Excess kurtosis: Meaning, Criticisms & Real-World Uses

What Is Excess Kurtosis?

Excess kurtosis is a statistical measure that quantifies the "tailedness" of a probability distribution compared to a normal distribution. As a key concept within Financial Statistics, it indicates whether the distribution of data, such as investment returns, has more or fewer extreme observations (outliers) than would be expected under a normal, bell-shaped curve. A positive excess kurtosis suggests that a distribution has heavier tails and a sharper peak than a normal distribution, implying a greater likelihood of extreme positive or negative outcomes. Conversely, a negative excess kurtosis indicates lighter tails and a flatter peak, meaning fewer extreme events.

History and Origin

The foundational concept of kurtosis was introduced by Karl Pearson in 1905, building on earlier statistical work. Pearson defined kurtosis as a measure related to the fourth moment of a distribution, using it to describe the "flat-toppedness" or "peakedness" of a curve relative to the normal curve.³ He introduced terms like "leptokurtic" (more peaked than normal) and "platykurtic" (flatter than normal).² The specific concept of excess kurtosis emerged as a way to standardize this measure, typically by subtracting 3 from Pearson's original kurtosis coefficient (β₂). This adjustment makes the normal distribution, a common benchmark in finance and statistics, have an excess kurtosis of zero. This standardization simplifies comparison, as a positive or negative value directly indicates a deviation from the normal distribution's tail characteristics.

Key Takeaways

Excess kurtosis measures the "tailedness" of a probability distribution relative to a normal distribution.
A positive excess kurtosis, known as leptokurtic, implies a higher probability of extreme outcomes (fat tails).
A negative excess kurtosis, known as platykurtic, suggests fewer extreme outcomes (thin tails).
A distribution with zero excess kurtosis is mesokurtic, exhibiting tail behavior similar to a normal distribution.
In finance, excess kurtosis is crucial for assessing tail risk in investment returns.

Formula and Calculation

The formula for excess kurtosis ((\gamma_2)) is derived from the fourth standardized moment of a distribution.

\gamma_2 = \frac{E[(X - \mu)^4]}{\sigma^4} - 3

Where:

(E[(X - \mu)^4]) represents the fourth moment about the mean ((\mu)). This is calculated as the expected value of the difference between each data point (X) and the mean, raised to the fourth power.
(\sigma^4) is the fourth power of the standard deviation, which is also the square of the variance.
The subtraction of 3 adjusts the raw kurtosis value so that a normal distribution has an excess kurtosis of zero.

Interpreting the Excess Kurtosis

Interpreting excess kurtosis involves understanding how the shape of a distribution's tails compares to that of a normal distribution.

Positive Excess Kurtosis (Leptokurtic): A distribution with positive excess kurtosis is characterized by heavier tails and often a higher, narrower peak than a normal distribution. This indicates that extreme values, both significantly positive and significantly negative, occur more frequently than predicted by a normal distribution. In financial contexts, this implies a greater probability of very large gains or very large losses.
¹ Negative Excess Kurtosis (Platykurtic): A distribution with negative excess kurtosis has lighter tails and a broader, flatter peak compared to a normal distribution. This suggests that extreme values are less common, and the data points are more concentrated around the mean. For investors, this might imply a lower likelihood of extreme price swings.
Zero Excess Kurtosis (Mesokurtic): A distribution with zero excess kurtosis, like the normal distribution, has tails that are neither too heavy nor too light. It serves as a benchmark for comparison, indicating a "normal" frequency of extreme events.

Understanding excess kurtosis helps analysts gauge the probability of outliers, which is particularly relevant in areas like risk management.

Hypothetical Example

Consider two hypothetical investment portfolios, Portfolio A and Portfolio B, with historical daily returns over a year. Both portfolios have the same average daily return and standard deviation.

Portfolio A (Leptokurtic): After calculating the excess kurtosis for Portfolio A's daily returns, the value is +1.5. This positive excess kurtosis suggests that Portfolio A's returns have heavier tails than a normal distribution. This means that while its average return and volatility might seem moderate, it has experienced, or is more prone to experiencing, more extreme daily gains or losses than a normally distributed asset would suggest. For instance, there might be a few days with exceptionally large percentage drops or surges.

Portfolio B (Platykurtic): The excess kurtosis for Portfolio B's daily returns is -0.8. This negative excess kurtosis indicates that Portfolio B's returns have lighter tails than a normal distribution. This implies that extreme daily returns (both positive and negative) are less frequent compared to what a normal distribution would predict. The returns are more tightly clustered around the mean, suggesting a more consistent, albeit potentially less exciting, performance profile without significant outliers.

An investor might prefer Portfolio B for its relative predictability and lower incidence of extreme movements, even if both portfolios have the same overall historical standard deviation.

