Use data to estimate the skewness and kurtosis of a random variable and interpret the results.

FRM® Part I · Quantitative Analysis · Sample Moments · FRM1.QA.5.h

The idea

Sample skewness averages the cubed standardised deviations. Each standardised deviation is z = (x − mean) ÷ sd. Skewness = Σz³ ÷ n. Negative values signal a longer or heavier left tail, meaning large losses are more extreme than large gains, as often seen in equity returns. Sample kurtosis averages the fourth powers of the standardised deviations. Kurtosis = Σz⁴ ÷ n. It equals 3 for a normal distribution. Excess kurtosis is kurtosis minus 3, and positive values mean fat tails with more extreme outcomes than normal. Small samples make both statistics noisy, and a single outlier can dominate them. For normal data their standard errors are approximately as follows. SEskew = √(6 ÷ n) and SEkurt = √(24 ÷ n). The exam point is to remember to subtract 3 for excess kurtosis and to read the sign of skewness correctly.

Formulas

Sample skewness

Skewness = Σ((xᵢ − x̄) ÷ s)³ ÷ n

xᵢ: each observation
x̄: the sample mean
s: the sample standard deviation
n: the number of observations

Negative values signal a heavier left tail. For normal data its standard error is about √(6 ÷ n).

Sample kurtosis

Kurtosis = Σ((xᵢ − x̄) ÷ s)⁴ ÷ n

xᵢ: each observation
x̄: the sample mean
s: the sample standard deviation
n: the number of observations

Equals 3 for normal data, so subtract 3 for excess kurtosis. For normal data the standard error of excess kurtosis is about √(24 ÷ n).

A practice question

From the question bank, with the reason each wrong option is wrong.

The sample kurtosis of a series of daily returns is 6.2. What is the best interpretation?

Excess kurtosis is 3.2, so tails are fatter than normal (kurtosis 3) and extreme moves are more likely. - correct
Its excess kurtosis is 6.2, so tails are fatter than normal.Why not: Kurtosis already includes the normal's 3; the excess kurtosis is 6.2 - 3 = 3.2.
Its tails are thinner than normal, because the value is above zero.Why not: Zero is the benchmark for excess kurtosis; for kurtosis the benchmark is 3, and above it means fatter tails.
It is negatively skewed, because kurtosis above 3 means a longer left tail.Why not: Kurtosis measures tail weight in both directions and does not indicate the sign of skewness.

Kurtosis is the average fourth power of standardised deviations, equal to 3 for a normal distribution. Excess kurtosis = kurtosis - 3. A value of 6.2 means leptokurtic, fat-tailed returns, which a normal VaR model understates.

A harder one

Eight daily returns (in %) are -4.0, -1.0, 0.5, 1.0, 1.5, 2.0, 2.5 and 3.0. The sample skewness is the average of the cubed standardised deviations, using the divisor-n standard deviation. What is it, closest to?

-0.93Why not: Standardises with the n - 1 standard deviation, which is not the stated divisor.
-1.14 - correct
-9.09Why not: Adds the cubed standardised deviations without dividing by n.
1.14Why not: Drops the sign: the large negative return dominates the sum, so the skewness is negative.

Mean = 0.6875, divisor-n variance = 4.496, standard deviation = 2.120. Averaging the cubed z-scores gives -1.136: a long left tail from the -4.0 return.

The hardest tier

A sample of returns has a standard deviation of 2%, a third central sample moment of -12 (%³) and a fourth central sample moment of 80 (%^4). What are the sample skewness and excess kurtosis?

-1.5 and 5.0Why not: Reports the kurtosis (5.0) as if it were the excess kurtosis; it needs 3 subtracted.
-3.0 and 2.0Why not: Divides the third moment by sigma² instead of sigma³.
-6.0 and 17.0Why not: Divides each moment by too low a power of sigma: sigma for skewness and sigma² for kurtosis.
-1.5 and 2.0 - correct

Skewness = m3/sigma³ = -12/8 = -1.5. Kurtosis = m4/sigma^4 = 80/16 = 5.0, so excess kurtosis = 2.0.

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