Skewness and kurtosis

What Is Skewness and Kurtosis?

Skewness and kurtosis are statistical measures used to describe the shape of a probability distribution in statistical analysis, particularly in finance, to assess the characteristics of investment return data. While the mean and variance describe the central tendency and dispersion of data, respectively, skewness and kurtosis provide insight into the asymmetry and "tailedness" of the distribution. Understanding skewness and kurtosis is crucial because financial returns rarely follow a perfect normal distribution, which is symmetrical and has a specific kurtosis.

Skewness measures the asymmetry of a probability distribution. A distribution is skewed if its tails are not balanced. Positive skewness indicates a tail that extends further to the right, meaning more extreme positive values, while negative skewness indicates a tail that extends further to the left, implying more extreme negative values.

Kurtosis measures the "tailedness" of a probability distribution, indicating the frequency of extreme values (outliers) relative to the tails of a normal distribution. A distribution with high kurtosis has fatter tails and a sharper peak than a normal distribution, suggesting a greater probability of extreme positive or negative outcomes. Conversely, a distribution with low kurtosis has thinner tails and a flatter peak.

History and Origin

The concepts of moment in statistics, from which skewness and kurtosis are derived, have roots in the work of English mathematician and biostatistician Karl Pearson. Pearson, often regarded as one of the founders of modern statistics, adapted the concept of moments from mechanics in the late 19th century to describe the characteristics of statistical distributions that deviated from symmetry. His work, particularly in developing the "method of moments" for curve fitting asymmetrical distributions, laid the foundation for the standardized measures of skewness and kurtosis that are used today. He introduced these ideas as part of his broader efforts to create a new statistical system capable of interpreting empirical data that did not conform to the then-prevalent assumption of normality.

Key Takeaways

• Skewness measures the asymmetry of a distribution, indicating whether data points are concentrated more to one side.
• Kurtosis quantifies the "tailedness" of a distribution, reflecting the likelihood of extreme outliers.
• Positive skewness suggests more frequent small losses and a few large gains, while negative skewness indicates more frequent small gains and a few large losses.
• High kurtosis (leptokurtic) implies a greater probability of extreme events, both positive and negative, compared to a normal distribution.
• In finance, understanding skewness and kurtosis helps investors assess non-normal risk characteristics of assets and portfolios beyond just mean and variance.

Formula and Calculation

Skewness and kurtosis are calculated using the third and fourth standardized moments of a distribution, respectively. The formulas generally rely on the sample data.

For a dataset (X = {x_1, x_2, ..., x_n}) with sample mean (\bar{x}) and standard deviation (s):

Sample Skewness ((g_1)):

For a population, the formula simplifies slightly by replacing the denominator with (\sigma^3) and the coefficient with (N). This measure can be positive, negative, or zero.

Sample Kurtosis ((g_2)):

The value of 3 is subtracted from the kurtosis calculation to make the kurtosis of a normal distribution equal to 0. This is known as excess kurtosis. If the result is positive, the distribution is leptokurtic (fat tails, sharp peak). If negative, it is platykurtic (thin tails, flat peak). A mesokurtic distribution (like the normal distribution) has an excess kurtosis of 0.

Interpreting Skewness and Kurtosis

Interpreting skewness and kurtosis in finance provides a deeper understanding of potential outcomes for investments than simply looking at average returns and volatility.

• Interpreting Skewness:

  • Positive Skewness: A positively skewed distribution has a longer right tail. In investment returns, this means a higher probability of small losses or small gains, but also a higher chance of large, infrequent positive returns. Investors often prefer positive skewness as it suggests a greater upside potential with limited downside.
  • Negative Skewness: A negatively skewed distribution has a longer left tail. This indicates a higher probability of small gains or small losses, but also a higher chance of large, infrequent negative returns. Many financial assets, particularly during market downturns, exhibit negative skewness, implying that extreme losses are more likely than extreme gains. Investors generally dislike negative skewness due to the increased tail risk.
  • Zero Skewness: A perfectly symmetrical distribution, like the theoretical normal distribution, has zero skewness.

