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Outline why the first two moments may not be enough to describe distributions that are non-normal.

FRM® Part I · Quantitative Analysis · Measuring Returns, Volatility, and Correlation · FRM1.QA.12.c

The idea

A normal distribution is completely described by its mean and variance, so for normal returns those two moments tell you everything, including the probability of any loss. Real returns are not normal. Skewness, the standardised third moment, measures asymmetry. A negative value means a longer left tail, with large losses more likely than equally large gains. Kurtosis, the standardised fourth moment, equals 3 for a normal distribution, and the gap from 3 is the excess: Excess kurtosis = kurtosis − 3. A positive excess kurtosis means fat tails, so extreme moves occur far more often than the normal model predicts. Two distributions can share the same mean and variance and still carry very different tail risk. That is why a normal VaR can understate losses at high confidence levels. High kurtosis also puts more mass near the centre, so a normal VaR can overstate risk at lower confidence levels.

A practice question

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

For a normal distribution, the skewness and the kurtosis (not excess kurtosis) are, respectively:

  1. 0 and 0Why not: Gives the excess kurtosis of a normal distribution, not its kurtosis.
  2. 3 and 0Why not: Swaps the two: skewness is 0 and kurtosis is 3.
  3. 0 and 3 - correct
  4. 1 and 1Why not: Treats the normal as having unit values for the higher moments, which has no basis.

A normal distribution is symmetric, so skewness is 0, and its kurtosis is 3, so excess kurtosis (kurtosis minus 3) is 0.

A harder one

Funds X and Y have identical monthly mean returns and standard deviations. X has skewness near zero and excess kurtosis near zero. Y has skewness of -1.2 and excess kurtosis of 5. An analyst using only the first two moments rates the funds equally risky. Which statement is most accurate?

  1. Y is safer, because its higher kurtosis means returns cluster more tightly around the mean.Why not: Confuses a taller peak with thinner tails; positive excess kurtosis means fatter tails.
  2. Y is likely to suffer larger extreme losses than X, which the first two moments do not reveal. - correct
  3. Y has fewer large losses than gains, because negative skewness means the right tail is longer.Why not: Reverses the sign of skewness: negative skewness is a longer left tail.
  4. The funds have the same tail risk, because the mean and standard deviation fully determine any return distribution.Why not: That holds only for the normal distribution; higher moments matter otherwise.

Negative skewness and positive excess kurtosis point to a long, fat left tail. The first two moments are blind to both, so equal mean and variance do not imply equal tail risk.

The hardest tier

Daily returns have mean zero and come from a mixture. With probability 90% the day is calm and returns are normal with standard deviation 1%; with probability 10% the day is stressed and returns are normal with standard deviation 3%. The kurtosis of the mixture is closest to:

  1. 3.00Why not: Assumes the mixture is normal because each component is normal.
  2. 2.78Why not: Computes E[sigma^4] / (E[sigma²])² and omits the factor of 3 for normal components.
  3. 8.33 - correct
  4. 13.02Why not: Standardises with the average standard deviation (1.2%) instead of the standard deviation implied by the average variance.

Variance = 0.9 x 1 + 0.1 x 9 = 1.8 (in %²). Fourth moment = 3 x (0.9 x 1 + 0.1 x 81) = 27. Kurtosis = 27 / 1.8² = 8.33. Mixing volatility regimes fattens the tails even though each component is normal.

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