Introduction

In the world of finance, understanding the characteristics of probability distributions is essential for effective risk assessment and decision-making. The first two moments, namely the mean $(\mu)$ and variance $(\sigma^2)$, are commonly used to describe distributions. However, when dealing with non-normal distributions, relying solely on these moments can lead to inadequate insights. This chapter delves into the limitations of using only the mean and variance to depict non-normal distributions and explores scenarios where additional measures are needed.

Limitations of Mean and Variance for Non-Normal Distributions

While the mean provides a central tendency measure and the variance indicates the dispersion of data points around the mean, non-normal distributions can exhibit behaviors that escape characterization through these moments alone. Non-normal distributions might possess asymmetry, heavy tails, or unusual shapes that cannot be fully captured by the mean and variance.

For instance, consider the case of financial returns data. A stock’s return distribution might have heavy tails, indicating the possibility of extreme events occurring more frequently than in a normal distribution. In such cases, the mean and variance fail to convey the potential impact of these extreme events on portfolio performance.

Skewness and Kurtosis as Additional Measures

To address the limitations of the mean and variance, additional statistical measures come into play. Skewness and kurtosis are two such measures that provide insights into distribution shape and tail behavior.

• Skewnees: Skewness indicates the asymmetry of a distribution. Positive skewness implies a longer right tail, while negative skewness indicates a longer left tail. In financial terms, positive skewness might indicate that the potential for positive returns is higher than for negative returns, which can be seen in certain option strategies.

• Kurtosis: Kurtosis measures the thickness of tails relative to a normal distribution. High kurtosis suggests heavy tails, indicating the potential for extreme events. Low kurtosis suggests light tails, indicating a more concentrated distribution. In finance, understanding kurtosis is crucial for managing tail risk, especially in portfolios where extreme events can significantly impact performance.