Folio Lab/Documentation

Skewness

Skewness measures the asymmetry of the return distribution around its mean. It is the third standardized central moment and reveals whether extreme returns are more likely to occur on the positive or negative side, capturing a dimension of risk that variance alone cannot describe.

Overview

Mean-variance analysis assumes that returns are normally distributed and therefore symmetric. In reality, financial returns exhibit significant skewness -- most equity returns are negatively skewed, meaning that large negative returns occur more frequently than a normal distribution would predict. This asymmetry has profound implications for risk management and portfolio construction.

Investors generally prefer positive skewness (more upside surprises) and dislike negative skewness (more downside surprises). Kraus and Litzenberger (1976) showed that if investors have preferences over skewness, then assets with negative skewness should command a risk premium. This implies that mean-variance optimization, which ignores skewness, may produce suboptimal portfolios for risk-averse investors.

Skewness is particularly important for evaluating strategies that involve option-like payoffs. For example, selling put options generates positively skewed P&L most of the time (small gains from premium collection) but is exposed to severe negative skewness during market crashes. Similarly, momentum strategies tend to exhibit negative skewness due to occasional sharp reversals.

Mathematical Formulation

Sample Skewness

The sample skewness (third standardized central moment) of a return series is:

where is the demeaned return at time , is the sample mean, and is the sample standard deviation. The cube preserves the sign of deviations, so negative deviations contribute negatively and positive deviations contribute positively.

Population Skewness

The population skewness is defined using the third central moment:

where is the third central moment. For a normal distribution, exactly.

Bias-Corrected (Fisher) Skewness

The bias-corrected estimator, commonly used in statistical software, adjusts for finite sample bias:

This correction becomes negligible for large samples but is important when working with short return histories.

Interpretation

Negative Skewness ()

The left tail is longer or fatter than the right tail. Large negative returns are more extreme and/or more frequent than large positive returns. Most equity return distributions exhibit negative skewness, particularly during periods including market crashes. The mean is typically less than the median.

Positive Skewness ()

The right tail is longer or fatter than the left tail. Large positive returns are more extreme and/or more frequent than large negative returns. This is typical of long option positions, venture capital investments, and lottery-like payoffs. The mean is typically greater than the median.

Zero Skewness ()

The distribution is symmetric. The normal distribution has zero skewness, as do all other symmetric distributions (e.g., Student's t). Note that zero skewness does not imply normality -- a distribution can be symmetric but still have heavy tails (excess kurtosis).

Financial Implications

The negative skewness observed in most equity markets has several important consequences:

  • VaR underestimation: Normal-distribution-based VaR systematically underestimates tail risk when the true distribution is negatively skewed.
  • Sharpe Ratio misleading: Strategies with negative skewness can appear attractive on a Sharpe Ratio basis while concealing significant crash risk.
  • Diversification limits: During crises, correlations tend to increase and skewness becomes more negative, reducing the effectiveness of diversification precisely when it is most needed.
  • Skewness premium:Assets with more negative skewness tend to offer higher average returns as compensation for bearing this asymmetric risk (Kraus & Litzenberger, 1976; Harvey & Siddique, 2000).

Advantages & Limitations

Advantages

  • Captures asymmetry: Reveals directional bias in the return distribution that variance cannot detect.
  • Risk decomposition: Helps explain why some portfolios with similar volatility have very different downside risk profiles.
  • Strategy evaluation: Essential for evaluating option-selling strategies, momentum, and other payoffs with asymmetric returns.
  • Standard statistic: Well-understood, easily computed, and available in all statistical and financial software packages.

Limitations

  • Estimation uncertainty: Skewness is a third-moment statistic and has much higher sampling variability than the mean or variance.
  • Outlier sensitivity: A single extreme observation can dramatically change the estimated skewness due to the cubic power.
  • Time-varying: Skewness is not constant over time; it tends to become more negative during market stress periods.
  • Not sufficient alone: Skewness must be considered jointly with kurtosis and other moments for a complete picture of distributional shape.

References

  1. Kraus, A., & Litzenberger, R. H. (1976). "Skewness Preference and the Valuation of Risk Assets." The Journal of Finance, 31(4), 1085-1100.
  2. Harvey, C. R., & Siddique, A. (2000). "Conditional Skewness in Asset Pricing Tests." The Journal of Finance, 55(3), 1263-1295.
  3. Cont, R. (2001). "Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues." Quantitative Finance, 1(2), 223-236.
  4. Jondeau, E., & Rockinger, M. (2003). "Conditional Volatility, Skewness, and Kurtosis: Existence, Persistence, and Comovements." Journal of Economic Dynamics and Control, 27(10), 1699-1737.
  5. Scott, R. C., & Horvath, P. A. (1980). "On the Direction of Preference for Moments of Higher Order Than the Variance." The Journal of Finance, 35(4), 915-919.