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
.