Math Atom · M-C08

Skewness and fat tails

8 minMath AtomProbability

By the end you will be able to

  1. 1Recognise a skewed distribution from a picture, say which way it leans, and where the mean, median and peak sit.
  2. 2Calculate skewness and kurtosis of a simple distribution as averages of cubed and fourth-power z-scores.
  3. 3Explain kurtosis, excess kurtosis and fat tails, with the normal (kurtosis 3) as the benchmark.
  4. 4Compare how often extreme moves happen under a normal and a fat-tailed distribution with the same standard deviation.

Why you need it

Mean and standard deviation describe the middle of a distribution. Risk lives in the edges. Two distributions can share a mean and a standard deviation and still disagree, by a factor of a hundred, about how often a disaster happens.

This Core Path lesson uses it:

It builds on M-C06: The normal distribution, which this atom uses as the benchmark.

The idea

Pictures first

Skew: which way does it lean? A normal curve is symmetric: the left half mirrors the right. Many real distributions are lopsided. Drag the slider and watch the mean, the median and the peak come apart.

Bend the curve

skewness 1

Long tail to the

this distribution (mean 0, sd 1) normal, same mean and sd
-3-2-10123meanmedianpeak

standard deviations from the mean

A long right tail: most outcomes are a little below average, while a few are far above it. The mean is dragged towards the tail, past the median.

The side with the long tail names the skew. A long right tail is positive skew: a lottery ticket, or the size of an insurance claim. A long left tail is negative skew: steady small gains with the occasional large loss.

Fat tails: how wild are the edges? Now keep the curve symmetric and change something subtler. Here are two curves with exactly the same mean and standard deviation.

Same mean, same standard deviation:— — normal—— fat-tailed
taller, narrower peakthinner shouldersfatter tails-4-2024

standard deviations from the mean

The fat-tailed curve takes probability from the “shoulders” and moves it to the middle and to the far ends. Most days are calmer than the normal suggests; a few are far wilder. Those fat tails barely show in the picture, but they decide how often a disaster comes.

A day the normal curve cannot explain

On 19 October 1987 the Dow Jones Industrial Average fell 22.6% in a single day. Suppose daily moves were normal with a standard deviation of about 1% (an assumption for illustration). That fall is about 22.6 standard deviations, and the normal curve puts its probability at about 1 in 10113. For comparison, there have been only about 3.5 × 1012 trading days’ worth of time since the universe began. The crash happened. The normal tails were wrong.

The rules

Both measures start from the z-score of M-C06: The normal distribution: how many standard deviations an outcome is from the mean. Then they raise it to a power and average.

Why powers 3 and 4? Look at what they do to outcomes further and further from the mean:

z (sd from mean)z³ (skewness)z⁴ (kurtosis)
0.50.1250.0625
111
2816
32781
464256

Outcomes close to the mean hardly count; far ones dominate. The fourth power is even more extreme, which is why kurtosis is a tail measure.

The rules, for reference

Skew(X)=E ⁣[(X−μσ)3]Kurt(X)=E ⁣[(X−μσ)4]\text{Skew}(X) = \mathbb{E}\!\left[\left(\tfrac{X-\mu}{\sigma}\right)^3\right] \qquad \text{Kurt}(X) = \mathbb{E}\!\left[\left(\tfrac{X-\mu}{\sigma}\right)^4\right]
excess kurtosis=Kurt(X)−3normal: skew 0, Kurt 3\text{excess kurtosis} = \text{Kurt}(X) - 3 \qquad \text{normal: skew } 0,\ \text{Kurt } 3

From data, the calculator’s skew(list) and kurt(list) give the sample versions (as in Excel), and kurt reports excess kurtosis. Estimates from a short series are noisy, because a single extreme day can swing them.

Try it

Both curves have mean 0 and standard deviation 1. Turn the tail dial ν down to fatten the tails, and move k to ask about bigger and bigger moves. Things to try: at k = 2 the two curves nearly agree; at k = 4 and k = 5 they part company completely. Then press “Another ten years” a few times and count the red days.

Try itNormal against fat tails

normal (sd 1) fat-tailed (sd 1, ν = 4) beyond ±k
-6-4-20246

Standard deviations from the mean. Look closely: the fat-tailed curve is taller in the middle, thinner in the shoulders, and higher far out. At this scale the tails look tiny.

The right tail, magnified: each gridline down is 10 times rarer
110⁻²10⁻⁴10⁻⁶10⁻⁸0123456
A move beyond ±4 sdNormalFat-tailed, ν = 4
Probability on a given day0.0063%0.48%
Expected days in 10 years (2,520)0.1612.13
About once every63 years208 days
How many times more often176×
Excess kurtosis0infinite

Ten years of illustrative days from each curve (simulation 1).

Normal world0 days beyond ±4 sd
Fat-tailed world, ν = 411 days beyond ±4 sd

Each thin line is one day, its height the day's move in standard deviations. Both worlds have the same standard deviation. Red days are beyond ±4.

The fat-tailed curve is a Student t distribution rescaled to standard deviation 1, a common simple model. The simulated days use a seeded random number generator and are illustrative, not market history.

Worked examples

1. Same mean and spread, opposite skew

Step 1 of 4
  1. 1

    The set-up

    Strategy A gains 1 with probability 0.9 and loses 9 with probability 0.1. Strategy B is the mirror image: it loses 1 with probability 0.9 and gains 9 with probability 0.1.

2. A rare storm raises kurtosis

Step 1 of 4
  1. 1

    The set-up

    OutcomeProbability
    −50.01
    −10.49
    10.49
    50.01

3. How often does a 4-sigma day come?

Step 1 of 4
  1. 1

    The set-up

    Compare a normal distribution with the fat-tailed model at ν = 4, both with standard deviation 1. How often is a day’s move more than 4 standard deviations, up or down? Assume 252 trading days a year.

Common mistakes

Each line below is a tempting slip. Spot what is wrong, then check.

Reaction

Check what stuck

Question 1 · objective 1

A strategy makes a small gain on most days and, very rarely, a large loss. Its distribution is…

Question 2 · objective 1

A distribution has a long right tail, like insurance claim sizes. Which is usually the largest?

Question 3 · objective 2

X is −1 or +1, each with probability 0.5 (mean 0, sd 1). What is its kurtosis (not the excess)?

Question 4 · objective 4

In the fat-tailed model with ν = 4 (sd 1), a move beyond ±4 sd is how many times more likely than under the normal? Nearest whole number.

Question 5 · objective 3

A set of daily returns has an excess kurtosis of 5. What does that tell you?

Glossary

Skewness
Lopsidedness. Skewness is the average cubed z-score, E[((X − μ)/σ)³]: positive for a long right tail, negative for a long left tail, zero if symmetric.
Median
The middle value: half the outcomes are below it and half above.
Kurtosis
The average fourth-power z-score, E[((X − μ)/σ)⁴]. Dominated by extreme outcomes, so it measures how heavy the tails are. The normal has 3.
Excess kurtosis
Kurtosis minus 3, so the normal scores 0. Positive excess kurtosis means fatter tails than the normal.
Fat tails
Tails that hold more probability than a normal curve with the same standard deviation: extreme outcomes happen more often than the normal says.
4-sigma move
Shorthand for a move of a given number of standard deviations: a 4-sigma day is a move of 4σ or more from the mean.

Previous · M-C06

The normal distribution

Next · M-C09

Quantiles