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Coskewness

A third-moment measure that quantifies how an asset's returns interact with squared market returns, enhancing traditional mean-variance portfolio optimization.

Why Coskewness?

Traditional mean-variance optimization only considers the first two moments of return distributions—expected returns and covariances. However, investors typically prefer positive skewness (larger probabilities of extreme positive returns) and avoid negative skewness. Coskewness extends portfolio theory to include these preferences by measuring how an asset's returns vary with squared market returns.

Assets with positive coskewness tend to perform well when market volatility increases, providing a hedge against market turbulence and commanding lower risk premiums. Conversely, assets with negative coskewness typically suffer during high market volatility periods and require higher expected returns to compensate investors.

1. Mathematical Definition

1.1 Beyond Variance: The Third Moment

Skewness measures the asymmetry of a probability distribution. For a return series $r$ , skewness is defined as:

Skew(r) = \frac{\mathbb{E}[(r - \mu)^3]}{[\mathbb{E}[(r - \mu)^2]]^{3/2}} = \frac{\mathbb{E}[(r - \mu)^3]}{\sigma^3}

Where $\mu$ is the mean return and $\sigma$ is the standard deviation. Positive skewness indicates an asymmetric tail extending toward more positive values, while negative skewness indicates an asymmetric tail extending toward more negative values.

1.2 Coskewness Definition

Coskewness extends this concept to measure the relationship between an asset's return and squared market returns:

s_{i,m} = \frac{\mathbb{E}[(r_i - \mu_i)(r_m - \mu_m)^2]}{\sigma_i \sigma_m^2}

Where $r_i$ and $r_m$ are the returns of asset $i$ and the market, $\mu_i$ and $\mu_m$ are their respective means, and $\sigma_i$ and $\sigma_m$ are their standard deviations.

In an unstandardized form, coskewness can be expressed as:

Coskew(r_i, r_m) = \mathbb{E}[(r_i - \mu_i)(r_m - \mu_m)^2]

1.3 Matrix Representation

For a portfolio of $n$ assets, coskewness can be represented using a three-dimensional matrix:

S_{i,j,k} = \mathbb{E}[(r_i - \mu_i)(r_j - \mu_j)(r_k - \mu_k)]

The coskewness of a portfolio with weights $w$ can then be calculated as:

s_p = \sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{k=1}^{n} w_i w_j w_k S_{i,j,k}

In practice, we often focus on the coskewness between each asset and the market portfolio, which simplifies the calculation.

1.4 Estimation via Regression

Coskewness can also be estimated using regression analysis:

r_i = \alpha + \beta r_m + \gamma r_m^2 + \epsilon

Where $\gamma$ captures the sensitivity of asset returns to squared market returns (coskewness), $\beta$ is the traditional beta (systematic risk), and $\alpha$ is the intercept.

1.5 Financial Significance

Coskewness is financially significant for several reasons:

Pricing impact: Assets with negative coskewness (which tend to perform poorly when market volatility increases) command higher risk premiums in equilibrium, as demonstrated in three-moment asset pricing models.
Diversification benefits: Including assets with positive coskewness can improve a portfolio's risk profile beyond what mean-variance optimization alone would achieve.
Crisis hedging: Assets with positive coskewness can serve as partial hedges during market turbulence, as they tend to perform relatively better when market volatility spikes.

2. Default Parameters

Parameter	Default	Description
window	`252`	Rolling window length in trading days (~1 year)
standardized	`True`	Whether to use standardized or raw coskewness
method	`"kraus-litzenberger"`	Estimation method (kraus-litzenberger, harvey-siddique)
min_periods	`60`	Minimum observations required for estimation
rf	`0.0`	Risk-free rate subtracted from returns before estimation

3. Implementation Considerations

When implementing coskewness in portfolio optimization:

Data requirements: Accurate coskewness estimation requires substantial historical data, as third-moment statistics are more sensitive to sampling error than means and variances.
Estimation method: The regression-based approach (Kraus-Litzenberger) is generally more stable and interpretable than direct calculation, especially for small samples.
Time-variation: Coskewness tends to vary over time, particularly during market regime changes. Rolling-window or GARCH-based approaches can capture this time variation.
Portfolio optimization: Including coskewness in portfolio optimization requires solving a cubic programming problem, which is more complex than quadratic programming used in mean-variance optimization.
Standardization: Standardized coskewness is comparable across assets and time periods, while raw coskewness depends on the scale of returns.

4. Advantages and Limitations

Advantages

Captures asymmetric risk not reflected in traditional mean-variance analysis.
Provides insights into asset behavior during market volatility spikes.
Helps identify potential hedges against market turbulence.
Improves portfolio performance by including investor preferences for positive skewness.
Addresses empirical anomalies unexplained by traditional CAPM.

Limitations

Requires substantial historical data for reliable estimation.
More sensitive to outliers than mean and variance estimates.
Complex to implement in portfolio optimization frameworks.
Time-varying nature makes it challenging to use for long-term asset allocation.
Limited consensus on the best estimation methodology.

5. 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.
Christoffersen, P., Feunou, B., Jacobs, K., & Turnbull, S. (2021). Option-Based Estimation of the Price of Coskewness and Cokurtosis Risk. Journal of Financial and Quantitative Analysis, 56(1), 65-91.
Boudt, K., Cornilly, D., & Verdonck, T. (2020). A coskewness shrinkage approach for estimating the skewness of linear combinations of random variables. Journal of Financial Econometrics, 18(1), 1-23.
Guidolin, M., & Timmermann, A. (2008). International asset allocation under regime switching, skew, and kurtosis preferences. The Review of Financial Studies, 21(2), 889-935.
Martellini, L., & Ziemann, V. (2010). Improved estimates of higher-order comoments and implications for portfolio selection. The Review of Financial Studies, 23(4), 1467-1502.