Fat tails, skewness and kurtosis: why risk models cannot assume a normal world
The simplest picture of returns is the normal distribution, fully described by a mean and a standard deviation. Real return series are rarely that tidy, and the gap is largest where risk managers look: the far tail. This guide covers the measures of that gap and why they matter for value at risk. It is typically met in FRM® Part I quantitative analysis; the FRM landing page lists the other guides.
The first four moments in plain words
A moment summarises one feature of a distribution.
- Mean: the centre, the average return.
- Variance: the typical size of deviations from the mean; its square root is the volatility.
- Skewness: whether the distribution leans to one side.
- Kurtosis: how much probability sits in the extreme tails.
Skewness and kurtosis defined
With mean m, standard deviation s and N observations, the standardised moments are:
- Skewness = average of (x - m)^3, divided by s^3
- Kurtosis = average of (x - m)^4, divided by s^4
- Excess kurtosis = kurtosis - 3
Cubing keeps the sign of each deviation, so skewness is positive with a long right tail and negative with a long left tail. A normal distribution has skewness 0 and kurtosis 3, so excess kurtosis 0.
Negative skew is the worrying sign: many small gains and the occasional large loss. Positive excess kurtosis is called leptokurtic, or fat-tailed: against a normal distribution of the same variance, a higher peak, thinner shoulders and more probability far out in both tails.
What fat tails do to VaR
A normal VaR is mean minus z times standard deviation, with z about 1.645 at 95%, 2.326 at 99% and 3.090 at 99.9%. If the true tail is fatter, the formula understates the loss at high confidence levels. At moderate levels the effect can reverse, because a fat-tailed distribution with the same variance has less mass in the shoulders. The backtesting guide covers how understatement shows up as too many exceptions.
Worked example: an invented fund
Take 1,000 invented daily returns, in percent, for a fictional fund, generated in Python (seed 7) as a mixture: 90% of days from a normal with mean 0.05 and standard deviation 0.8, 10% from a normal with mean 0 and standard deviation 2.6. It is not market data. All figures were computed and asserted.
The sample statistics:
| Statistic | Value |
|---|---|
| Mean | 0.070% |
| Standard deviation (N - 1) | 1.077% |
| Skewness | 0.11 |
| Kurtosis | 8.31 |
| Excess kurtosis | 5.31 |
Skewness is near zero because the mixture is symmetric, yet excess kurtosis is large: they are separate properties. Now count days more than k standard deviations below the mean, against what a normal with the same mean and standard deviation expects in 1,000 days:
| Threshold | Observed days | Normal expects |
|---|---|---|
| Below 2 sd | 18 | 22.8 |
| Below 3 sd | 9 | 1.35 |
| Below 4 sd | 5 | 0.03 |
There are fewer moderate days than the normal predicts and many more extreme ones. A 4-sd daily loss is roughly a once-in-a-century event for a normal model; here it happened five times.
One-day VaR, as a loss in percent:
| Confidence | Normal VaR | Empirical VaR |
|---|---|---|
| 95% | 1.70 | 1.43 |
| 99% | 2.44 | 2.79 |
| 99.9% | 3.26 | 5.20 |
The normal model overstates the 95% figure and understates the 99% and 99.9% figures, by about 13% and 37%. The empirical 99.9% figure is the second-worst of 1,000 days, so it is very noisy: it shows direction, not a precise number.
Sample-estimation noise
Cubed and fourth-power deviations let a handful of extreme days dominate. For a truly normal sample of N observations the standard errors are roughly the square root of 6/N for skewness and of 24/N for excess kurtosis: about 0.08 and 0.15 at N = 1,000, and larger with fat tails. Splitting the example into four blocks of 250 days gives skewness from -0.24 to 0.63 and excess kurtosis from 4.7 to 6.2, from one identical process. Do not read much into a skewness of 0.1 or -0.2 from a year of daily data.
Usual mistakes
- Confusing kurtosis and excess kurtosis. A normal distribution has kurtosis 3 and excess kurtosis 0. Check which one a question or software reports; some packages report the excess figure.
- Getting the sign of skew wrong. Negative skew means a long left tail (large losses), not that most returns are negative.
- Equating kurtosis with high volatility. Kurtosis describes shape, not scale.
- Assuming skewness and kurtosis move together. A symmetric distribution can have very high kurtosis, as above.
How the exam typically tests it
Questions typically give a skewness or kurtosis figure and ask you to interpret it, to say whether a normal VaR is likely to be too high or too low, or to separate kurtosis from excess kurtosis.
Common questions
Is excess kurtosis the same as kurtosis? No. Excess kurtosis is kurtosis minus 3, so that the normal distribution scores zero. A kurtosis of 5 means an excess kurtosis of 2.
Does a fat-tailed distribution always give a higher VaR than a normal one? No. At high confidence levels such as 99% and above it usually does, for the same mean and variance. At lower levels such as 90% or 95% it can be lower, as the example shows.
What does negative skewness mean for a portfolio? The left tail is longer, so large losses are more likely than equally large gains.
Sources
- GARP, the FRM programme: what the exam is and its two parts.
The exam and the syllabus can change. Check GARP's current details before you book. Charta's practice questions for this topic let you test the sign and level checks above.