Leptokurtic

Leptokurtic distributions are statistical distributions where there are extreme points (or outliers) along the X axis, resulting in a higher kurtosis than found in a normal distribution. These extreme values (outliers) are also evidence of “fat tails” relative to the normal distribution’s tail. A distribution is leptokurtic when the kurtosis value is a large positive number.

It is difficult to compare tail behavior via density plots because, even when the tails are thicker than the normal distribution, the values are close to zero as shown in the left panel graph below.  On the other hand, tail thickness relative to the normal distribution is easily seen in a normal quantile-quantile plot; see the right panel below

The prefix "lepto-" means thin or skinny. The “thin and skinny” appearance of leptokurtic distributions is actually a consequence of the outliers, which stretches the X axis of the plot, making the bulk of the data appear to occupy a narrow vertical strip. "Kurtosis" means “arched” or “bulging” based on its Greek origins, but these terms do not characterize distributions correctly. Rather, leptokurtic distributions have heavy tails, or outliers, when compared to mesokurtic or platykurtic distributions.

When analyzing historical returns, kurtosis helps the investor gauge an asset's level of risk. A leptokurtic distribution means that the investor will experience occasional large fluctuations (e.g. five or more standard deviations from the mean) more often than predicted by the normal distribution.

Leptokurtosis can impact how analysts estimate value at risk (VaR). An investor using a normal distribution to estimate VaR may overestimate at low levels of significance, but might underestimate at high levels of significance if the return distribution is leptokurtic.  This is the result of the leptokurtic distribution having a fatter tail, where returns far from the mean (e.g., five or more standard deviations) are more likely than VaR calculations based on the normal distribution would predict. The fat tail means that risk is coming from the extreme observations that occur occasionally. Therefore, conservative investors should avoid this type of return distribution.

To determine the classification of kurtosis, a normal distribution is used as a comparison point. The usual kurtosis coefficient is the average of the standardized data values (z values), each taken to the fourth power. For a normal distribution, this quantity is an estimate of the number 3.0, so often leptokurtic distributions are characterized by a kurtosis coefficient that is greater than 3.0. Equivalently, the term “excess kurtosis” refers to the kurtosis coefficient minus 3.0, so that leptokurtic distributions are also characterized by excess kurtosis greater than 0.0.

Distributions with higher kurtosis are more heavy-tailed; i.e., more prone to serious outliers.

While leptokurtosis refers to greater outlier potential, mesokurtosis and playkurtosis describe lesser outlier potential. Mesokurtic distributions have kurtosis near 3.0, meaning that their outlier character is similar to that of the normal distribution. Platykurtic distributions have kurtosis less than 3.0, thus exhibit less kurtosis than a normal distribution, implying that their outlier characteristic is less extreme than that of the normal distribution.