Skewed Distribution

What Is Skewed Distribution?

A skewed distribution is a statistical term in statistics and probability that describes a lack of symmetry in a dataset. Unlike a symmetrical distribution, where data points are evenly distributed around the mean, a skewed distribution has a longer "tail" on one side, indicating that the data is concentrated on the opposite side. This unevenness means the majority of the observations fall on either the lower or higher end of the distribution, influencing the relationship between the mean, median, and mode. Understanding skewed distribution is crucial in data analysis for correctly interpreting the characteristics of financial data, such as investment returns or asset prices.

History and Origin

The concept of skewness as a measure of a distribution's asymmetry was formalized by prominent statisticians in the late 19th and early 20th centuries. While descriptive statistics have roots extending further back, Karl Pearson, an English mathematician and biostatistician, is widely credited with defining the Pearson's coefficient of skewness around 1895. His work significantly contributed to the systematic study of probability distribution and laid the groundwork for modern quantitative analysis techniques, including those applied in finance.

Key Takeaways

• A skewed distribution indicates that data points are not symmetrically distributed around the mean.
• Positive Skew (Right-skewed): The tail is on the right, meaning the majority of data values (and the mode and median) are on the left of the mean.
• Negative Skew (Left-skewed): The tail is on the left, meaning the majority of data values (and the mode and median) are on the right of the mean.
• Skewness is a critical factor in risk management and portfolio performance analysis, especially when assessing potential extreme outcomes.
• It is one of the "higher moments" of a distribution, alongside variance and kurtosis, providing a more complete picture of data characteristics than standard deviation alone.

Formula and Calculation

Skewness is often quantified using the Pearson's moment coefficient of skewness, which is based on the third standardized moment of a distribution.

The formula for the sample skewness ($g_1$) is:

Where:

• (n) = the number of data points
• (X_i) = the (i)-th data point
• (\bar{X}) = the sample mean of the data
• (s) = the sample standard deviation of the data

Interpreting the Skewed Distribution

The interpretation of a skewed distribution depends on the direction and magnitude of the skew.

• Zero Skew: A distribution with zero skew is perfectly symmetrical, such as a normal distribution. In this case, the mean, median, and mode are all equal.
• Positive Skew (Right Skew):
  • The tail extends to the right (positive side).
  • The mean is greater than the median, and the median is greater than the mode (Mean > Median > Mode).
  • This indicates a concentration of data points on the lower end, with fewer, higher values pulling the mean upwards. In finance, this could represent returns with a high frequency of small gains but a few very large gains.
• Negative Skew (Left Skew):
  • The tail extends to the left (negative side).
  • The mean is less than the median, and the median is less than the mode (Mean < Median < Mode).
  • This signifies a concentration of data points on the higher end, with fewer, lower values (or outliers) pulling the mean downwards. For investment returns, this could imply a frequent occurrence of small losses or break-even scenarios, but a few rare, significant losses.

Understanding the skew provides insights into the potential for extreme outcomes that might not be captured by measures like standard deviation alone.

Hypothetical Example

Consider two hypothetical investment portfolios, Portfolio A and Portfolio B, both with an average annual return (mean) of 8% and a standard deviation of 15% over the past decade.

• Portfolio A (Positively Skewed Returns): Imagine Portfolio A had many years with returns between 5% and 10%, but a few years with exceptionally high returns (e.g., +40%, +50%) that pulled the average up. This would result in a positive skew. An investor might experience consistent moderate gains, but the occasional very large gain.
• Portfolio B (Negatively Skewed Returns): Suppose Portfolio B also frequently had returns between 5% and 10%, but instead of large gains, it experienced a few years with significant losses (e.g., -30%, -45%). These large negative outliers would pull the average return down relative to the most frequent returns, leading to a negative skew. An investor might experience consistent moderate gains but faces a higher probability of rare, large losses.

Even with the same mean and standard deviation, the skewed distribution of returns indicates very different risk profiles for these portfolios.

