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Outline the four population moments that are commonly used.

FRM® Part I · Quantitative Analysis · Random Variables · FRM1.QA.2.c

The idea

The four commonly used population moments describe a distribution's location, spread, asymmetry and tail weight. The first moment is the mean, the probability-weighted average of the outcomes: Mean = E(X). The second central moment is the variance, the average of the squared deviations: Variance = E((X − μ)²). Its square root is the standard deviation σ. Skewness is the standardised third central moment: Skewness = E((X − μ)³) ÷ σ³. It is zero for symmetric distributions and negative when the left tail is longer. Kurtosis is the standardised fourth central moment: Kurtosis = E((X − μ)⁴) ÷ σ⁴. It equals 3 for the normal distribution. Excess kurtosis is kurtosis minus 3. Higher kurtosis means fatter tails and more extreme outcomes than a normal model implies. A common slip is to forget the division by σ³ or σ⁴, or to forget to subtract 3 when excess kurtosis is asked.

Formulas

Population skewness

Skewness = E[(X − μ)³] ÷ σ³

X
the random variable
μ
its mean
σ
its standard deviation

Zero for a symmetric distribution, negative when the left tail is longer. Forgetting to divide by σ³ is the usual slip.

Population kurtosis

Kurtosis = E[(X − μ)⁴] ÷ σ⁴

X
the random variable
μ
its mean
σ
its standard deviation

Equals 3 for a normal distribution. Subtract 3 when excess kurtosis is asked.

A practice question

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

Which list gives the four population moments most commonly used to describe a distribution?

  1. Mean, median, mode and variance.Why not: Median and mode are location measures, not moments; the third and fourth moments are skewness and kurtosis.
  2. Mean, variance, covariance and correlation.Why not: Covariance and correlation are bivariate measures of dependence, not the third and fourth moments of one variable.
  3. Mean, variance, skewness and kurtosis. - correct
  4. Mean, standard deviation, value at risk and expected shortfall.Why not: Value at risk and expected shortfall are quantile-based risk measures, not population moments.

The first four moments measure location (mean), dispersion (variance), asymmetry (skewness, the standardised third moment) and tail weight (kurtosis, the standardised fourth moment).

A harder one

A random return R (percent) takes the values -4, 0, 2 and 6 with probabilities 0.20, 0.30, 0.30 and 0.20. What is the variance of R, in percent squared?

  1. 11.6Why not: Reports the second raw moment E[R²] without subtracting the squared mean.
  2. 10.6 - correct
  3. 12.6Why not: Adds the squared mean to E[R²] instead of subtracting it.
  4. 3.26Why not: Reports the standard deviation, the square root of the variance.

Mean = 1.0. E[R²] = 16(0.2) + 0 + 4(0.3) + 36(0.2) = 11.6. Variance = 11.6 - 1.0² = 10.6.

The hardest tier

A random variable X has E[X] = 1, E[X²] = 5 and E[X³] = 17. Its third central moment is E[(X - 1)³] = E[X³] - 3 E[X] E[X²] + 2 E[X]³. What is the skewness of X?

  1. 2.13Why not: Divides the raw third moment 17 by sigma³ = 8 instead of using the third central moment.
  2. 1.00Why not: Divides the third central moment 4 by the variance 4 instead of by sigma³.
  3. 0.50 - correct
  4. 0.06Why not: Divides the third central moment 4 by the variance cubed (64) instead of the standard deviation cubed.

Variance = 5 - 1 = 4, so sigma = 2. Third central moment = 17 - 3(1)(5) + 2 = 4. Skewness = 4 / 2³ = 0.50.

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