Skewness kurtosis

What Is Skewness Kurtosis?

Skewness kurtosis refers to two distinct statistical measures, skewness and kurtosis, that describe the shape of a probability distribution of a dataset. These concepts are fundamental to statistical finance and quantitative finance, providing insights beyond simple measures of mean and volatility. While volatility quantifies the dispersion of data around the mean, skewness and kurtosis offer a more granular understanding of the data's distributional characteristics.

Skewness measures the asymmetry of the data distribution around its average. A symmetrical distribution, like a normal distribution, has zero skewness, meaning its left and right sides are mirror images. Positive skewness indicates a longer or fatter tail on the right side of the distribution, suggesting a greater likelihood of extreme positive values. Conversely, negative skewness implies a longer or fatter tail on the left, indicating a higher probability of extreme negative values.

Kurtosis measures the "tailedness" of a distribution, indicating the presence of extreme values or outliers. It quantifies how much data resides in the tails relative to a normal distribution. High kurtosis (leptokurtic) means fatter tails and a sharper peak, suggesting a higher probability of infrequent but extreme observations. Low kurtosis (platykurtic) signifies thinner tails and a flatter peak, implying fewer extreme outliers.

Together, skewness and kurtosis provide critical information about the risk characteristics of financial data, particularly asset return distributions, which often deviate from a simple normal distribution.

History and Origin

The foundational concepts of skewness and kurtosis are largely attributed to the pioneering work of Karl Pearson, a prominent English mathematician and biostatistician. Pearson, often regarded as one of the founders of modern statistics, introduced the method of moments of a distribution in the late 19th and early 20th centuries as a means to describe and analyze asymmetrical frequency distributions⁷.

His work in the late 1800s and early 1900s, particularly his development of the Pearson product-moment correlation coefficient and the chi-square distribution, laid much of the groundwork for quantitative data analysis. Pearson defined kurtosis as a measure of the flatness of the top of a symmetric distribution relative to a normal curve. While the interpretation of kurtosis has evolved, Pearson's contributions established these statistical measures as essential tools for understanding the shape of data.

Key Takeaways

Skewness quantifies the asymmetry of a data distribution, indicating whether extreme values are more likely on the positive or negative side.
Kurtosis measures the "tailedness" of a distribution, revealing the likelihood and extremity of outliers.
Both skewness and kurtosis are higher-order moments of a distribution that offer insights into data shape beyond mean and standard deviation.
In finance, skewness kurtosis helps investors assess the potential for extreme gains or losses and is crucial for risk management and portfolio management.
Understanding these measures is vital because financial asset returns frequently exhibit non-normal distributions, with fatter tails and asymmetry.

Formula and Calculation

Skewness and kurtosis are calculated using the third and fourth standardized moments of a probability distribution, respectively. For a sample dataset, the formulas are:

Sample Skewness ($g_1$):
$g_1 = \frac{\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^3}{\left(\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2\right)^{3/2}} = \frac{m_3}{m_2^{3/2}}$
Where:

$n$ = number of data points
$x_i$ = individual data point
$\bar{x}$ = sample mean
$m_k$ = the k-th central moment, calculated as $\frac{1}{n} \sum_{i=1}^{{n} (x_i - \bar{x})}k$

Sample Kurtosis ($g_2$):
$g_2 = \frac{\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^4}{\left(\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2\right)^2} - 3 = \frac{m_4}{m_2^2} - 3$
The subtraction of 3 in the kurtosis formula is to obtain the excess kurtosis. A normal distribution has a theoretical kurtosis of 3, so subtracting 3 normalizes the value, making a normal distribution have an excess kurtosis of 0.

Interpreting Skewness and Kurtosis

Interpreting skewness kurtosis is essential for understanding the underlying characteristics of data, especially in financial markets.

For skewness:

Zero Skewness: Indicates a perfectly symmetrical distribution, where the mean, median, and mode are approximately equal. This is characteristic of a normal distribution.
Positive Skewness: The right tail of the distribution is longer and fatter, pulling the mean to the right of the median and mode. In finance, this implies a higher probability of small losses and a lower, but still present, probability of large gains.
Negative Skewness: The left tail is longer and fatter, pulling the mean to the left. This suggests a higher probability of small gains and a lower, but significant, probability of large losses. Investors typically prefer assets with positive skewness, as it means potential for upside surprises, while negative skewness points to "tail risk" on the downside.

