## What Is Kurtosis?

Kurtosis is a statistical measure used to describe a characteristic of a dataset. When normally distributed data is plotted on a graph, it generally takes the form of an upsidedown bell. This is called the bell curve. The plotted data that are furthest from the mean of the data usually form the tails on each side of the curve. Kurtosis indicates how much data resides in the tails.

Distributions with a large kurtosis have more tail data than normally distributed data, which appears to bring the tails in toward the mean. Distributions with low kurtosis have fewer tail data, which appears to push the tails of the bell curve away from the mean.

For investors, high kurtosis of the return distribution curve implies that there have been many price fluctuations in the past (positive or negative) away from the average returns for the investment. So, an investor might experience extreme price fluctuations with an investment with high kurtosis. This phenomenon is known as *kurtosis risk*.

### Key Takeaways

- Kurtosis describes the "fatness" of the tails found in probability distributions.
- There are three kurtosis categories—mesokurtic (normal), platykurtic (less than normal), and leptokurtic (more than normal).
- Kurtosis risk is a measurement of how often an investment's price moves dramatically.
- A curve's kurtosis characteristic tells you how much kurtosis risk the investment you're evaluating has.

#### Kurtosis

## Understanding Kurtosis

Kurtosis is a measure of the combined weight of a distribution's tails relative to the center of the distribution curve (the mean). For example, when a set of approximately normal data is graphed via a histogram, it shows a bell peak, with most of the data residing 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 sharply peaked with low kurtosis, and a distribution can have a lower peak with high kurtosis. Thus, kurtosis measures "tailedness," not "peakedness."

## Formula and Calculation of Kurtosis

### Calculating With Spreadsheets

There are several different methods for calculating kurtosis. The simplest way is to use the Excel or Google Sheets formula. For instance, assume you have the following sample data: 4, 5, 6, 3, 4, 5, 6, 7, 5, and 8 residing in cells A1 through A10 on your spreadsheet. The spreadsheets use this formula for calculating kurtosis:

{ [ n ( n + 1 ) / ( n - 1 ) ( n - 2 ) ( n - 3 ) ] [ Σ ( xi - x̄ ) / s ]4 } - { [ 3 ( n - 1 ) 2 ] / [ ( n - 2 ) ( n - 3 ) ] }

However, we'll use the following formula in Google Sheets, which calculates it for us, assuming the data resides in cells A1 through A10:

=KURT(A1:A10)

The result is a kurtosis of -0.1518, indicating the curve has lighter tails and is platykurtic.

### Calculating By Hand

Calculating kurtosis by hand is a lengthy endeavor, and takes several steps to get to the results. We'll use new data points and limit their number to simplify the calculation. The new data points are 27, 13, 17, 57, 113, and 25.

It's important to note that a sample size should be much larger than this; we are using six numbers to reduce the calculation steps. A good rule of thumb is to use 30% of your data for populations under 1,000. For larger populations, you can use 10%.

First, you need to calculate the mean. Add up the numbers and divide by six to get 42. Next, use the following formulas to calculate two sums, s2 (the square of the deviation from the mean) and s4 (the square of the deviation from the mean squared). Note—these numbers do not represent standard deviation; they represent the variance of each data point.

- s2 = Σ ( y
_{i}- ȳ )^{2} - s4 = Σ ( y
_{i}- ȳ )^{4}

Where:

- y
_{i}= the i^{th}variable of the sample - ȳ = the mean

To get s2, use each variable, subtract the mean, and then square the result. Add all of the results together:

- (27 - 42)
^{2}= (-15)^{2}= 225 - (13 - 42)
^{2}= (-29)^{2}= 841 - (17 - 42)
^{2}= (-25)^{2}= 625 - (57 - 42)
^{2}= (15)^{2}= 225 - (113 - 42)
^{2}= (71)^{2}= 5041 - (25 - 42)
^{2}= (-17)^{2}= 289 - 225 + 841 + 625+ 225 + 5,041 + 289 = 7,246

