The Correlation & Covariance Matrix
Properties of a "good" correlation matrix...
Under certain assumptions about the dependence structure (so-called "elliptical distributions", examples include the multivariate normal and student t distributions), the correlation coefficient can also be viewed as the cosine of the angle between the two vectors of samples drawn from the two random variables. The cosine of this angle can be calculated as the dot product.
Customized Correlation Matrizes
Sometimes, it is desirable to manually change or even build from scratch a correlation matrix.
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Using correlation matrizes in simulations: Cholesky decomposition versus Eigenvalue matrices.
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The Covariance Matrix
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Estimation of Correlations & Covariances
The estimation of the covariance / correlation matrix is an important topic in portfolio construction respectively asset allocation ("Garbage in, Garbage out").
Historical correlations and covariances are suject to estimation errors. Optimizers like for example classical Markowitz mean-variance optimization are known to be "estimation error maximizers". Various approaches exist to tackle the estimation issues. One approach is of statistical nature and based on Bayesian statistics. Various effective estimators exist...
- James Stein Estimator for expected returns
- Ledoit Wolf Estimator for correlations
- Jorion Estimator for expected returns & correlations
You can download a spreadsheet illustrating the Ledoit Wolf estimator in the download section.
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