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Contrast the shape of the normal distribution with the way returns on risky financial assets, equities for example, are typically distributed.

FRM® Part I · Valuation and Risk Models · Measures of Financial Risk · FRM1.VR.1.b

The idea

The normal distribution is symmetric, with skewness 0 and kurtosis 3, and is fully described by its mean and variance. Returns on equities and other risky assets depart from it. They show excess kurtosis (kurtosis well above 3, meaning fat tails and a taller, narrower centre), often negative skewness (a longer, heavier left tail, with more weight on large losses), and volatility clustering, where turbulent days follow turbulent days. Moves of four or five standard deviations, which a normal model says should occur once in decades or centuries, appear in real data far more often. Skewness = E[(R − μ)³] ÷ σ³. Kurtosis = E[(R − μ)⁴] ÷ σ⁴. Excess kurtosis is kurtosis minus 3. The Jarque-Bera statistic tests normality, where S is the sample skewness, K the sample kurtosis and n the number of observations. JB = n/6 × (S² + (K − 3)²/4). In large samples it is compared with a chi-square distribution with 2 degrees of freedom. Exam point: when returns have fatter tails than the normal, normal-based VaR can understate far-tail losses.

Formulas

Excess kurtosis of returns

excess kurtosis = E[(R − μ)⁴] ÷ σ⁴ − 3

R
the return
μ
the mean return
σ
the standard deviation of returns

A normal distribution has kurtosis 3 and so excess kurtosis 0; fat-tailed equity returns are well above. The trap is dividing by σ² or σ³ instead of σ⁴, or forgetting to subtract 3.

Jarque-Bera test statistic for normality

JB = (n ÷ 6) × [S² + (K − 3)² ÷ 4]

n
the number of observations
S
the sample skewness
K
the sample kurtosis, not excess kurtosis

Compare it with a chi-square with 2 degrees of freedom; a large value rejects normality. The usual slips are using K instead of K − 3, or leaving out the division by 4.

A practice question

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

Which pair of skewness and kurtosis values describes a normal distribution?

  1. Skewness 0 and kurtosis 3. - correct
  2. Skewness 0 and kurtosis 0.Why not: Confuses kurtosis with excess kurtosis; excess kurtosis is 0 for a normal distribution, but kurtosis itself is 3.
  3. Skewness 3 and kurtosis 0.Why not: Swaps the two values; a normal distribution is symmetric, so its skewness is zero.
  4. Skewness 1 and kurtosis 3.Why not: Carries over the standard deviation of a standard normal as the skewness; a symmetric distribution has skewness 0.

A normal distribution is symmetric (skewness 0) and has kurtosis 3, which is why risk managers look at excess kurtosis, kurtosis minus 3, to measure tail heaviness.

A harder one

A sample of n = 1,000 daily returns has skewness -0.4 and kurtosis 5.2. The 5% critical value of a chi-square with 2 degrees of freedom is 5.99. The Jarque-Bera statistic, JB = n/6 x [S² + (K - 3)² / 4], is closest to:

  1. 833Why not: Omits the division by 4 on the excess kurtosis term, overstating the statistic.
  2. 228 - correct
  3. 1153Why not: Uses kurtosis (5.2) instead of excess kurtosis (K - 3), which would flag even a normal sample.
  4. 27Why not: Uses only the skewness term and ignores the kurtosis term, which carries most of the departure here.

JB = 1000/6 x [(-0.4)² + (5.2 - 3)² / 4] = 166.67 x (0.16 + 1.21) = 228.3. This is far above 5.99, so normality is rejected, consistent with fat-tailed returns.

The hardest tier

A risk manager assumes that daily equity returns are normally distributed and uses 252 trading days per year. Under this assumption, a one-day loss of at least 4 standard deviations would be expected about once every:

  1. 125 years - correct
  2. 63 yearsWhy not: Uses the two-tailed probability, counting gains of 4 standard deviations as well as losses.
  3. 87 yearsWhy not: Converts days to years with 365 rather than the 252 trading days the problem uses.
  4. 2.9 yearsWhy not: Uses the 3-standard-deviation tail probability of 0.00135 instead of 4 standard deviations.

P(Z < -4) = 3.17e-05, so one such loss in 31,574 days = 125 years. Real equity markets have produced moves of this size or larger several times in a few decades, which is evidence of fat tails.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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