Skewness and Kurtosis

Mean and standard deviation describe the centre and spread of returns, but they miss two features that matter a lot for risk: asymmetry and tails. Skewness measures whether a distribution leans toward large gains or large losses. Kurtosis measures how heavy the tails are, meaning how often extreme values occur. Two strategies with the same average return and volatility can have very different skewness and kurtosis, and very different chances of a devastating loss.

Skewness#

skewness = E[(X - μ)³] / σ³

Equity indices typically show negative skewness in daily returns: large down days are more common than equally large up days.

Kurtosis#

kurtosis = E[(X - μ)⁴] / σ⁴
excess kurtosis = kurtosis - 3

Daily stock returns often have excess kurtosis well above 0, sometimes 5 to 20 or more for individual stocks. See Fat Tails.

Comparing two strategies#

Why skewness matters to traders#

• Psychology: negative skew strategies feel good most of the time, which can lead to oversizing before a big loss. See Overconfidence.
• Positive skew strategies require patience through many small losses. See Losing and Winning Streaks.
• Risk measures: volatility based measures such as the Sharpe ratio can flatter negatively skewed strategies. See Sharpe Ratio and Sortino Ratio.
• Pricing: investors tend to overpay for positive skew (lottery like assets), which may lower their returns.

Estimating skewness and kurtosis#

Sample skewness and kurtosis are very sensitive to outliers and need large samples to be reliable. A single extreme day can change kurtosis dramatically. Robust alternatives use quantiles, such as comparing the distance from the median to the 95th and 5th percentiles. See Outliers and Robust Statistics.

Using them in risk management#

Skewness of leveraged and option positions#

Buying options creates positive skew: the most you lose is the premium, while gains can be large. Selling options creates negative skew. Leverage does not change skewness by itself, but it magnifies the size of tail losses, so a negatively skewed strategy run with leverage can produce losses large enough to end a trading account. See Leverage and Risk of Ruin.

Frequently asked questions#

What is skewness in trading?#

A measure of asymmetry in returns; negative skewness means occasional large losses, positive skewness means occasional large gains.

What is kurtosis?#

A measure of how heavy a distribution's tails are; high kurtosis means extreme values occur more often than in a normal distribution.

Why do skewness and kurtosis matter?#

Because strategies with the same average and volatility can have very different risks of large losses, which these measures help reveal.

Next, learn how mixing regimes creates fat tails in Empirical and Mixture Distributions.