Skewness

Skewness: Definition, Formula, Example, and FAQs

What Is Skewness?

Skewness is a statistical measure that quantifies the asymmetry of a probability distribution. In the realm of statistical finance and data analysis, it indicates the extent to which a distribution of data deviates from a symmetrical shape, such as the normal distribution. A symmetrical distribution has an equal number of data points on either side of its mean. Skewness reveals the direction and magnitude of this asymmetry, providing insight into the likelihood of extreme values (outliers) in one tail of the distribution versus the other. This measure is crucial for understanding the characteristics of financial data, where symmetrical distributions are rare.

History and Origin

The concept of skewness, as a formal statistical measure, was significantly developed by Karl Pearson in the late 19th and early 20th centuries. Pearson, an English mathematician and biostatistician, is credited with establishing the discipline of mathematical statistics. His work on asymmetrical frequency curves, particularly his 1895 paper "Contributions to the Mathematical Theory of Evolution, II: Skew Variation in Homogeneous Material," laid the groundwork for understanding and quantifying skewness. He realized that many natural and biological phenomena did not fit the symmetrical bell curve of the normal distribution and sought methods to describe these "skew" variations. Pearson's method of moments, from which skewness is derived, allowed for a more comprehensive analysis of diverse data shapes.⁵

Key Takeaways

Skewness measures the asymmetry of a data distribution.
Positive skewness indicates a longer right tail, suggesting more frequent small losses and a few large gains in financial returns.
Negative skewness indicates a longer left tail, implying more frequent small gains and a few large losses.
It provides crucial insights beyond mean and standard deviation for risk assessment in finance.
Financial professionals use skewness to evaluate investment opportunities and portfolio risks.

Formula and Calculation

Skewness is typically calculated as the third standardized moment of a distribution. The most common formula for population skewness ((\gamma_1)) is:

\gamma_1 = \frac{E[(X - \mu)^3]}{\sigma^3}

Where:

(X) represents the random variable.
(\mu) is the mean of the distribution.
(E) denotes the expected value operator.
(\sigma) is the standard deviation of the distribution.

For a sample, the formula for sample skewness ((g_1)) is:

g_1 = \frac{n}{(n-1)(n-2)} \sum_{i=1}^{n} \left( \frac{x_i - \bar{x}}{s} \right)^3

Where:

(n) is the number of data points in the sample.
(x_i) is the individual data point.
(\bar{x}) is the sample mean.
(s) is the sample standard deviation.

Interpreting Skewness

The sign and magnitude of skewness provide valuable information about a distribution's shape.

Zero Skewness: A value of zero indicates a perfectly symmetrical distribution, where the mean, median, and mode are all equal. The normal distribution is an example of a symmetrical distribution with zero skewness.
Positive Skewness (Right-Skewed): A positive skewness value means the tail on the right side of the distribution is longer or fatter than the left tail. This implies that most of the data points are concentrated on the left side (lower values), with a few extreme high values pulling the mean to the right of the median. In finance, this could suggest a higher probability of small losses and a lower, but still present, probability of large gains.
Negative Skewness (Left-Skewed): A negative skewness value means the tail on the left side of the distribution is longer or fatter than the right tail. This indicates that most data points are concentrated on the right side (higher values), with a few extreme low values pulling the mean to the left of the median. In financial contexts, negative skewness might imply a higher probability of small gains and a lower, but still present, probability of large losses.

Understanding skewness helps investors assess the potential upside and downside of an investment strategy beyond just its average return and volatility.

Hypothetical Example

Consider two hypothetical investment portfolios, Portfolio A and Portfolio B, with the same average monthly return of 0.5% and the same monthly standard deviation (volatility) of 2%. Without considering skewness, they might appear equally attractive from a risk-return perspective.

However, if we calculate their skewness:

Portfolio A has a skewness of +0.8. This positive skewness suggests that Portfolio A's returns are mostly concentrated on the lower side, but it occasionally experiences significantly large positive returns. An investor in Portfolio A might see many small negative or slightly positive months, but the infrequent large gains make up for it over time.
Portfolio B has a skewness of -0.9. This negative skewness indicates that Portfolio B's returns are primarily clustered on the higher side, but it is susceptible to infrequent, very large negative returns. An investor in Portfolio B might experience consistent small positive returns, but there's a risk of a substantial loss event.

An investor focused on avoiding large drawdowns might prefer Portfolio A despite the smaller, more frequent negative returns, as the potential for extreme positive returns balances the risk. Conversely, an investor seeking steady small gains might initially favor Portfolio B but would need to be acutely aware of its exposure to large, unexpected losses. This example highlights how skewness provides critical context for portfolio management decisions.