• Interpreting Kurtosis:

  • Leptokurtic (Kurtosis > 3 or Excess Kurtosis > 0): A leptokurtic distribution has fatter tails and a higher, narrower peak than a normal distribution. This suggests that extreme outcomes (both positive and negative) are more probable than implied by a normal distribution. In finance, this means there's a greater chance of very large gains or very large losses. Investors with a strong aversion to extreme losses might avoid assets with very high kurtosis, as it signals increased susceptibility to large swings.
  • Mesokurtic (Kurtosis = 3 or Excess Kurtosis = 0): This describes a distribution with a kurtosis similar to that of a normal distribution. It serves as a benchmark for comparison.
  • Platykurtic (Kurtosis < 3 or Excess Kurtosis < 0): A platykurtic distribution has thinner tails and a flatter, broader peak than a normal distribution. This suggests that extreme outcomes are less probable than implied by a normal distribution.

Together, skewness and kurtosis offer valuable insights into the shape of a dataset's distribution, complementing measures like variance or standard deviation by highlighting the prevalence and direction of extreme events.

Hypothetical Example

Consider two hypothetical investment portfolios, Portfolio A and Portfolio B, both with the same historical average annual return of 8% and a standard deviation of 15%.

Portfolio A (Positively Skewed, Leptokurtic):

• Skewness: 0.8 (Positive)
• Kurtosis (Excess): 1.5 (Positive, Leptokurtic)

This suggests that Portfolio A's returns are mostly clustered on the left (lower end) with a long tail extending to the right. While the average is 8%, investors might experience many small losses or modest gains, but there's a noticeable chance of infrequent, very large positive returns. The high excess kurtosis implies that these large positive returns, along with potentially large negative returns, are more likely than a normal distribution would predict. An investor seeking lottery-like payouts might find this attractive, despite the higher propensity for minor underperformance.

Portfolio B (Negatively Skewed, Leptokurtic):

• Skewness: -1.2 (Negative)
• Kurtosis (Excess): 2.0 (Positive, Leptokurtic)

Portfolio B's returns are concentrated on the right (higher end) with a long tail extending to the left. This implies that while the average is 8%, there's a higher chance of experiencing small gains, but also a greater likelihood of infrequent, very large negative returns. The even higher excess kurtosis reinforces the notion of increased probability of significant outliers on both ends, with the negative skew highlighting the greater risk of substantial losses. A risk-averse investor using Value at Risk (VaR) calculations would find Portfolio B particularly concerning due to the increased tail risk on the downside.

Even though both portfolios have the same mean and standard deviation, their skewness and kurtosis reveal vastly different risk profiles regarding extreme outcomes.

Practical Applications

Skewness and kurtosis are integral to advanced investment analysis and financial modeling, moving beyond the limitations of mean-variance analysis alone.

• Portfolio Management: Modern portfolio management often considers higher moments when constructing portfolios. Investors generally prefer portfolios with positive skewness and lower kurtosis, as this implies more upside potential and fewer extreme negative events. Incorporating skewness and kurtosis into portfolio optimization models helps allocate assets to better align with investor preferences for specific return distribution shapes.
• Risk Management: These measures are critical for understanding and quantifying tail risk. High kurtosis, for instance, signals that large market movements are more likely than a normal distribution would suggest, requiring more robust risk control measures. Financial institutions use these insights to stress-test portfolios and determine appropriate capital reserves.
• Derivatives Pricing: The Black-Scholes model, which assumes normally distributed returns, often struggles to accurately price options far out of the money. The observed "volatility smile" or "skew" in option markets is partly attributed to the market's pricing of non-normal skewness and kurtosis in underlying asset returns, as investors demand higher premiums for protection against extreme events.
• Algorithmic Trading: In quantitative trading strategies, particularly those analyzing high-frequency market data, real-time calculations of skewness and kurtosis can signal shifts in market dynamics, influencing entry and exit points or risk exposure.

