FRM Exam Part I · Measuring Return, Volatility, and Correlation

Normality, Skewness and Kurtosis of Returns Explained

Updated 11 October 2026 · Fact-checked

Returns are normal only if skewness is 0 and kurtosis is 3 (excess kurtosis 0). Financial returns usually show fat tails, meaning kurtosis above 3. Compute sample skewness S and kurtosis K, then use the Jarque-Bera statistic JB = (n ÷ 6) × [S² + (K − 3)² ÷ 4]. Compare it with the chi-squared critical value, 2 degrees of freedom.

Understand Normality, Skewness and Kurtosis of Returns

The normal distribution is symmetric and has thin tails. Many risk models assume returns follow it. The question is whether real returns do. You check this using the third and fourth standardized moments.

Skewness measures asymmetry. It is the average of cubed standardized deviations: S = E[(R − μ)³] ÷ σ³. A normal distribution has skewness 0. Negative skewness means a longer left tail: large losses are more likely than equally large gains. Equity returns often show negative skewness.

Kurtosis measures tail heaviness. It is K = E[(R − μ)⁴] ÷ σ⁴. A normal distribution has kurtosis 3. Excess kurtosis is K − 3. A positive value means fat tails (leptokurtic): extreme outcomes occur more often than a normal distribution predicts, and the peak is usually more pointed too. This is why normal-based VaR can understate tail risk.

The Jarque-Bera (JB) test checks both at once. The null hypothesis is that returns are normal, so S = 0 and excess kurtosis = 0. The statistic grows as skewness or excess kurtosis moves away from zero. Under the null, JB follows a chi-squared distribution with 2 degrees of freedom. The 5% critical value is about 5.99. If JB exceeds it, you reject normality.

In practice, daily returns on equities, currencies and commodities nearly always reject normality because of fat tails. Sample size matters: with large n, even small departures are detected.

Key formulas to remember

Skewness: S = [Σ(Rᵢ − R̄)³ ÷ n] ÷ σ̂³; Normal = 0. Negative means a longer left tail. Use the same σ̂ definition as your data set gives.
Kurtosis: K = [Σ(Rᵢ − R̄)⁴ ÷ n] ÷ σ̂⁴; Normal = 3. Above 3 means fat tails (leptokurtic).
Excess kurtosis: Excess kurtosis = K − 3; Normal = 0. Many questions give K and ask you to subtract 3.
Jarque-Bera statistic: JB = (n ÷ 6) × [S² + (K − 3)² ÷ 4]; Null: returns are normal. Compare with chi-squared, 2 degrees of freedom.
Critical value: χ²(2) at 5% ≈ 5.99; at 1% ≈ 9.21; Reject normality if JB is greater than the critical value.

How to solve Normality, Skewness and Kurtosis of Returns questions

Use this order for any question on normality, skewness and kurtosis.

1Identify what is given: raw returns, or already-computed S and K, and the sample size n.
2If you only have returns, compute the mean, then the standard deviation, then the standardized third and fourth moments.
3Interpret skewness: zero is symmetric, negative is a long left tail, positive is a long right tail.
4Interpret kurtosis against 3. Compute excess kurtosis as K − 3 if needed.
5For a Jarque-Bera question, plug S, K and n into JB = (n ÷ 6) × [S² + (K − 3)² ÷ 4].
6Compare JB with the chi-squared critical value (2 df): 5.99 at 5%, 9.21 at 1%.
7State the conclusion: reject or fail to reject normality, and link it to risk, such as understated tail risk in normal VaR.

Quickest way: Fast Jarque-Bera check

When to use it: When S, K and n are given and you must decide whether normality is rejected.

Check whether K is given as kurtosis or as excess kurtosis. Subtract 3 only if it is kurtosis.
Square S and square the excess kurtosis, then divide the second square by 4.
Add the two terms and multiply by n ÷ 6.
Compare with 5.99 for a 5% test. Skip any further work once the answer is clear.
If the options are only about meaning, remember: K above 3 means fat tails, negative S means a long left tail.

