Skewness and kurtosis: why market crashes are more frequent than the normal distribution says

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Many investment metrics, such as volatility, the Sharpe ratio or the simple value at risk, act as if returns were normally distributed. They are not. Skewness and kurtosis measure how far the distribution deviates from that: skewness shows whether large swings tend to go down or up, kurtosis how often extreme swings occur at all. For equities, this usually means that crashes are far more frequent than the bell curve predicts.

The four moments of a distribution

A return distribution can be roughly described with four metrics: the mean shows its location, the standard deviation its width. The third metric, skewness, measures asymmetry; the fourth, kurtosis, measures how much weight lies in the tails. For a normal distribution, skewness is 0 and kurtosis is 3. That is why excess kurtosis, i.e. kurtosis minus 3, is usually reported.

Worked example: one crash month changes everything

For small samples, bias-corrected estimators are used to offset the distortion caused by having few values. Even so, skewness and kurtosis from twelve values are highly uncertain: a single month determines almost the entire result. The metrics only become meaningful with many years of data, and especially when crises are included. Without the slump month, the series would be almost symmetrical; with it, it is strongly left-skewed: a single month tips the whole picture.

Calculating it yourself: the bias-corrected estimators

The simple formulas above (using means over the sample) underestimate skewness and kurtosis when there are few data points. Statistics packages and spreadsheets therefore use bias-corrected estimators. For n returns with the sample moments g₁ (skewness) and g₂ (excess kurtosis):

In the example above, the simple moments give a skewness of −2.67 and an excess kurtosis of 5.80; corrected, they are −3.07 and 9.95. In spreadsheets, the SKEW and KURT functions (SCHIEFE and KURT in German Excel) return exactly these corrected values. If you compare metrics from different programs, check which variant is used. Some report kurtosis without subtracting 3.

The standard errors are large: for normally distributed data, the standard error of skewness is roughly √(6 / n), and that of kurtosis √(24 / n). From 120 monthly returns, this gives about 0.22 for skewness and 0.45 for kurtosis. Small deviations from zero are therefore no proof of a skewed distribution. The Jarque-Bera test combines both into a single test for normality.

How rare are crashes really?

The normal distribution makes extreme days extremely unlikely. A fall of five standard deviations in a single day should practically never happen. If you use a distribution with fat tails instead, such as a Student's t distribution with five degrees of freedom (scaled to the same volatility), things look very different:

Market experience is closer to the second figure. Mandelbrot (1963) was already proposing to replace the normal distribution for price changes with a family of distributions with particularly pronounced fat tails; in his dissertation on stock prices, Fama (1965) confirmed that extreme moves are much more frequent than under the normal distribution. Cont (2001) summarises the stylised facts recognised today: fat tails, volatility clustering, hardly any autocorrelation in the returns themselves, but asymmetry. He also shows how these properties invalidate many common statistical methods. Taleb (2007) popularised the consequences of rare extreme events.

One reason for fat tails is fluctuating volatility: even if returns were normally distributed on each individual day, but with volatility sometimes low and sometimes high, the mixture would produce fat tails. Models such as GARCH capture exactly this (Bollerslev 1986).

What research shows: skewness has a price

Investors like positive skewness (the small chance of a large gain) and avoid negative skewness (the small risk of a large loss). Kraus and Litzenberger (1976) therefore extended the CAPM to include skewness. Harvey and Siddique (2000) tested this empirically: conditional, systematic skewness helps explain the differences in expected returns across assets, even alongside the size and value factors, and is associated with a premium of 3.60% a year on average.

Dependence between markets is asymmetric too: Longin and Solnik (2001) find that the correlation of international stock markets rises in bear markets, but not in bull markets. Diversification therefore helps less precisely when you need it most. More on this in the article correlation and diversification.

For advanced readers: value at risk with skewness and kurtosis

Parametric value at risk uses the quantile of the normal distribution, i.e. z = −1.645 at 95%. The expansion by Cornish and Fisher (1938) adjusts this quantile for skewness S and excess kurtosis K. Favre and Galeano (2002) turned this into the modified value at risk and optimised it for hedge funds and other non-normally distributed investments.

The adjustment is an approximation and unreliable for very large skewness or kurtosis; the quantiles can then even end up in the wrong order. Alternatives are historical simulation (the empirical quantile), fat-tailed distributions such as the Student's t distribution, or extreme value theory, which models only the tails. Expected shortfall, the average of the losses beyond the VaR, also reacts more strongly to fat tails than the VaR itself.

What this means for investors

• Volatility underestimates crash risk: two portfolios with the same volatility can have very different loss risks. Also look at the maximum drawdown and the worst months.
• A high Sharpe ratio with negative skewness is suspicious: strategies that sell insurance look excellent in calm years.
• Simulations with fat tails: for retirement planning, a Monte Carlo simulation that bootstraps from real returns or uses a Student's t distribution is more cautious than one based on the normal distribution.
• Long data series: skewness and kurtosis from a few calm years are deceptive. Crises have to be included in the data.

For monthly returns, skewness and kurtosis are usually less pronounced than for daily returns, because daily fluctuations partly offset each other. Over the years, however, the big crises remain visible as outliers in monthly and annual returns.

How Portfolio Lab calculates it

In the Backtest, the metrics include the skewness and (excess) kurtosis of the period returns, both as bias-corrected sample estimators. Alongside them are the historical value at risk 95% (the 5% quantile of the period returns) and the CVaR 95% (the average of the returns that were as bad or worse). Besides bootstrap and the normal distribution, the Monte Carlo simulation offers a Student's t distribution with five degrees of freedom, scaled to the measured volatility. Details are in the metrics methodology.

This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.