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### Finding The Most Important Principal Component

Suppose we have a set of  series of returns (or losses, …). A principal component is a set of exposures (and a principal component series is a series of returns) corresponding to an eigenvector of the relevant  covariance matrix, . Eigenvectors satisfy the vector equation  for some scalar .

Typically principal components are identified in practice using suitable software packages designed to identify eigenvectors and eigenvalues, applied to the relevant covariance matrix, , e.g. using using Nematrian web services functions that target principal components, i.e. MnPrincipalComponents, MnPrincipalComponentsSizes and MnPrincipalComponentsWeights.

However,  for the first, i.e. most important, principal component there is a conceptually simpler approach as follows.

We note that any vector can be written as a combination of the eigenvectors of a matrix, and that these eigenvectors can be chosen to be orthonormal (if suitably chosen if some eigenvalues take the same value) so we can write any vector, , of active positions as the sum of positions, , in the relevant eigenvectors, , i.e. as:

Then, . If we order the eigenvectors (principal components) so that the most important ones are first, i.e.  then  is maximised, subject to , if  and . Thus, we can identify the most important principal component by reference to the set of positions of unit magnitude that exhibit the largest risk (here equated with ex-ante tracking error/variance/VaR).

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