What is the Jarque-Bera test and how do you use it to check if financial returns are normal?

The Jarque-Bera (JB) test is exactly that — a joint test of whether a sample's skewness and excess kurtosis are consistent with a normal distribution. It is one of the most commonly referenced normality tests in FRM because financial return distributions routinely violate normality.

The JB Statistic

JB = (n/6) x [S^2 + (K-3)^2 / 4]

Where:

• n = number of observations
• S = sample skewness
• K = sample kurtosis (some formulas use excess kurtosis K-3 directly)

For a normal distribution: S = 0 and K = 3 (excess kurtosis = 0). So if the data is perfectly normal, JB = 0.

Under H0 (normality), JB follows a chi-squared distribution with 2 degrees of freedom.

Worked Example

Dunmore Capital's risk team analyzes 500 daily returns of their emerging markets fund and finds:

• Sample skewness: S = -0.45 (negative skew — left tail is longer)
• Sample kurtosis: K = 5.20 (leptokurtic — fat tails)

JB = (500/6) x [(-0.45)^2 + (5.20 - 3)^2 / 4]

JB = 83.33 x [0.2025 + 4.84/4]

JB = 83.33 x [0.2025 + 1.21]

JB = 83.33 x 1.4125

= 117.7

Critical value: chi^2(0.05, 2) = 5.99

Since 117.7 >> 5.99, we decisively reject normality. The returns exhibit both significant negative skewness and excess kurtosis.

Why It Matters for Risk Management

1. VaR models that assume normality will underestimate tail risk if JB rejects normality
2. Negative skewness means extreme losses are more likely than a normal distribution predicts
3. Excess kurtosis means both extreme gains and losses occur more frequently
4. If JB rejects, consider using t-distributions, EVT, or historical simulation instead of parametric normal VaR

Exam tip: The JB test is always two-tailed (JB >= 0) and uses 2 degrees of freedom regardless of sample size.

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