For kurtosis:

Mesokurtic (Excess Kurtosis = 0): The distribution has a tail structure similar to a normal distribution.
Leptokurtic (Excess Kurtosis > 0): The distribution has fatter tails and a sharper peak than a normal distribution. This indicates a higher probability of extreme events (both positive and negative), which means more outliers. In investing, this implies a higher risk of severe downturns or exceptional windfalls.
Platykurtic (Excess Kurtosis < 0): The distribution has thinner tails and a flatter peak than a normal distribution. This suggests fewer extreme outliers, implying a more stable and predictable pattern of returns.

These measures help in data analysis by providing context for evaluating the likelihood and magnitude of unusual observations, which simple standard deviation alone cannot capture.

Hypothetical Example

Consider two hypothetical investment portfolios, Portfolio A and Portfolio B, with identical average annual return and standard deviation over a decade.

Portfolio A (Positively Skewed, Leptokurtic):
Let's say its returns, when plotted, show a concentration of values around the mean, but with a noticeable long tail stretching to the right and some infrequent, very large positive returns. There are also a few significant negative returns, creating slightly fatter tails overall compared to a normal distribution.

Skewness: +0.8 (positive skew)
Excess Kurtosis: +2.5 (leptokurtic)

This suggests Portfolio A frequently delivers small positive returns, but occasionally provides very large positive returns. However, the leptokurtic nature also indicates a higher propensity for extreme negative returns, even if less frequent. An investor might consider this for high-growth investment strategies where large upside potential is sought, alongside an understanding of magnified tail risk.

Portfolio B (Negatively Skewed, Platykurtic):
This portfolio's returns are more spread out, with a flatter peak, and a longer tail extending to the left, indicating a greater chance of significant negative returns. Its tails are thinner than a normal distribution's.

Skewness: -0.5 (negative skew)
Excess Kurtosis: -0.7 (platykurtic)

Portfolio B might deliver consistent, moderate positive returns, but occasionally experiences substantial drawdowns. The platykurtic nature suggests that extreme events (both positive and negative) are less likely than with a normal distribution. This profile might appeal to conservative investors prioritizing capital preservation and consistent, low-variance returns, but it still highlights the potential for downside surprises implied by the negative skewness.

This example illustrates how skewness kurtosis provides crucial insights into the "shape" of investment returns, helping investors gauge the true nature of risk beyond simple average return and standard deviation.

Practical Applications

Skewness kurtosis measures are integral to modern financial data analysis and risk management. Their practical applications span various areas in finance:

Portfolio Construction and Optimization: Traditional portfolio management often relies on the assumption of normally distributed returns. However, real-world asset returns, especially in financial markets, are frequently skewed and have fat tails. By incorporating skewness and kurtosis, investors can build portfolios that better account for extreme events and asymmetrical risks, moving beyond basic mean-variance optimization. For example, investors seeking to minimize the chance of large losses might prefer portfolios with positive skewness and lower kurtosis.
Risk Assessment and Stress Testing: These measures help quantify "tail risk," the likelihood of extreme positive or negative outcomes. Financial institutions and regulators use skewness and kurtosis to perform stress tests and calculate measures like Value at Risk (VaR) more accurately, especially for complex instruments or markets prone to sudden, large movements. The Federal Reserve Board, for instance, focuses on assessing vulnerabilities, noting that "elevated valuation pressures may increase the possibility of outsized drops in asset prices" in its Financial Stability Report⁶. This reflects the concern for extreme tail events that skewness and kurtosis help identify.
Option Pricing: Option pricing models often need to account for non-normal return distributions. The presence of skewness and kurtosis in underlying asset prices influences implied volatility and the pricing of out-of-the-money options, leading to phenomena like the "volatility smile" or "smirk."
Hedge Fund Strategies: Many hedge fund investment strategies actively seek to exploit or manage higher-moment risks. Strategies like "tail risk hedging" are explicitly designed to profit from or protect against extreme negative outcomes, requiring a deep understanding of skewness and kurtosis. Academic research, such as "Higher-Moment Risk" by Gormsen and Jensen, highlights how "higher-moment risk peaks during seemingly calm periods where variance is low and past returns are high," underscoring their importance for sophisticated investors⁵.
Regulatory Oversight: Regulators increasingly recognize the importance of these higher moments for systemic risk. Basel III, for example, emphasizes robust risk management practices, requiring banks to hold more capital to withstand financial stress and economic downturns, implicitly encouraging consideration of extreme outcomes not fully captured by simpler variance measures.

