var(r) = eVeT+ var(re) (3.7)whereVis the covariance matrix and eis the factor exposure vector. Also
note from Equation 3.7 that the variance of the return is expressed as a sim-
ple sum of two terms. The first term is the variance due to the common fac-
tors, and the second term is the idiosyncratic/specific variance. Also, given
that the standard deviation is the square root of variance, Equation 3.7 may
also be written as
(3.8)One can easily remember the formula by drawing parallels between this and
the Pythagorean theorem from high school geometry. The standard devia-
tions may be represented as the sides of a right-angled triangle as shown in
Figure 3.1. In practice, it turns out that the specific variance is the smaller
component of the total variance, and a significant portion of the total vari-
ance is explained by the common factor variance. Note that key to the eval-
uation of the common factor variance is the knowledge of the covariance
matrix of factor returns.
So, how is the covariance matrix calculated in practice? If we have a
sample of past historic factor returns, then it is a simple matter of using the
formulas in the appendix of the first chapter to evaluate each of the entries
of the covariance matrix. The question therefore now becomes, how do we
get a sample of past historic factor returns? To do this, we first write out
the linear equations for the return of each stock with known stock returns,
treating the factor returns as unknown variables. Next, we solve this system
of equations to obtain an estimate of the factor and specific returns. We
now have the past factor returns that may be used to estimate the covariance
matrix.
In other words, the covariance matrix can be deduced from the factor
returns. The converse of this statement is also true. Knowledge of the
σσσret^222 =+cf specificFactor Models 41
FIGURE 3.1 The Risk Diagram.σcommon factorσspecificσtotal