that the matrix has a square root; that is, V=B^2 for some B, where V,Bare
matrices.
Consider the situation where we are required to evaluate the covariance
between the returns of securities AandB. Let eAandeBbe the factor expo-
sure vectors for the two stocks. Adapting the formula for variance previously
discussed, we have the covariance as
(3.9)
We can therefore calculate the covariance between all the securities in our
universe and make them entries in a covariance matrix. This matrix would
come in handy to evaluate correlations between securities. Note that if the
total universe of securities is about 5000 stocks, then the covariance matrix
for the list is a square matrix with 25 million entries. Calculating the vari-
ance and covariance of each stock pair individually by sampling past data
can be a tedious endeavor. Armed with the factor exposure vectors and the
factor covariance matrix, the covariance and correlations between securities
may be calculated readily. Thus, the use of the factor covariance matrix re-
duces the complexity of evaluating the correlations between securities in a
dramatic way. Even so, the full and complete covariance matrix for all the
stocks in the universe is given by
CovMatrix = XVXT (3.10)It is also prudent to be aware of certain potential issues when working with
the factor covariance matrix. For example, it is not uncommon in invest-
ment circles to hear someone say, “But you don’t want to be mining the co-
variance matrix!” Let us examine what they mean by that. Mininghere
refers to data mining, albeit with a negative connotation. The word is used
synonymously with bias, indicating that since the covariance matrix is de-
duced from historical data, the values are a reflection of the past and may
not hold going forward. While the empirical observation has been that the
covariance matrices are relatively stable, it is still subject to the fact that the
values used in the covariance matrix may not be exact, and it may be useful
to do some sensitivity analysis on applications where we use the covariance
matrix. It is probably worthwhile to also bear in mind that along with the
covariance matrix, the specific variance is also backward looking and is
subject to the “mining syndrome.”
Another issue that is commonly cited with regard to the covariance ma-
trix and its use is the underlying assumption of the Gaussian distribution for
the computed variance. The so-called fat tails that are ubiquitous in the vari-
ance of returns in financial time series are not accounted for by the model.
cov( , )rr eVeAB= A BTFactor Models 43