Limitations and Criticisms

While skewness kurtosis provides valuable insights, they are not without limitations and have faced criticisms:

Sensitivity to Outliers: Both skewness and kurtosis, especially the traditional Pearson measures, are highly sensitive to extreme outliers. A single aberrant data point can significantly distort their values, potentially leading to misleading interpretations of the overall distribution's shape. This non-robustness means that their calculation can be easily "spoiled by extreme outliers"⁴.
Misconception of "Peakedness": A persistent misconception, particularly concerning kurtosis, is that it directly measures the "peakedness" or flatness of a distribution's central part. However, as numerous statisticians have clarified, kurtosis primarily measures the "tailedness"—the presence and extremity of outliers—and tells "virtually nothing about the shape of the peak". Th³is misunderstanding can lead to incorrect conclusions about data concentration or spread.
² Interpretive Ambiguity: While the formulas are precise, the qualitative interpretation of the magnitudes of skewness and kurtosis can sometimes be subjective without a clear benchmark. What constitutes "high" or "low" might vary across different datasets and contexts.
Not a Complete Picture: Even with skewness and kurtosis, a distribution is not fully characterized. Other factors, such as multimodality (having multiple peaks), are not explicitly captured by these two measures, potentially leading to an incomplete understanding of complex probability distribution shapes.
¹ Data Requirements: Accurate calculation of higher moments like skewness and kurtosis often requires large datasets. For small samples, these measures can be highly unstable and unreliable, making it challenging to derive robust insights.

Despite these criticisms, when used judiciously and in conjunction with other statistical tools, skewness kurtosis remain powerful instruments in data analysis, particularly in fields like finance where understanding extreme events is paramount for effective risk management.

Skewness Kurtosis vs. Standard Deviation

While skewness kurtosis and standard deviation all describe aspects of a data distribution, they address different characteristics. Standard deviation is a measure of dispersion, quantifying the typical deviation of data points from the mean. It provides a good indication of the overall spread or volatility of a dataset. For a normal distribution, the mean and standard deviation are sufficient to describe the entire distribution.

However, many real-world datasets, especially in financial markets, do not perfectly follow a normal distribution. This is where skewness and kurtosis become essential. Skewness, the third moment, describes the asymmetry of the distribution, indicating whether it leans to one side. Kurtosis, the fourth moment, describes the "tailedness" of the distribution, highlighting the presence and severity of outliers.

Essentially, standard deviation tells you "how wide" the data is, whereas skewness tells you "which way" it leans, and kurtosis tells you "how heavy" its tails are. For investors, ignoring skewness and kurtosis and relying solely on standard deviation can lead to an underestimation of true risk, particularly the potential for large, unexpected losses or gains.

FAQs

What is the difference between skewness and kurtosis?

Skewness measures the asymmetry of a probability distribution, indicating whether the tail of the distribution is longer on one side (positive or negative). Kurtosis, on the other hand, measures the "tailedness" of a distribution, or the degree to which extreme values (outliers) are present compared to a normal distribution.

Why are skewness and kurtosis important in finance?

In finance, skewness and kurtosis are crucial for assessing risk beyond traditional volatility. Financial return distributions often exhibit fat tails and asymmetry, meaning extreme events are more common than a normal distribution would suggest. These measures help investors better understand the potential for large losses or gains and are vital for robust portfolio management and investment strategies.

Can skewness and kurtosis be negative?

Yes, skewness can be negative, indicating a longer tail on the left side of the distribution, suggesting a greater likelihood of extreme negative values. Excess kurtosis can also be negative (platykurtic), meaning the distribution has thinner tails and fewer extreme outliers than a normal distribution.

How do skewness and kurtosis relate to the normal distribution?

The normal distribution serves as a benchmark for both skewness and kurtosis. A perfect normal distribution has a skewness of 0 and an excess kurtosis of 0. Deviations from these values indicate how much a given distribution differs from normality in terms of symmetry and tail behavior, which is critical information in data analysis.

What is kurtosis risk?

Kurtosis risk refers to the increased potential for extreme return outcomes (both positive and negative) in a financial asset or portfolio when its return distribution exhibits high kurtosis (leptokurtic). This means that while severe losses are rare, they are more likely and potentially more damaging than suggested by standard deviation alone.