FINANCE Corporate financial policy and R and D Management

We can expand this model to include Kfactors. The total excess return
equation for a multiple-factor model becomes:

(8.14)

whereXjk= risk exposure of security jto factor k

= rate of return on factor k

Note that when K= 1, the MFM equation reduces to the earlier single-factor
version—the CAPM addressed in the previous chapter.
When a portfolio consists of only one security, equation (8.13) de-
scribes its excess return. But most portfolios comprise many securities,
each representing a proportion, or weight, of the total portfolio. When
weights hp 1 , hp 2 ,...,hpNreflect the proportions of Nsecurities in portfolio P,
we express the excess return in the following equation:

(8.15)

where

This equation includes the risk from all sources and lays the groundwork
for further MFM analysis.

Risk Prediction with Multiple-Factor Models

Investors look at the variance of their total portfolios to provide a compre-
hensive assessment of risk. To calculate the variance of a portfolio, you
need to calculate the covariances of all the constituent components. With-
out the framework of a multiple-factor model, estimating the covariance of
each asset with every other asset is computationally burdensome and sub-
ject to significant estimation errors. Let us examine the risk structure of the
BARRA MFM (see Figure 8.1).
An MFM simplifies these calculations dramatically. This results from
replacing individual company profiles with categories defined by common
characteristics (factors). Since the specific risk is assumed to be uncorre-

XhXPk Pj jk

j

N

=

=

∑

1

rXfhu ̃pPkk ̃ ̃

k

K

Pj j

j

N

=+

==

∑∑

11

f ̃

k

rXfu ̃jjkkj ̃ ̃

k

K

=+

=

∑

1

220 THE USE OF FINANCIAL INFORMATION IN THE RISK AND RETURN OF EQUITY