Limitations and Criticisms

While valuable, relying solely on skewness and kurtosis has its limitations in financial analysis:

• Computational Complexity: Incorporating higher-order moments (like skewness and kurtosis) into advanced portfolio management and risk models can significantly increase computational complexity. Calculating co-skewness and co-kurtosis matrices for large portfolios presents substantial challenges in terms of computation, memory storage, and algorithmic design.²
• Data Sensitivity and Estimation Error: Skewness and kurtosis are highly sensitive to outliers. Small changes in extreme data points can drastically alter their values. This makes them prone to significant estimation error, especially with limited historical market data or rapidly changing market conditions. The accuracy of these measures depends heavily on the quality and quantity of data.
• Interpretation Challenges: While the conceptual interpretation is straightforward, applying these measures in real-world scenarios, particularly for non-financial professionals, can be challenging. Their implications for investment decisions, especially when considered alongside other metrics like mean and variance, require a sophisticated understanding.
• Limited Applicability in Some Models: Early empirical evidence in asset allocation suggested that, in some static settings, the mean-variance criterion alone yielded allocations similar to those obtained from direct utility optimization, implying that higher moments might not always play a significantly distinguishable role in practice, except in extreme deviations from normality or highly leveraged portfolios.¹ This doesn't negate their importance in specific contexts but highlights that their impact isn't universally profound across all financial models or investor types.
• Non-Stationarity: Financial time series often exhibit non-stationarity, meaning their statistical properties (including skewness and kurtosis) change over time. Using historical moments to predict future behavior can be misleading if the underlying data-generating process evolves.

Skewness vs. Standard Deviation

While both skewness and standard deviation are measures that describe aspects of a data set's distribution, they capture fundamentally different characteristics. Understanding their distinction is critical in investment analysis.

Standard deviation, a measure of volatility, quantifies how much individual data points typically deviate from the average. A higher standard deviation indicates greater risk (more spread out data). Skewness, on the other hand, tells us where that deviation is more likely to occur in terms of extreme values. A portfolio with a high standard deviation might be risky, but if it also has positive skewness, it implies that the extreme deviations are more likely to be positive, which could be desirable for some investors. Conversely, a high standard deviation combined with negative skewness suggests a higher probability of significant downside events. They are complementary measures, with standard deviation indicating the amount of variability and skewness describing its directional bias.

FAQs

Q1: Why are skewness and kurtosis important in finance if mean and variance are commonly used?
A1: While mean and variance (or standard deviation) describe the average return and overall volatility, they assume a symmetrical distribution. Financial returns, however, are rarely perfectly symmetrical and often have fatter tails than a normal distribution. Skewness and kurtosis capture these non-normal characteristics, providing critical insights into the likelihood and magnitude of extreme positive or negative events, which are crucial for assessing true risk and designing robust portfolios.

Q2: Can skewness and kurtosis be used for individual stocks?
A2: Yes, skewness and kurtosis can be calculated for individual stock returns. Analyzing these measures for single stocks helps investors understand the unique probability distribution of their returns. For example, some growth stocks might exhibit positive skewness (lottery-ticket like returns), while certain stable dividend stocks might have less extreme tails (lower kurtosis). This information informs individual stock selection within a broader portfolio management strategy.

Q3: Do investors prefer positive or negative skewness?
A3: Most investors generally prefer positive skewness. This is because positive skewness implies that while small losses or gains may be common, there is a higher probability of experiencing large, infrequent positive returns. Conversely, negative skewness indicates a greater chance of large, infrequent negative returns, which represents a significant tail risk that most investors seek to avoid.

Q4: What does high kurtosis mean for investors?
A4: High kurtosis, or leptokurtosis, means that a distribution has fatter tails and a sharper peak compared to a normal distribution. For investors, this translates to a higher probability of extreme outcomes, both positive and negative. While it signals the potential for large gains, it also means a greater risk of significant losses. Assets with high kurtosis are often associated with increased risk during market crises or periods of high uncertainty.