18-MultiFactorModels Page 536 Wednesday, February 4, 2004 1:10 PM
536 The Mathematics of Financial Modeling and Investment Managementtechniques. In general one finds the entire set of N returns as one factor
plus a number of additional factors as we have seen in Chapter 12.
Another statistical technique for determining factors is principal
components analysis (PCA). As explained in Chapter 12, PCA is imple-
mented by computing the eigenvalues of the estimated variance-covari-
ance matrix. As shown in the study by Plerou et al., the distribution of
eigenvalues typically follows that of a random matrix with the excep-
tion of a number of outliers. These outliers are the eigenvalues and the
corresponding eigenvectors that form the factors.
PCA (as well as factor analysis) is a powerful statistical technique with
a deep economic interpretation. To see this point, let’s analyze the largest
eigenvalues and the corresponding eigenvectors. The largest eigenvalue cor-
responds to an eigenvector whose components are all approximately equal
to 1/N. Therefore, the largest eigenvalue corresponds to the entire market.
The other large eigenvalues correspond to eigenvectors that have only a
subset of components different from zero. The important finding is that
these eigenvectors correspond to specific market sectors. In fact, the assets
corresponding to the nonzero components of the largest eigenvectors corre-
spond with good approximation to the Standard & Poor’s market sector
classification. Exhibit 18.2 shows the results obtained by performing PCA
on the correlation matrix of the S&P 500 stocks in the period January 2,
2001–September 19, 2003. The ten largest eigenvalues correspond with
good approximation to ten sectors of the Standard & Poor’s classification.^4
That the ten largest eigenvalues correspond to ten sectors of the
Standard and Poor’s classification is a powerful and somewhat surpris-
ing result in empirical financial econometrics. Performing PCA on a
large aggregate of stock prices, we find that the information-carrying
eigenvalues identify stable subsets of the market that correspond to
meaningful sectors. It is an important theoretical-empirical finding that
lends support to the use of factor analysis in financial econometrics.
The eigenvector corresponding to the largest eigenvalue identifies
the entire market. Note that this eigenvector is a totally different con-
cept than the “market portfolio” of the CAPM. In fact, the market port-
folio of the CAPM, which is obtained as a General Equilibrium Theory
and not as a factor model, includes all investable assets and not only
stocks. Performing PCA on a large aggregate of stock prices one obtains
a multiplicity of factors. In principle, on a very large sample, the two
methods—factor analysis and PCA—yield the same result. On a finite
sample, however, results might differ significantly. Note that both factor
analysis and PCA tend to solve the problems of the sample limitations.(^4) The details of the methodology to arrive at these results can be found in Plerou, et
al., “Random Matrix Approach to Cross Correlations in Financial Data.”