Finding The Most Important Principal
Component
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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).