Abstract
We compare various methodologies to estimate the covariance matrix in a fixed-income portfolio. Adopting a statistical approach for the robust estimation of the covariance matrix, we compared the Shrinkage (SH), the Nonlinear Shrinkage (NSH), the Minimum Covariance Determinant (MCD) and the Minimum Regularised Covariance Determinant (MRCD) estimators against the sample covariance matrix, here employed as a benchmark. The comparison was run in an application aimed at individuating the principal components of the US term structure curve. The contribution of the work mainly resides in the fact that we give a freshly new application of the MRCD and the NS robust covariance estimators within the fixed-income framework. Results confirm that, likewise financial portfolios, also fixed-income portfolios can benefit of using robust statistical methodologies for the estimation of the covariance matrix.
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Notes
- 1.
As explained in Boudt et al. (2017), S* is a transformation of the sample covariance matrix SU(H), obtained as: \( S^{*} = \Lambda ^{1/2} Q^{{\prime }} S_{U} \left( H \right)Q\Lambda ^{1/2} \).
- 2.
Principal Component Analysis (Hotelling 1933) is a dimension reduction technique that works on a covariance (or correlation) matrix identifying the volatility factors that drive the time series under investigation. The PCA relies on the spectral decomposition of the covariance matrix \( \Sigma \):
$$ \Sigma = {\text{G}}\Omega {\text{G}}^{{\prime }} $$where G is the square matrix of the eigenvalues of \( \Sigma \), and \( \Omega \) is a diagonal matrix filled with the eigenvalues of the covariance matrix. The principal components are given by the normalized eigenvectors, ranked in descendant order according to the size of related eigenvalues. This because the total variance is equal to the sum of all the eigenvalues, so that the size of a single eigenvalues is the percentage of total variance explained. As a limited number of eigenvalues is usually enough to explain at least the 99% of total variation, the reduction of the covariance matrix can be performed by retaining only the eigenvalues that explain a certain threshold of the variance, eliminating the others.
- 3.
The sample and MCD methodologies are estimated using MATLAB embedded functions. The Shrinkage code is from Ledoit and Wolf (2004b), while the Nonlinear Shrinkage is estimated with the nlshrink R package, https://rdrr.io/cran/nlshrink/man/nlshrink.html, and the MRCD is from the KU Leuven webpage, https://wis.kuleuven.be/stat/robust/Programs/MRCD.
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Neffelli, M., Resta, M. (2018). A Comparison of Estimation Techniques for the Covariance Matrix in a Fixed-Income Framework. In: Mili, M., Samaniego Medina, R., di Pietro, F. (eds) New Methods in Fixed Income Modeling. Contributions to Management Science. Springer, Cham. https://doi.org/10.1007/978-3-319-95285-7_6
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