Deriving the principal components of two uncorrelated return series

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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 .


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