Excess Kurtosis

What Is Excess Kurtosis?

The term excess kurtosis refers to a metric used in statistics and probability theory comparing the kurtosis coefficient with that of a normal distribution. Kurtosis is a statistical measure that is used to describe the size of the tails on a distribution. Excess kurtosis helps determine how much risk is involved in a specific investment. It signals that the probability of obtaining an extreme outcome or value from the event in question is higher than would be found in a probabilistically normal distribution of outcomes.

Understanding Excess Kurtosis

Kurtosis measures how fat a distribution's tail is when compared to the center of the distribution. The tails of a distribution measure the number of events that occurred outside of the normal range. Unlike skewness, kurtosis measures either tail's extreme values. Excess kurtosis means the distribution of event outcomes have lots of instances of outlier results, causing fat tails on the bell-shaped distribution curve. Normal distributions have a kurtosis of three. Excess kurtosis can, therefore, be calculated by subtracting kurtosis by three.

Since normal distributions have a kurtosis of three, excess kurtosis can be calculated by subtracting kurtosis by three.

Excess kurtosis is an important tool in finance and, more specifically, in risk management. With excess kurtosis, any event in question is prone to extreme outcomes. It is an important consideration to take when examining historical returns from a particular stock or portfolio. The higher the kurtosis coefficient is above the normal level — or the fatter the tails on the return distribution graph — the more likely that future returns will be either extremely large or extremely small. Stock prices with a higher likelihood of outliers on either the positive or negative side of the mean closing price can be said to have either positive or negative skewness, which can be related to kurtosis.

Types of Excess Kurtosis

The values of excess kurtosis can be either negative or positive. When the value of an excess kurtosis is negative, the distribution is called platykurtic. This kind of distribution has a tail that's thinner than a normal distribution. When applied to investment returns, platykurtic distributions — those with negative excess kurtosis — generally produce results that won't be very extreme, which are great for investors who don't want to take a lot of risk.

When excess kurtosis is positive, it has a leptokurtic distribution. The tails on this distribution is heavier than that of a normal distribution, indicating a heavy degree of risk. The returns on an investment with a leptokurtic distribution or positive excess kurtosis will likely have extreme values. Investors who are willing and able to take a lot of risk will probably want to invest in a vehicle with a positive excess kurtosis.

Excess kurtosis can be at or near zero as well, so the chance of an extreme outcome is rare. This is known as a mesokurtic distribution. The tails of this kind of distribution is similar to that of a normal distribution.

Example of Excess Kurtosis

Let's use a hypothetical example of excess kurtosis. If you track the closing value of stock ABC every day for a year, you will have a record of how often the stock closed at a given value. If you build a graph with the closing values along the X-axis and the number of instances of that closing value that occurred along the Y-axis of a graph, you will create a bell-shaped curve showing the distribution of the stock's closing values. If there are a high number of occurrences for just a few closing prices, the graph will have a very slender and steep bell-shaped curve. If the closing values vary widely, the bell will have a wider shape with less steep sides. The tails of this bell will show you how often heavily deviated closing prices occurred, as graphs with lots of outliers will have thicker tails coming off each side of the bell.