calculates covariance, coskewness and cokurtosis matrices using linear shrinkage between the sample estimator and a structured estimator
M2.shrink(R, targets = 1, f = NULL) M3.shrink(R, targets = 1, f = NULL, unbiasedMSE = FALSE, as.mat = TRUE) M4.shrink(R, targets = 1, f = NULL, as.mat = TRUE)
| R | an xts, vector, matrix, data frame, timeSeries or zoo object of asset returns |
|---|---|
| targets | vector of integers selecting the target matrices to shrink to. The first four structures are, in order: 'independent marginals', 'independent and identical marginals', 'observed 1-factor model' and 'constant correlation'. See Details. |
| f | vector or matrix with observations of the factor, to be used with target 3. See Details. |
| unbiasedMSE | TRUE/FALSE whether to use a correction to have an unbiased estimator for the marginal skewness values, in case of targets 1 and/or 2, 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 |
The coskewness and cokurtosis matrices are defined as the matrices of dimension p x p^2 and p x p^3 containing the third and fourth order central moments. They are useful for measuring nonlinear dependence between different assets of the portfolio and computing modified VaR and modified ES of a portfolio.
Shrinkage estimation for the covariance matrix was popularized by Ledoit and Wolf (2003, 2004). An extension to coskewness and cokurtosis matrices by Martellini and Ziemann (2010) uses the 1-factor and constant-correlation structured comoment matrices as targets. In Boudt, Cornilly and Verdonck (2017) the framework of single-target shrinkage for the coskewness and cokurtosis matrices is extended to a multi-target setting, making it possible to include several target matrices at once. Also, an option to enhance small sample performance for coskewness estimation was proposed, resulting in the option 'unbiasedMSE' present in the 'M3.shrink' function.
The first four target matrices of the 'M2.shrink', 'M3.shrink' and 'M4.shrink' correspond to the models 'independent marginals', 'independent and identical marginals', 'observed 1-factor model' and 'constant correlation'. Coskewness shrinkage includes two more options, target 5 corresponds to the latent 1-factor model proposed in Simaan (1993) and target 6 is the coskewness matrix under central-symmetry, a matrix full of zeros. For more details on the targets, we refer to Boudt, Cornilly and Verdonck (2017) and the supplementary appendix.
If f is a matrix containing multiple factors, then the shrinkage estimator will use each factor in a seperate single-factor model and use multi-target shrinkage to all targets matrices at once.
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
Ledoit, Olivier and Wolf, Michael. 2003. Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of empirical finance, 10(5), 603-621.
Ledoit, Olivier and Wolf, Michael. 2004. A well-conditioned estimator for large-dimensional covariance matrices. Journal of multivariate analysis, 88(2), 365-411.
Martellini, Lionel and Ziemann, V\"olker. 2010. Improved estimates of higher-order comoments and implications for portfolio selection. Review of Financial Studies, 23(4), 1467-1502.
Simaan, Yusif. 1993. Portfolio selection and asset pricing: three-parameter framework. Management Science, 39(5), 68-577.
CoMoments StructuredMoments EWMAMoments MCA
data(edhec) # construct an underlying factor (market-factor, observed factor, PCA, ...) f <- rowSums(edhec) # multi-target shrinkage with targets 1, 3 and 4 # as.mat = F' would speed up calculations in higher dimensions targets <- c(1, 3, 4) sigma <- M2.shrink(edhec, targets, f)$M2sh m3 <- M3.shrink(edhec, targets, f)$M3sh m4 <- M4.shrink(edhec, targets, f)$M4sh # compute equal-weighted portfolio modified ES mu <- colMeans(edhec) p <- length(mu) ES(p = 0.95, portfolio_method = "component", weights = rep(1 / p, p), mu = mu, sigma = sigma, m3 = m3, m4 = m4)#> $MES #> [1] 0.03418537 #> #> $contribution #> [1] 0.0052837933 -0.0002137705 0.0035375067 0.0072299562 0.0010344300 #> [6] 0.0034968193 0.0032865974 0.0017118492 0.0036208050 0.0014452126 #> [11] 0.0029590724 -0.0025796747 0.0033727718 #> #> $pct_contrib_MES #> [1] 0.154563005 -0.006253274 0.103480140 0.211492707 0.030259437 #> [6] 0.102289940 0.096140469 0.050075493 0.105916804 0.042275764 #> [11] 0.086559616 -0.075461369 0.098661268 #># compare to sample method sigma <- cov(edhec) m3 <- M3.MM(edhec) m4 <- M4.MM(edhec) ES(p = 0.95, portfolio_method = "component", weights = rep(1 / p, p), mu = mu, sigma = sigma, m3 = m3, m4 = m4)#> $MES #> [1] 0.03241585 #> #> $contribution #> [1] 0.0074750513 -0.0028125166 0.0039422674 0.0069376579 0.0008077760 #> [6] 0.0037114666 0.0043125937 0.0007173036 0.0036152960 0.0013693293 #> [11] 0.0037650911 -0.0048178690 0.0033924063 #> #> $pct_contrib_MES #> [1] 0.23059863 -0.08676361 0.12161541 0.21402052 0.02491917 0.11449542 #> [7] 0.13303965 0.02212817 0.11152864 0.04224258 0.11614968 -0.14862694 #> [13] 0.10465269 #>