portfolio is also along the same lines. If we are able to evaluate the net fac-
tor exposure for the portfolio, then we can treat it as a single security and
evaluate the common factor variance using the formula discussed in the pre-
vious section. Let us therefore concentrate our efforts on the evaluation of
the factor exposure for the portfolio. The factor exposure of the portfolio is
simply the weighted sum of the factor exposures of all the securities in it.
The weight to be used for the exposure of each security is determined by the
weight of the security in the portfolio.
Let us consider a portfolio composed of two securities, AandB, with
exposure vectors given by eAandeB. Let the weights of the two securities in
the portfolio be hAandhB, respectively. The exposure vector of the portfo-
lio is given as
ep=hAeA+hBeB (3.11)Assuming a two-factor model and writing out the formula in matrix form,
we have
Factor Models 45
TALKING POINTS
Bear in mind that the APT model is a linear model; that is, the return
is a linear combination of factor returns. Is that a limitation of the
model? Some would argue that linearity in returns is probably valid
only in a fixed range of values for the factor exposures. As the values
stretch more and more to the extremes, linearity leads to progressively
poor predictions of asset returns. This is akin to quite a few phenom-
ena encountered in the physical sciences where linear relationships are
valid only in a certain operating range.
With that said, if you happen to find yourself in a finance confer-
ence, in the middle of a discussion on APT at cocktail hour, here is
something to try out. Look to a distance, put a finger to your cheek
and proclaim wistfully, “I wonder what the consequences are of ne-
glecting the potentially inherent nonlinearities in the modeling
process... .”
In all likelihood, this will cause a whole lot of other people to also
wonder with you. If, however, someone were to probe further and ask
you to elaborate on your musings, you could always pretend to recog-
nize someone at a distance, smile politely, and excuse yourself, saying
that you need to mingle.