Common mistakes in Normality, Skewness and Kurtosis of Returns

Using kurtosis K instead of excess kurtosis K − 3 in the Jarque-Bera formula.
The formula looks like it asks for K, and the subtraction is easy to forget.
Fix: Always write (K − 3) first. A normal series must give JB = 0.
Saying a kurtosis of 3 means fat tails.
Students mix up kurtosis and excess kurtosis.
Fix: Kurtosis 3 is normal. Fat tails need kurtosis above 3, or excess kurtosis above 0.
Reading negative skewness as a negative average return.
The word negative is linked with losses.
Fix: Skewness describes tail asymmetry, not the mean. Negative skewness means a longer left tail.
Using the wrong degrees of freedom, such as 1 or n − 1.
Other tests use those values.
Fix: JB tests two restrictions (skewness and excess kurtosis), so it uses 2 degrees of freedom.
Concluding that normality is accepted when JB is below the critical value.
Confusing failing to reject with proving the null.
Fix: Say you fail to reject normality. The data give no evidence against it at that significance level.

Worked examples

Example 1

A sample of 250 daily returns has skewness −0.40 and kurtosis 5.2. Test normality using the Jarque-Bera statistic at the 5% level (critical value 5.99).

Show the solution

Excess kurtosis = 5.2 − 3 = 2.2.
S² = (−0.40)² = 0.16.
(K − 3)² ÷ 4 = 4.84 ÷ 4 = 1.21.
Sum = 0.16 + 1.21 = 1.37.
JB = (250 ÷ 6) × 1.37 = 41.6667 × 1.37 = 57.08.
57.08 is greater than 5.99, so reject the null.

Answer: JB ≈ 57.08, above 5.99. Reject normality; returns show negative skew and fat tails.

Example 2

A sample of 60 monthly returns has skewness 0.30 and kurtosis 3.4. Is normality rejected at the 5% level? What does the kurtosis suggest?

Show the solution

Excess kurtosis = 3.4 − 3 = 0.4.
S² = 0.09.
(K − 3)² ÷ 4 = 0.16 ÷ 4 = 0.04.
Sum = 0.09 + 0.04 = 0.13.
JB = (60 ÷ 6) × 0.13 = 10 × 0.13 = 1.30.
1.30 is less than 5.99, so do not reject the null.

Answer: JB = 1.30, below 5.99. Fail to reject normality. Kurtosis is slightly above 3, but the departure is not statistically significant with this sample.

Exam tips

Read carefully whether the question gives kurtosis or excess kurtosis. Examiners use both.
Memorize 5.99 (5%) and 9.21 (1%) for chi-squared with 2 degrees of freedom.
Expect conceptual options linking fat tails to understated VaR under a normal assumption.
The JB calculation is short. Do it on the calculator in one pass: square, add, multiply by n ÷ 6.
Remember that larger n makes JB larger for the same S and K, so large samples reject normality more easily.

Practice questions from Measuring Return, Volatility, and Correlation

Normality, Skewness and Kurtosis of Returns in other exams

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

Normality, Skewness and Kurtosis of Returns: frequently asked questions

Are financial returns normally distributed?

Usually not. Empirical returns tend to have fat tails (excess kurtosis above 0) and often negative skewness. Longer horizons, such as monthly returns, look closer to normal than daily returns.

What is excess kurtosis?

It is kurtosis minus 3. A normal distribution has excess kurtosis of 0. Positive values mean fatter tails than the normal, so extreme moves are more frequent.

How do I read the Jarque-Bera result?

The null hypothesis is that returns are normal. If JB exceeds the chi-squared critical value with 2 degrees of freedom, such as 5.99 at 5%, you reject normality. If not, you fail to reject it.

Why do fat tails matter for risk management?

A normal model assigns very low probability to extreme losses. If real returns have fat tails, measures like normal VaR can understate the chance and size of large losses.