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### Deriving the principal components of two uncorrelated return series

Suppose the returns on two uncorrelated series are and . It is assumed that we want an analytical solution rather than a numerical solution (a numerical solution can be found using Nematrian web services functions that target principal components, i.e. MnPrincipalComponents, MnPrincipalComponentsSizes and MnPrincipalComponentsWeights).

For a two series problem, if the covariance matrix is then the principal components are associated with the eigenvectors and eigenvalues of the covariance matrix, i.e. with values of that satisfy, for some vector the equation . The therefore satisfy the following equations:  This means that and , i.e. (since for a covariance matrix): In this instance, the two series are uncorrelated and therefore . The quadratic then becomes , i.e. or . The (population) covariance matrix is , which thus has two eigenvalues and and associated eigenvectors which are of the form and respectively for arbitrary and . The first principal component is associated with whichever of and is the larger, and the second principal component with the other one.

If we want principal components that are orthonormal return series and portfolio exposures that correspond to these principal components then the portfolio exposures must have for and 2, which means in this instance that and . If we choose then the resulting return series are merely de-meaned versions of the original series, i.e. are are and or vice-versa depending on whether or .