Eigenvalue...........groooooooan!

[Charles Albert]

*PCA involves transforming the original correlation matrix to derive factor
loadings and eigenvalues. The loadings are in the form of a matrix with the
same dimensions as the original correlation matrix. For example, if there
were 30 rows and 30 columns in the original correlation matrix, the
loadings matrix will also have 30 rows and 30 columns. Each column of the
loadings matrix represents a latent factor that explains a portion of the
movement of the underlying portfolio. Each row of the loadings matrix
corresponds to an asset contained in the original investment universe.
Where an asset intersects a column we observe the sensitivity – or
“loading” – of that asset on a particular factor. By design, each factor is
independent of the other factors, which means they have zero correlation
with each other, and represent independent bets in the portfolio.*

*When PCA is applied to portfolio analysis, columns in the loadings matrix
are referred to as ‘principal portfolios’ because they are composed of long
or short positions in the constituent assets. In this way, the factor
loadings can be interpreted as asset “weights” in that factor’s “principal
portfolio.” The returns from these principal portfolios can be observed by
applying the weight vector to the asset returns, and, because they are
independent, the returns to each principal portfolio will have a
correlation of exactly 0 to one another. We can “project” the returns for
principal portfolios back through time just as we can for any other type of
portfolio.*

*Each principal portfolio has a corresponding eigenvalue, which describes
the proportion of total portfolio standardized variance attributable to
that factor. When the correlation matrix is used in the analysis, the sum
of standardized variances is equal to the number of variables or assets in
the universe under analysis; when the covariance matrix is used, the
eigenvalues sum to the total portfolio variance.*



*http://www.investresolve.com/blog/the-case-for-tactical-alpha-part-2-the-fundamental-flaw-of-grinolds-fundamental-law/
<http://www.investresolve.com/blog/the-case-for-tactical-alpha-part-2-the-fundamental-flaw-of-grinolds-fundamental-law/>*




A LIfe,


A Life,


My Kingdom for a Life!




love


cfa
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