Practical Applications

Excess kurtosis is a vital tool in financial modeling and analysis, particularly within the realm of risk management and portfolio management.

Risk Assessment: Investors and analysts use excess kurtosis to assess the true risk profile of assets and portfolios. A high positive excess kurtosis indicates increased tail risk, meaning a greater likelihood of experiencing rare, but significant, price movements. This is critical for understanding potential downside risk beyond what volatility measures like standard deviation alone can convey. The presence of kurtosis risk, or "fat tails," implies that assuming a normal distribution can lead to an underestimation of potential losses during extreme market events.
Option Pricing: In quantitative finance, models like the Black-Scholes model traditionally assume that asset returns are normally distributed. However, real-world asset returns often exhibit positive excess kurtosis, leading to the phenomenon of "volatility smiles" or "smirks" in options markets, where out-of-the-money options are priced higher than the Black-Scholes model would predict. This reflects the market's pricing in of the greater probability of extreme events.
Portfolio Diversification: Understanding the excess kurtosis of individual assets helps in constructing more robust portfolios. Combining assets with different tail characteristics can potentially help mitigate overall portfolio tail risk, aiming for a more mesokurtic portfolio return distribution.
Market Analysis: Economists and financial researchers often analyze the kurtosis of various market indices and asset classes to understand broader market dynamics and the frequency of extreme events. Federal Reserve Bank of St. Louis research, for instance, delves into the distribution of asset returns and their implications for financial markets.

Limitations and Criticisms

While excess kurtosis provides valuable insights into the tails of a distribution, it is not without limitations or criticisms.

One common misconception is that kurtosis measures the "peakedness" of a distribution, when its primary focus is on the tails. A high peak can occur with either fat or thin tails, making "tailedness" the more accurate interpretation of excess kurtosis. Another limitation arises when dealing with financial time series data, where returns are often not independent and identically distributed, which can complicate the interpretation of statistical moments like kurtosis.

Furthermore, relying solely on excess kurtosis can be misleading if other distributional properties, such as skewness (the measure of asymmetry), are ignored. A distribution might have high kurtosis but be heavily skewed to one side, impacting the nature of the extreme events. Historically, models that underestimated the true kurtosis of financial markets have faced significant challenges. For example, the Long-Term Capital Management (LTCM) hedge fund experienced severe losses in the late 1990s, partly due to its models understating the kurtosis (and thus the extreme risks) of financial securities. This highlights the danger of models that assume normal distributions when actual market data exhibit "fat tails." Benoit Mandelbrot, a prominent mathematician, notably criticized the extensive reliance on the normal distribution in modern finance, including models like the Capital Asset Pricing Model, precisely because of the prevalence of kurtosis in real-world financial data.

Excess Kurtosis vs. Skewness

Excess kurtosis and skewness are both statistical measures that describe the shape of a probability distribution, but they focus on different aspects.

Excess Kurtosis primarily measures the "tailedness" of a distribution. It tells you how much data is concentrated in the tails relative to a normal distribution, indicating the frequency and magnitude of extreme observations. A positive excess kurtosis means more extreme events, while a negative value means fewer.

Skewness, on the other hand, measures the asymmetry of a distribution. A perfectly symmetrical distribution has zero skewness, meaning its left and right sides are mirror images. Positive skewness indicates a longer tail on the right side of the distribution, implying more frequent small losses and a few large gains. Negative skewness indicates a longer tail on the left, suggesting more frequent small gains and a few large losses.

While both are crucial for understanding the full shape of a distribution beyond just its mean and standard deviation, excess kurtosis quantifies the extremity of events, whereas skewness describes the direction of the distribution's asymmetry.

FAQs

What does positive excess kurtosis mean in finance?

In finance, positive excess kurtosis (leptokurtic distribution) means that an investment's returns have "fat tails." This implies that very large positive or negative returns occur more frequently than would be expected if the returns followed a normal distribution. It signals a higher likelihood of extreme events, often associated with increased tail risk.

How is excess kurtosis different from kurtosis?

"Kurtosis" can sometimes refer to the raw fourth moment of a distribution, where a normal distribution has a kurtosis of 3. "Excess kurtosis" is simply this raw kurtosis minus 3. This adjustment makes the normal distribution have an excess kurtosis of 0, providing a more intuitive benchmark for comparing distributions and assessing their "tailedness" relative to the familiar bell curve.

Why is excess kurtosis important for investors?

Excess kurtosis is important for investors because it helps them understand the true nature of risk, particularly the risk of extreme losses or gains, which standard volatility measures like standard deviation might not fully capture. By identifying distributions with high excess kurtosis, investors can better assess tail risk and potentially adjust their portfolio management strategies to account for the higher probability of rare, impactful events.