Practical Applications

Skewed distributions are prevalent in finance and have several practical applications:

• Financial Modeling: In financial modeling, assuming that asset returns are normally distributed can lead to significant underestimation of risk, especially tail risk. Incorporating skewness allows for more realistic models that account for the observed asymmetry in market volatility and returns.
• Option Pricing: The "volatility skew" or "volatility smirk" observed in options markets is a direct manifestation of a skewed implied probability distribution of future asset prices. Options traders often monitor volatility skew to gauge market sentiment and assess the relative pricing of options across different strike prices and expiration dates. For instance, in equity markets, out-of-the-money put options often have higher implied volatility than out-of-the-money call options, indicating investor demand for downside protection.⁶ Reuters reported on how options traders brace for volatility skew during earnings seasons.⁵
• Risk Management: Investors and fund managers use skewness to understand the potential for extreme gains or losses. A negatively skewed distribution of returns, for example, signals a higher likelihood of infrequent but substantial losses, which is critical for risk management and capital allocation.
• Portfolio Construction: When evaluating investments, investors may prefer positively skewed returns (more frequent small losses/gains, but occasional large gains) over negatively skewed returns (more frequent small gains, but occasional large losses), even if the mean and variance are similar. This preference reflects an aversion to large downside surprises. Understanding different facets of risk, including those beyond mean and variance, can lead to more robust portfolio construction.⁴

Limitations and Criticisms

While skewness provides valuable insights, it also has limitations:

• Sensitivity to Outliers: Skewness is highly sensitive to extreme values or outliers. A single large outlier can significantly distort the skewness coefficient, potentially misrepresenting the overall shape of the distribution.
• Insufficient Alone: Skewness, as a single statistical measure, does not fully describe the shape of a distribution. It must be considered alongside other statistical moments like mean, variance, and especially kurtosis, which describes the "tailedness" of the distribution. Distributions can be highly skewed and also exhibit "fat tails," meaning extreme events are more probable than a normal distribution would suggest.³
• Difficulty in Interpretation for Non-Experts: While the concept of skewness is straightforward, its implications for complex financial instruments or portfolios can be challenging for non-experts to interpret, particularly in the context of multi-asset portfolio performance.
• Assumptions in Models: Many traditional financial models, such as the Capital Asset Pricing Model (CAPM) and the Black-Scholes option pricing model, assume that asset returns follow a normal distribution, which by definition has zero skewness. When actual returns are significantly skewed, the outputs of these models may be inaccurate, leading to mispricing of assets or misjudgment of risk. Research highlights the importance of considering non-normal distributions, including skewness and fat tails, for more accurate risk management and portfolio selection.²

Skewed Distribution vs. Normal Distribution

The primary distinction between a skewed distribution and a normal distribution lies in their symmetry.

While the normal distribution is a convenient and widely used assumption in many statistical and financial theories due to its mathematical simplicity, real-world financial data, particularly investment returns, often exhibit significant skewness. This means that large positive or negative deviations from the mean are more frequent or pronounced on one side of the distribution, making the skewed distribution a more accurate representation for many financial phenomena.

##¹ FAQs

Why is skewed distribution important in finance?

Skewed distribution is important in finance because investment returns and asset prices rarely conform to perfectly symmetrical patterns. Understanding skewness helps investors and analysts assess potential upside and downside risks that are not captured by traditional measures like standard deviation. For example, a negatively skewed return distribution suggests a higher chance of infrequent but substantial losses, which is critical for accurate risk management.

Can a distribution be both skewed and have "fat tails"?

Yes, a distribution can be both skewed and have "fat tails." Skewness refers to the asymmetry of the distribution, while "fat tails" (measured by kurtosis) indicate that extreme events are more probable than predicted by a normal distribution. Many financial datasets, such as stock returns, exhibit both characteristics, meaning they are asymmetrical and have a higher likelihood of extreme gains or losses.

How does skewness affect investment decisions?

Skewness influences investment decisions by providing a more complete picture of risk beyond just average returns and volatility. Investors may prefer assets with positive skew, where the potential for large gains outweighs the risk of smaller losses. Conversely, negatively skewed assets, despite potentially high average returns, might be less attractive due to the increased probability of severe downside events. Recognizing skewed distribution patterns allows for more informed portfolio construction and risk management strategies.