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# Kurtosis Definition

## What Is Kurtosis?

Like skewness, kurtosis is a statistical measure that is used to describe distribution. Whereas skewness differentiates extreme values in one versus the other tail, kurtosis measures extreme values in either tail. Distributions with large kurtosis exhibit tail data exceeding the tails of the normal distribution (e.g., five or more standard deviations from the mean). Distributions with low kurtosis exhibit tail data that are generally less extreme than the tails of the normal distribution.

For investors, high kurtosis of the return distribution implies the investor will experience occasional extreme returns (either positive or negative), more extreme than the usual + or - three standard deviations from the mean that is predicted by the normal distribution of returns. This phenomenon is known as kurtosis risk.

### Key Takeaways

• Kurtosis describes the "fatness" of the tails found in probability distributions.
• The normal distribution has a kurtosis of exactly 3.0, and is known as mesokurtic.
• "Fat tails" are seen in distributions with a kurtosis greater than 3.0, and is known as leptokurtic.
• Stock prices have been described as having fat tails.
• Platykurtic distributions have skinner tails, with a kurtosis of less than 3.0
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## Understanding Kurtosis

Kurtosis is a measure of the combined weight of a distribution's tails relative to the center of the distribution. When a set of approximately normal data is graphed via a histogram, it shows a bell peak and most data within three standard deviations (plus or minus) of the mean. However, when high kurtosis is present, the tails extend farther than the three standard deviations of the normal bell-curved distribution.

Kurtosis is sometimes confused with a measure of the peakedness of a distribution. However, kurtosis is a measure that describes the shape of a distribution's tails in relation to its overall shape. A distribution can be infinitely peaked with low kurtosis, and a distribution can be perfectly flat-topped with infinite kurtosis. Thus, kurtosis measures "tailedness," not "peakedness."

## Types of Kurtosis

There are three categories of kurtosis that can be displayed by a set of data. All measures of kurtosis are compared against a standard normal distribution, or bell curve.

### Meoskurtic (kurtosis = 3.0)

The first category of kurtosis is a mesokurtic distribution. This distribution has a kurtosis statistic similar to that of the normal distribution, meaning the extreme value characteristic of the distribution is similar to that of a normal distribution.

### Leptokurtic (kurtosis > 3.0)

The second category is a leptokurtic distribution. Any distribution that is leptokurtic displays greater kurtosis than a mesokurtic distribution. Characteristics of this distribution is one with long tails (outliers.) The prefix of "lepto-" means "skinny," making the shape of a leptokurtic distribution easier to remember. The "skinniness" of a leptokurtic distribution is a consequence of the outliers, which stretch the horizontal axis of the histogram graph, making the bulk of the data appear in a narrow ("skinny") vertical range.

Thus leptokurtic distributions are sometimes characterized as "concentrated toward the mean," but the more relevant issue (especially for investors) is there are occasional extreme outliers that cause this "concentration" appearance. Examples of leptokurtic distributions are the T-distributions with small degrees of freedom.

While a leptokurtic distribution may be "skinny" in the center, it also features "fat tails".

### Platykurtic (kurtosis < 3.0)

The final type of distribution is a platykurtic distribution. These types of distributions have short tails (paucity of outliers.) The prefix of "platy-" means "broad," and it is meant to describe a short and broad-looking peak, but this is an historical error. Uniform distributions are platykurtic and have broad peaks, but the beta (.5,1) distribution is also platykurtic and has an infinitely pointy peak.

The reason both these distributions are platykurtic is their extreme values are less than that of the normal distribution. For investors, platykurtic return distributions are stable and predictable, in the sense that there will rarely (if ever) be extreme (outlier) returns.

## Why Is Kurtosis Important?

Kurtosis explains how often observations in some data set fall in the tails vs. the center of a probability distribution. In finance and investing, excess kurtosis is interpreted as a type of risk, known as "tail risk," or the chance of a loss occurring due to a rare event, as predicted by a probability distribution. If such events turn out to be more common than predicted by a distribution, the tails are said to be "fat." Tail events have had experts questions the true probability distribution of returns for investable assets - and now many believe that the normal distribution is not a correct template.

## What Is Excess Kurtosis?

Excess kurtosis compares the kurtosis coefficient with that of a normal distribution. Since normal distributions have a kurtosis of three, excess kurtosis can be calculated by subtracting kurtosis by three. Thus, a kurtosis of more than 3.0 will have positive excess kurtosis and be leptokurtic, with relatively fat tails and skinny center. If there is instead a kurtosis of less than 3.0, you have negative excess kurtosis, resulting in a platykurtic distribution with smaller tails and a broader center.

## Is Kurtosis the Same As Skewness?

No. Kurtosis measures how much of a probability distribution is centered around the middle (mean) vs. the tails. Skewness instead measures the relative symmetry of a distribution around the mean.

## The Bottom Line

Kurtosis describes how much of a probability distribution falls in the tails versus its center. In a normal distribution, the kurtosis is equal to three. Positive or negative excess kurtosis will then change the shape of the distribution, accordingly. For investors, kurtosis is important in understanding tail risk, or how frequently "infrequent" events occur given one's assumption about the distribution of price returns.

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