Kurtosis

Kurtosis is a statistical measure of how much of a distribution's variation comes from rare, extreme values instead of frequent, moderate ones. A distribution with high kurtosis has heavy tails: observations far from the average occur more often than they would under a normal distribution. It is the fourth standardised moment of a distribution, following the mean, the variance and skewness.

In finance, kurtosis matters because asset returns are heavy-tailed. Daily moves in shares, currencies and commodities cluster near zero most of the time but occasionally jump by amounts that a normal model treats as practically impossible. Risk managers, option traders and portfolio builders use kurtosis to describe this property, commonly called fat tails.

For a random variable $X$, kurtosis is defined as:

• $\mu$ is the mean of $X$.
• $\sigma$ is the standard deviation of $X$.
• $E[\cdot]$ denotes the expected value, or average.

Because deviations are raised to the fourth power, large deviations dominate the result. The normal distribution has a kurtosis of exactly 3, so it is common to report excess kurtosis, which subtracts 3:

Distributions are labelled by this figure. Those with positive excess kurtosis are leptokurtic, those with negative excess kurtosis are platykurtic, and those with none are mesokurtic. Software packages differ on whether the word kurtosis means the raw or the excess figure, so the convention should always be checked.

Take ten daily returns, in percent: eight are 0, one is +10 and one is -10. The mean is 0. The variance is the average squared deviation, $(100 + 100) / 10 = 20$. The average fourth-power deviation is $(10{,}000 + 10{,}000) / 10 = 2{,}000$. Kurtosis is $2{,}000 / 20^{2} = 5$, an excess of 2 over the normal value. Now take ten returns of which five are +4.47 and five are -4.47. The variance is again about 20, but the average fourth power is about 400, giving a kurtosis of 1. Both series have the same volatility. The first concentrates its risk in two extreme days and the second spreads it evenly.

Daily returns on equity indices typically show substantial positive excess kurtosis, and the figure is higher still for individual shares and for intraday data. The standard illustration is 19 October 1987, when the Dow Jones Industrial Average fell 22.6 percent in a day. Under a normal distribution fitted to the index's usual volatility, a move of that size, more than twenty standard deviations, should essentially never happen. Events described as a six sigma event or a black swan event are the visible face of kurtosis.

Part of the effect comes from volatility that changes over time. A mix of calm periods and turbulent periods produces a heavy-tailed overall distribution even if returns within each period are close to normal. Models such as GARCH capture this clustering and account for much, though not all, of the excess kurtosis in daily data. Kurtosis tends to fall as the horizon lengthens: monthly returns are closer to normal than daily ones.

• Risk measurement. A value at risk figure computed on the assumption of normal returns understates potential losses when kurtosis is high. Risk teams use heavy-tailed alternatives such as Student's t-distribution, historical simulation or stress tests.
• Option pricing. Heavy tails make options far from the current price worth more than the basic Black-Scholes model implies. The market expresses this through the volatility smile.
• Strategy evaluation. Two strategies with the same Sharpe ratio can differ greatly in tail risk. Strategies that collect small, steady gains and occasionally suffer large losses, such as selling options, show high kurtosis together with negative skewness.
• Testing for normality. The Jarque-Bera test combines sample skewness and kurtosis to test whether data are consistent with a normal distribution.

Sample kurtosis is an unstable statistic. Since it depends on fourth powers, one or two observations can determine most of its value, and the estimate from a few years of data can change sharply when a single crisis day enters or leaves the sample. For some heavy-tailed distributions the theoretical fourth moment is infinite, in which case the sample figure never settles down however much data is collected.

Kurtosis is also often misdescribed as a measure of how peaked a distribution is. Modern statistics texts stress that it reflects the tails, not the shape of the centre. Finally, it does not distinguish gains from losses: it treats a large upward jump the same as a large fall. It is therefore read together with skewness, with a histogram of the data and with direct measures of downside such as maximum drawdown.