Kurtosis in Practice: Real-World Applications and Interpretations

What is Kurtosis?

Kurtosis is a statistical measure used to define a characteristic of a dataset. When plotting normally distributed data on a graph, the data typically takes the shape of a bell. The data that are furthest away from the average of the data are known as the tails. Kurtosis is a statistic used to describe the amount of data that is in the tails. Distributions with a high kurtosis tend to have more tails than those with a low kurtosis. This means that the tails of a bell curve tend to be further away from the average than those of a normal distribution.

Understanding Kurtosis

Kurtosis, on the other hand, is the sum of the tails of a distribution concerning the center (the mean) of the distribution. For example, if you plot a set of about normal data on a histogram (i.e., a bell peak), you’ll see that the bulk of the data is within 3 standard deviations (or less) of the average. However, if you look at the normal bell curve, you’ll notice that the tails extend beyond the 3 standard deviations in the normal bell curve.

Kurtosis is sometimes mistaken for a measure of peakness. Kurtosis measures the shape of the tails of a distribution about the distribution's overall shape. For example, a distribution may have a sharp peak with a kurtosis of 0.5, and a lower peak can be achieved with a higher kurtosis of 1.0. Kurtosis measures "tailedness," not "peakness."

Importance of Kurtosis

Kurtosis is the number of times an observation in a dataset falls in the tails of the probability distribution, rather than the center of the distribution. Kurtosis is often used in finance and investing as a measure of risk, which is the chance that a loss will occur because of a rare event. If the occurrence of a rare event is more frequent than the distribution predicts, the tails of the distribution are referred to as "fat" kurtosis.

Formula and Calculation of Kurtosis

Calculating with Spreadsheets

There are several various methods for calculating kurtosis. Here the simplest way is to use the Excel or Google Sheets formula. For illustration, 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:1

𝑛(𝑛+1)(𝑛−1)(𝑛−2)(𝑛−3)×(∑𝑥𝑖−𝑥ˉ𝑠)4−3(𝑛−1)2(𝑛−2)(𝑛−3)​(n−1)(n−2)(n−3)n(n+1)​×(∑sxi​−xˉ​)4−(n−2)(n−3)3(n−1)2​​

However, we will be using the following formula in Google Sheets, which calculates it for us, assuming the data resides in cells A1 through A10:2.

Excess Kurtosis

There is another method of calculating the kurtosis called 'excess kurtosis'. Since kurtosis is proportional to the normal distribution (with kurtosis value 3), excess kurtosis is often easier to work with. As the name suggests, it is the kurtosis value over the kurtosis value of the normal distribution. This means that for a normal distribution with any mean and variance, the excess kurtosis is always 0.

Excess kurtosis=The fourth momentσ4−3=n∑j=1(Xj−μ)4nσ4−3𝐸𝑥𝑐𝑒𝑠𝑠 𝑘𝑢𝑟𝑡𝑜𝑠𝑖𝑠=The fourth momentσ4−3=∑𝑗=1𝑛(𝑋𝑗−μ)4𝑛σ4−3

Uses of Kurtosis

Kurtosis is a financial term used to describe the risk of price volatility in an investment. Kurtosis measures how much volatility an investment's price has experienced regularly. A high kurtosis in the return distribution indicates that an investment will occasionally produce extreme returns. Be aware that high kurtosis can swing both ways, meaning high kurtosis means either high positive returns or extremely negative returns.

Let’s look at a stock with a kurtosis of high kurtosis. Suppose the average price of the stock is US$25.85 per share. The bell curve shows that the stock price has fluctuated widely and often enough for the bell curve to have heavy tails. This indicates that there is a high degree of volatility in the stock price. An investor should expect wide price swings frequently.

Types of Kurtosis

There are three types of kurtosis that a set of data can display: mesokurtic, leptokurtic, and platykurtic.

Mesokurtic (Kurtosis = 3.0)

The first category of kurtosis is mesokurtic distribution. This kurtosis corresponds to the normal distribution. This means that the extreme value of the distribution corresponds to the normal value. A stock with a mesokurtic distribution represents a medium risk level.

Leptokurtic (Kurtosis > 3.0)

The second category is leptokurtic distribution. This kurtosis of any distribution is greater than the kurtosis of mesokurtics. This distribution is represented as a curve that has long tails (the outliers). The "skinniness" of the distribution is due to the outliers stretching the horizontal axes of a histogram graph. This means that the majority of the data appears in a narrow vertical range.  When a stock has large price movements, it typically shows a high degree of risk but also the potential for higher returns.

Platykurtic (Kurtosis < 3.0)

The final type of distribution is platykurtic distribution. PlatyKurtic has short tails, i.e. fewer outliers. Playkurtic distributions have seen more stability than other curves as it has had extreme price movements occurrence in the past. This interprets into a less-than-moderate level of risk.

Applications of Kurtosis

Here are a few applications of kurtosis including the explanation of the interpretation of kurtosis:

Risk Evaluation in Finance:

Kurtosis helps in understanding the profiles that are at risk of financial securities. It contributes to assessing the tail risks that ascend due to the shape of distributions of returns. It provides insights into the probability of extreme investment outcomes.

Quality Control:

Kurtosis often plays an important in manufacturing for quality control processes. It helps in rectifying products that deviate meaningfully from the standard specifications.

Signal Processing:

Kurtosis is used in signal processing to recognize the faults in rotating machinery. It quantifies the 'tailedness' of the distribution of signal values, which can refer to the presence of anomalies or noise.

Investment Allocation Approach:

Another way Kurtosis can be applied is in the context of an investment allocation strategy. For instance, a value-focused portfolio manager might want to invest assets with a kurtosis of a negative, as a negative kurtosis indicates a flattening distribution with more infrequent small returns. On the other hand, a momentum-focused manager might want to focus on assets with a kurtosis of positive, with peak distributions of less infrequent but larger returns.

Psychometrics:

In psychology, kurtosis is used to assess the effects of non-normality on psychometric models, which can impact the interpretation of psychological tests and assessments.

Meteorology:

Kurtosis can be used to analyze weather data distributions, helping to predict extreme weather events by assessing the probability of outlier values in historical data.

Conclusion

In this article, we have discussed kurtosis/excess kurtosis, and how it describes the ' shape' of distribution, which is often overlooked by average, variance, and skewness, among others.

One of the most important metrics to use in finance when measuring tail-risk is kurtosis. Without tail-risk, alpha may be underestimated. Therefore, tail-risk/kurtosis risk evaluation is an integral part of the overall performance evaluation of financial securities.

There are many other metrics, such as kurtosis, that when used in combination with algorithmic trading, increase the value of trading strategies.