Kurtosis

Published Apr 29, 2024

Definition of Kurtosis

Kurtosis is a statistical measure that describes the shape of a distribution’s tails in relation to its overall shape. Specifically, it measures the “tailedness” of the probability distribution of a real-valued random variable. In simpler terms, kurtosis tells us how heavy or light the tails of a distribution are compared to a normal distribution. This is important because it gives insights into the behavior of the data, especially in terms of outliers, risks, and extremes.

Types of Kurtosis

There are three types of kurtosis, each characterizing a distinct distribution shape:

• Leptokurtic (Positive Kurtosis): A distribution is considered leptokurtic when its kurtosis is greater than that of a normal distribution, which has a kurtosis of 3. Leptokurtic distributions have heavier tails and a sharper peak, indicating a higher likelihood of extreme values.
• Mesokurtic: A mesokurtic distribution has the same kurtosis as a normal distribution. These distributions are characterized by tails similar in length to the normal distribution, suggesting a moderate likelihood of extreme values.
• Platykurtic (Negative Kurtosis): Distributions with kurtosis less than 3 are considered platykurtic. They have lighter tails and a flatter peak compared to a normal distribution, indicating a lower probability of extreme values.

Example

Consider the annual returns of two different investments over a decade. Investment A’s annual returns follow a distribution with high kurtosis, while Investment B’s returns follow a distribution with low kurtosis.

For Investment A, the high kurtosis means that it has a higher tendency to produce extreme returns (either very high or very low), compared to Investment B, which has fewer outliers and less extreme returns. This difference in distribution shapes can significantly impact investment risk and return profiles.

Why Kurtosis Matters

Understanding the kurtosis of a dataset is crucial for several reasons:

1. Risk Management: In finance, higher kurtosis means there’s a higher risk of investment returns far deviating from the average, which is critical for risk assessment and management.
2. Statistical Analysis: Kurtosis is used in statistical tests, data analysis, and in choosing the right models for data. For example, many statistical methods assume normality, and knowing the kurtosis helps in assessing this assumption.
3. Signal Processing: In engineering and sciences, kurtosis is used to analyze signals and background noise, helping to identify abnormalities or signal characteristics.

Frequently Asked Questions (FAQ)

How does kurtosis differ from skewness?

Kurtosis and skewness are both measures of a distribution’s shape. While kurtosis refers to the tailedness of the distribution or the propensity for extreme outliers, skewness measures the asymmetry of the distribution around its mean. In other words, skewness tells us if one tail is longer or fatter than the other, while kurtosis indicates how heavy the tails are in general.

Can kurtosis be negative?

Yes. Kurtosis can be negative, indicating a distribution with tails lighter than those of a normal distribution (platykurtic). However, when speaking in terms of excess kurtosis (kurtosis minus 3), positive values indicate heavy tails (leptokurtic), and negative values indicate light tails (platykurtic).

How is kurtosis useful in financial analysis?

In financial analysis, kurtosis is fundamental to understanding the risk profile of assets. Higher kurtosis can indicate a higher risk of outlier returns, both negative and positive. This helps financial analysts and investors make informed decisions by considering the likelihood of extreme outcomes in investment returns.

Understanding kurtosis provides deep insights into data behavior, especially in terms of outliers and extreme values, which is invaluable across various domains, including finance, economics, engineering, and statistical analysis.