Practical Applications

Skewness is a vital tool in quantitative analysis across various areas of financial markets:

Portfolio Construction and Risk Management: Investors and portfolio managers use skewness to understand the potential for extreme outcomes in their portfolios. For instance, an investment with positive skewness might be attractive because it offers a small chance of a very large gain, even if it has a higher probability of small losses. Conversely, an asset with negative skewness, while offering consistent small gains, might carry the risk of infrequent, significant losses.⁴ Understanding this helps in asset allocation decisions. Morningstar, for example, discusses how skewness helps characterize the shape of returns and understand the likelihood of extreme gains or losses in investment funds.³
Options Pricing: Skewness plays a crucial role in understanding the "volatility smile" or "volatility skew" observed in options markets. This phenomenon refers to the empirical observation that implied volatility for out-of-the-money options differs systematically from at-the-money options. The skew in implied volatilities reflects market participants' perception of the likelihood of extreme price movements and their aversion to negative skewness in underlying asset returns.
Hedge Fund Strategies: Many hedge fund strategies, particularly those involving options or derivatives, explicitly aim to generate positive skewness, meaning they seek strategies with limited downside and significant upside potential. These strategies might involve selling "tail risk" or constructing complex payoff structures.
Credit Risk Analysis: In credit risk, skewness can help assess the distribution of losses. A highly skewed distribution of losses (e.g., many small losses but a few very large defaults) requires different modeling and capital allocation approaches than a more symmetrical distribution.

Limitations and Criticisms

While skewness offers valuable insights, it also has limitations:

Sensitivity to Outliers: Skewness, as a third moment, is highly sensitive to extreme values or outliers in the data. A single unusually large or small observation can significantly distort the skewness measure, making it appear more extreme than the overall data trend suggests. This sensitivity can lead to unstable estimations, particularly with limited data.²
Interpretation in Isolation: Skewness should rarely be interpreted in isolation. It provides information about the asymmetry of a distribution, but it does not describe the "peakedness" or "fatness" of the tails. For a complete understanding of a distribution's shape, skewness must be considered alongside kurtosis, which measures the tails' characteristics and the concentration of data around the mean. Without kurtosis, the full picture of tail risk or opportunity remains incomplete. Research Affiliates highlights that relying solely on mean and standard deviation can be misleading, but higher moments like skewness and kurtosis also have limitations, including their sensitivity to outliers and unstable estimation.¹
Data Requirements: Accurate calculation of skewness often requires a sufficient amount of historical data. In financial markets, where conditions can change rapidly, historical skewness may not always be a reliable predictor of future distribution shapes.
Not a Causal Measure: Skewness describes the shape of past returns, but it does not explain why those returns occurred or guarantee that future returns will exhibit the same skewness.

Skewness vs. Kurtosis

Skewness and kurtosis are both statistical measures that describe the shape of a probability distribution, but they focus on different aspects. Skewness measures the asymmetry of the distribution, indicating whether one tail is longer or fatter than the other. A positively skewed distribution has a long right tail, while a negatively skewed one has a long left tail.

In contrast, kurtosis measures the "tailedness" or "peakedness" of a distribution. High kurtosis (leptokurtic) implies that a distribution has fatter tails and a sharper peak than a normal distribution, suggesting a higher probability of extreme events (both positive and negative). Low kurtosis (platykurtic) indicates thinner tails and a flatter peak, implying fewer extreme observations. While skewness tells you about the direction of extreme values, kurtosis tells you about their frequency and magnitude relative to the center. Both are crucial for a comprehensive understanding of risk in financial data.

FAQs

What does positive skewness mean for an investment?

Positive skewness for an investment generally means that the distribution of its returns has a longer or fatter tail on the right side. This implies that while the investment might experience frequent small losses or modest gains, there is a chance of infrequent but significant positive returns. Investors who seek lottery-ticket-like payoffs or who are comfortable with many small losses in exchange for the possibility of large gains might find positively skewed investments appealing.

Can skewness predict future returns?

Skewness itself is a descriptive statistic of past returns and does not directly predict future returns. However, understanding the historical skewness of an asset or portfolio can inform expectations about the shape of its future return distribution. For example, consistently negatively skewed assets might suggest a vulnerability to large, sudden drops, which investors should account for in their risk assessments.

Is zero skewness always desirable?

Not necessarily. While a zero skewness indicates a symmetrical distribution, like the normal distribution, it does not inherently mean it is the "best" or "safest" for all investment objectives. For some investors, a positively skewed distribution might be preferred if they are seeking outsized gains, even with the acceptance of more frequent small losses. The desirability of skewness depends entirely on an investor's preferences, risk tolerance, and specific investment strategy.

How does skewness relate to risk?

Skewness provides a nuanced view of risk that goes beyond volatility (standard deviation). While volatility measures the overall dispersion of returns, skewness indicates the probability of extreme positive or negative outcomes. A negatively skewed asset, for instance, implies a higher chance of large unexpected losses, which is a critical aspect of downside risk. Conversely, a positively skewed asset might appeal to those seeking infrequent, but substantial, upside potential.