Nevertheless, in the absence of any information whatsoever, the covari-
ance matrix serves as a critical piece of information to assess correlations be-
tween securities and for use in factor models. Careful use of the covariance
matrix can help keep the reconciliation process between the risk and return
of a large universe of securities tractable.
Example
Factor exposure for stock Ain two-factor model = (0.5, 0.75)
The specific variance on stock A= .0123
The factor covariance matrix for the two-factor model is =
The variance of return for the stock
=.0901 + .0123
=.1024
The square root of the variance is the standard deviation = .32, or 32%.
Thus, stock Ahas a volatility value of 32%.
Factor exposures for stock Bin the two-factor model = (0.75, 0.5)
Covariance between stocks
Application: Calculating the Risk on a Portfolio
In the earlier sections we discussed how the APT model may be used to cal-
culate the risk on a particular asset. Now we will focus on assessing the risk
of an entire portfolio. Just as with a single security, the risk can be ex-
pressed as a sum of two components; namely, common factor risk and spe-
cific risk. We will adopt the approach in which we reason out the formulas
for each of these components, leading in turn to the expression for the risk
in a portfolio.
Let us start with the common factor variance for a security, which can
be computed if we know the factor exposures for the security and the factor
covariance matrix. The logic to evaluate the common factor variance for the
AB and =[]
(^05075) =
0625 0225
0 225 1024
075
05
.. 0 0801
..
..
.
.
.
A=[]
(^05075) +
0625 0225
0225 1024
05
075
.. 0123
..
..
.
.
.
..
..
0625 0225
0225 1024
44 BACKGROUND MATERIAL