Coskewness
Coskewness measures the co-movement between an asset's returns and the squared returns of the market, capturing the asset's contribution to the asymmetry of the portfolio return distribution. Assets with negative coskewness tend to have large losses when the market is most volatile, making them riskier than standard beta would suggest.
Overview
Mean-variance analysis assumes that investors care only about the first two moments of the return distribution (mean and variance). However, real-world return distributions are often skewed, and investors have strong preferences over skewness: they prefer positive skewness (large upside potential with limited downside) and dislike negative skewness (limited upside with large downside risk).
Harvey and Siddique (2000) demonstrated that systematic skewness risk is priced in the cross-section of expected returns. Assets that contribute negatively to portfolio skewness -- those that tend to crash when the market is already volatile -- must offer higher expected returns as compensation. Coskewness is the third-order analog of covariance: while covariance measures co-movement in the second moment, coskewness measures co-movement in the third moment.
Mathematical Formulation
Skewness (Univariate)
The third standardized moment of a return distribution:
Positive skewness indicates a longer right tail (more frequent small losses, occasional large gains), while negative skewness indicates a longer left tail (more frequent small gains, occasional large losses).
Standardized Coskewness
The standardized coskewness between asset and market measures the tendency of the asset to have extreme returns when the market has large moves:
A negative value of means the asset tends to have negative returns when the market exhibits large volatility (whether up or down), contributing negative skewness to the portfolio.
Coskewness Tensor
For a portfolio of assets, the full coskewness structure is captured by a third-order tensor:
This tensor has elements (though symmetry reduces the number of unique elements). It captures all pairwise and three-way interactions in the third moment.
Portfolio Skewness
The skewness of a portfolio with weight vector involves a triple sum over the coskewness tensor:
This can be written compactly in matrix notation as where is the coskewness matrix and denotes the Kronecker product.
Sample Estimation
Given observations, the sample coskewness is:
Interpretation
| Coskewness Value | Interpretation |
|---|---|
| Asset tends to crash when the market is volatile. Adds negative skewness to the portfolio. Investors demand a higher return premium. | |
| Asset's return asymmetry is independent of market volatility. No systematic skewness contribution. | |
| Asset tends to perform well when the market is volatile. Adds positive skewness to the portfolio. Acts as a hedge and commands lower expected return. |
Advantages & Limitations
Advantages
- Beyond mean-variance: Captures important information about return asymmetry that variance alone cannot convey.
- Priced risk factor:Empirically shown to be priced in the cross-section of expected returns (Harvey & Siddique, 2000).
- Portfolio optimization: Enables mean-variance-skewness optimization for investors with skewness preferences.
- Tail risk insight:Provides information about an asset's behavior during extreme market conditions.
Limitations
- Estimation difficulty: Third moments require more data for reliable estimation than second moments; sample coskewness is very noisy.
- Dimensionality: The coskewness tensor has elements, making it impractical for large portfolios without simplifying structures.
- Non-stationarity: Coskewness relationships can change dramatically across market regimes.
- Model complexity: Incorporating coskewness into portfolio optimization requires solving higher-order polynomial programs.
References
- Harvey, C. R., & Siddique, A. (2000). "Conditional Skewness in Asset Pricing Tests." Journal of Finance, 55(3), 1263-1295.
- Kraus, A., & Litzenberger, R. H. (1976). "Skewness Preference and the Valuation of Risk Assets." Journal of Finance, 31(4), 1085-1100.