calculates coskewness and cokurtosis as the skewness and kurtosis of two assets with reference to one another.
CoSkewnessMatrix(R, ...) CoKurtosisMatrix(R, ...) CoVariance(Ra, Rb) CoSkewness(Ra, Rb) CoKurtosis(Ra, Rb) M3.MM(R, unbiased = FALSE, as.mat = TRUE, ...) M4.MM(R, as.mat = TRUE, ...)
| R | an xts, vector, matrix, data frame, timeSeries or zoo object of asset returns |
|---|---|
| … | any other passthru parameters |
| Ra | an xts, vector, matrix, data frame, timeSeries or zoo object of asset returns |
| Rb | an xts, vector, matrix, data frame, timeSeries or zoo object of index, benchmark, portfolio, or secondary asset returns to compare against |
| unbiased | TRUE/FALSE whether to use a correction to have an unbiased estimator, default FALSE |
| as.mat | TRUE/FALSE whether to return the full moment matrix or only the vector with the unique elements (the latter is advised for speed), default TRUE |
Ranaldo and Favre (2005) define coskewness and cokurtosis as the skewness and kurtosis of a given asset analysed with the skewness and kurtosis of the reference asset or portfolio. Adding an asset to a portfolio, such as a hedge fund with a significant level of coskewness to the portfolio, can increase or decrease the resulting portfolio's skewness. Similarly, adding a hedge fund with a positive cokurtosis coefficient will add kurtosis to the portfolio.
The co-moments are useful for measuring the marginal contribution of each asset to the portfolio's resulting risk. As such, comoments of asset return distribution should be useful as inputs for portfolio optimization in addition to the covariance matrix. Martellini and Ziemann (2007) point out that the problem of portfolio selection becomes one of selecting tangency points in four dimensions, incorporating expected return, second, third and fourth centered moments of asset returns.
Even outside of the optimization problem, measuring the co-moments should be a useful tool for evaluating whether or not an asset is likely to provide diversification potential to a portfolio, not only in terms of normal risk (i.e. volatility) but also the risk of assymetry (skewness) and extreme events (kurtosis).
Boudt, Kris, Brian G. Peterson, and Christophe Croux. 2008. Estimation and Decomposition of Downside Risk for Portfolios with Non-Normal Returns. Journal of Risk. Winter.
Boudt, Kris, Cornilly, Dries and Verdonck, Tim. 2017. A Coskewness Shrinkage Approach for Estimating the Skewness of Linear Combinations of Random Variables. Submitted. Available at SSRN: https://ssrn.com/abstract=2839781
Martellini, L., & Ziemann, V. 2010. Improved estimates of higher-order comoments and implications for portfolio selection. Review of Financial Studies, 23(4), 1467-1502.
Ranaldo, Angelo, and Laurent Favre Sr. 2005. How to Price Hedge Funds: From Two- to Four-Moment CAPM. SSRN eLibrary.
Scott, Robert C., and Philip A. Horvath. 1980. On the Direction of Preference for Moments of Higher Order than the Variance. Journal of Finance 35(4):915-919.
BetaCoSkewness BetaCoKurtosis
BetaCoMoments ShrinkageMoments EWMAMoments
StructuredMoments MCA
data(managers) CoVariance(managers[, "HAM2", drop=FALSE], managers[, "SP500 TR", drop=FALSE])#> [1] 0.0006641516CoSkewness(managers[, "HAM2", drop=FALSE], managers[, "SP500 TR", drop=FALSE])#> [1] -2.101289e-06CoKurtosis(managers[, "HAM2", drop=FALSE], managers[, "SP500 TR", drop=FALSE])#> [1] 2.579066e-06