To get s4, use each variable, subtract the mean, and raise the result to the fourth power. Add all of the results together:

- (27 - 42)
^{4}= (-15)^{4}= 50,625 - (13 - 42)
^{4}= (-29)^{4}= 707,281 - (17 - 42)
^{4}= (-25)^{4}= 390,625 - (57 - 42)
^{4}= (15)^{4}= 50,625 - (113 - 42)
^{4}= (71)^{4}= 25,411,681 - (25 - 42)
^{4}= (-17)^{4}= 83,521 - 50,625+707,281+390,625+50,625+25,411,681+83,521 = 26,694,358

So, our sums are:

- s2 = 7,246
- s4 = 26,694,358

Now, calculate m2 and m4, the second and fourth moments of the kurtosis formula:

- m2 = s2 / n, or 7,246 / 6 = 1,207.67
- m4 = s4 / n, or 26,694,358 / 6 = 4,449,059.67

We can now calculate kurtosis using a formula found in many statistics textbooks that assumes a perfectly normal distribution with kurtosis of zero:

k = ( m4 / m2^{2}) - 3

Where:

- k = kurtosis
- m4 = fourth moment
- m2 = second moment
- 4,449,059.67 / 1,458,466.83 = 3.05

So, the kurtosis for the sample variables is 3.05 - 3, or .05.

## Types of Kurtosis

There are three categories of kurtosis that a set of data can display—mesokurtic, leptokurtic, and platykurtic. All measures of kurtosis are compared against a normal distribution curve.

### Mesokurtic (kurtosis = 3.0)

The first category of kurtosis is mesokurtic distribution. This distribution has a kurtosis similar to that of the normal distribution, meaning the extreme value characteristic of the distribution is similar to that of a normal distribution. Therefore, a stock with a mesokurtic distribution generally depicts a moderate level of risk.

### Leptokurtic (kurtosis > 3.0)

The second category is leptokurtic distribution. Any distribution that is leptokurtic displays greater kurtosis than a mesokurtic distribution. This distribution appears as a curve one with long tails (outliers.) 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.

A stock with a leptokurtic distribution generally depicts a high level of risk but the possibility of higher returns because the stock has typically demonstrated large price movements.

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 platykurtic distribution. These types of distributions have short tails (fewer outliers.). Platykurtic distributions have demonstrated more stability than other curves because extreme price movements rarely occurred in the past. This translates into a less-than-moderate level of risk.

## Using Kurtosis

Kurtosis is used in financial analysis to measure an investment's risk of price volatility. Kurtosis risk differs from more commonly used measurements such as alpha, beta, r-squared, or the Sharpe ratio. Alpha measures excess return relative to a benchmark index, and beta measures the volatility a stock has compared to the broader market.

R-squared measures the percent of movement a portfolio or fund has that can be explained by a benchmark, and the Sharpe ratio compares return to risk. Kurtosis measures the amount of volatility an investment's price has experienced regularly.

For example, imagine a stock had an average price of $25.85 per share. If the stock's price swung widely and often enough, the bell curve would have heavy tails (high kurtosis). This means that there is a lot of variation in the stock price—an investor should anticipate wide price swings often.

If the stock had light tails (low kurtosis), the investor might anticipate that the stock price would swing widely only occasionally.

## Why Is Kurtosis Important?

Kurtosis explains how often observations in some data sets 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 are more common than predicted by a distribution, the tails are said to be "fat."

## What Is Excess Kurtosis?

Excess kurtosis compares the kurtosis coefficient with that of a normal distribution. Most normal distributions are assumed to have a kurtosis of three, so excess kurtosis would be more or less than three; however, some models assume a normal distribution has a kurtosis of zero, so excess kurtosis would be more or less than zero.

## Is Kurtosis the Same As Skewness?

No. Kurtosis measures how much of the data in a probability distribution are 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 instead of its center. In a normal distribution, the kurtosis is equal to three (or zero in some models). 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.