Skewness and kurtosis

What does skewness tell you?

Mean and standard deviation do not say whether the big moves are gains or losses. Squaring hides the sign. Skewness restores it: it measures how lopsided a distribution is around its mean. A distribution that is not symmetric is skewed .

Skewnessasymmetry of returns around the mean

• Symmetric (skew 0) Memorize mean = median = mode
• Positive skew Memorize long right tail: frequent small losses, a few large gains
  • Order mode < median < mean
  • Examples lottery-like payoffs, bought options
• Negative skew Memorize long left tail: frequent small gains, a few large losses
  • Order mean < median < mode
  • Examples selling insurance, equity index returns

For a unimodal skewed distribution, the order of the averages follows the tail: the mean is dragged toward the tail, the mode stays at the peak and the median falls between them.

For a given mean and standard deviation, investors prefer positive skew. Relative to the mean, positive skew means limited but frequent downside and rarer, unlimited-looking upside. Equity index returns are usually negatively skewed.

The measure itself is built from cubed deviations. Cubing, unlike squaring, keeps the sign, so a long right tail gives a positive number. For large samples (100 or more observations), the sample skewness is approximately:

$$\text{Skewness} \approx \frac{\frac{1}{n}\sum_{i=1}^{n} (X_i - \bar{X})^3}{s^3}$$

You will not be asked to compute it by hand. You need to read its sign.

What does kurtosis tell you?

Kurtosis measures the combined weight of the tails relative to the rest of the distribution: how often extreme returns, up or down, occur. The normal distribution has a kurtosis of 3. Because 3 is the benchmark, analysts usually work with excess kurtosis .

$$K_E = K - 3$$

Kurtosisweight of the tails relative to the normal (kurtosis 3)

• Mesokurtic Memorize kurtosis = 3, excess = 0, like the normal
• Leptokurtic Memorize kurtosis > 3, excess > 0, fat-tailed
  • Consequence extreme returns more often than the normal predicts
• Platykurtic Memorize kurtosis < 3, excess < 0, thin-tailed

A leptokurtic distribution is fat-tailed . A platykurtic one is thin-tailed . A mesokurtic distribution has tails like the normal.

The large-sample estimate of sample excess kurtosis uses fourth powers of the deviations:

$$K_E \approx \frac{\frac{1}{n}\sum_{i=1}^{n} (X_i - \bar{X})^4}{s^4} - 3$$

Slide the skew and watch the mean, median and mode reorder. Raise the excess kurtosis and watch the tails above and below the normal curve.

Why do shape measures matter for risk?

Negative skew plus fat tails is the combination that risk managers fear. Returns look steady for long stretches and then lose a lot at once, and a standard deviation computed from the whole series understates the chance. A model that assumes a normal distribution therefore underestimates the probability of a large loss, which is why value at risk from a normal curve can be too optimistic.

Exam tip Skew = which way the tail points. Kurtosis = how fat the tails are. A question that gives kurtosis may want excess kurtosis: subtract 3 first.

Go deeper Why the lognormal distribution is positively skewed

Stock prices modeled with compounding returns follow a lognormal distribution. It cannot fall below zero, but it has no ceiling, and compounding makes gains grow faster than losses shrink. The result is a long right tail and positive skew. This is covered in the simulation methods module.

Go deeper How the formulas work

Skewness averages the cubed deviations and divides by s³ so the answer has no units. Kurtosis averages fourth powers and divides by s⁴. Fourth powers magnify large deviations enormously: a deviation twice as large counts sixteen times as much, which is why kurtosis is driven by the tails and barely notices what happens near the mean. Statistical software often reports excess kurtosis under the plain label “kurtosis”.