The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 332 Wednesday, February 4, 2004 12:59 PM

332 The Mathematics of Financial Modeling and Investment Management

If the entries of the variance-covariance matrix are random but with

nonzero average, it can be demonstrated that a large eigenvalue still

appears. However, a small number of large eigenvalues also appear

while the bulk of the distribution resembles that of a random matrix.

The eigenvector corresponding to the largest eigenvalue includes all

components with all equal weights proportional to 1.

If we compute the distribution of the eigenvalues of the variance-

covariance matrix of the S&P 500 over a window of two years, we

obtain a distribution of eigenvalues which is close to the distribution of

a random matrix with some exception. In particular, the empirical dis-

tribution of eigenvalues fits well the theoretical distribution with the

exception of a small number of eigenvalues that have much larger val-

ues. Following the reasoning of Malevergne and Sornette, the existence

of a large eigenvalue with a corresponding eigenvector of 1s in a large

variance-covariance matrix arises naturally in cases where correlations

have a random distribution with a nonzero mean.

This analysis shows that there is little information in the variance-

covariance matrix of a large portfolio. Only a few eigenvalues carry

information while the others are simply the result of statistical fluctua-

tions in the sample correlation. Note that it is the entire matrix which is

responsible for the structure of eigenvalues, not just a few highly corre-

lated assets. This can be clearly seen in the case of a variance-covariance

matrix whose entries are all equal. Clearly there is no privileged correla-

tion between any couple of assets but a very large eigenvalue nevertheless

appears.

MULTIFACTOR MODELS

The analysis of the previous section demonstrates that modeling an

equity portfolio as a set of correlated random walks is only a rough

approximation. Though the random walk test cannot be rejected at the

level of individual securities and though there are significant empirical

correlations between securities, the global structure of large portfolios is

more intricate than a set of correlated random walks.

Failure in modeling log price processes as correlated random walks

might happen for several reasons: There might be nonlinearities in the

DGPs of price processes; dependence between log price processes might

not be linear. There might be structural changes (which are a discrete

form of nonlinearity). What is empirically ascertained is that the vari-

ance-covariance matrix of a large set of price processes is not stable and

that its eigenvalues have a distribution that resembles the distribution of