FINANCE Corporate financial policy and R and D Management

lated among the assets, only the factor variances and covariances need to
be calculated during model estimation (see Figure 8.2).
By using a multiple-factor model, we significantly reduce the number
of calculations. For example, in the U.S. Equity Model (US-E3), 65 fac-
tors capture the risk characteristics of equities. Moreover, since there
are fewer parameters to determine, they can be estimated with greater
precision.
We can easily derive the matrix algebra calculations that support and
link the above diagrams by using an MFM. From Figure 8.2, we start with
the MFM equation:

(8.16)

where = excess return on asset i

X= exposure coefficient on the factor

= factor return

u ̃ = specific return

f ̃

r ̃i

rXfu ̃i=+ ̃ ̃

Appendix 8.A 221

FIGURE 8.1 The Covariance Structure of Security Returns

V (i,j) = Cov [r (i), r(j)]

where V (i,j) = asset covariance matrix

i,j = individual stocks

~ ~

V(1,1) V(1,2)... V(1,N)

V(2,1) V(2,2)... V(2,N)

V(3,1) V(3,2)... V(3,N)

V(N,1) V(N,2)... V(N,N)

V =

..

.

..

.

..

.