancel out. The second central moment, the variance, is the first useful one.