Final_1.pdf

(3.12)

or

ep=hX (3.13)

That is, the exposure of a portfolio is given by the matrix product of the
holdings vector and the exposure matrix. Using the value in Equation 3.13
for the factor exposure of the portfolio, the common factor variance is given
by substituting the calculated exposure value:

(3.14)

Performing the substitutions and applying the matrix identity on the trans-
pose of a matrix product, we have

(3.15)

Let us now tackle the specific risk component. The specific risk is calculated
as the variance of the specific return. Now, the specific return on the port-
folio is the sum of the specific returns of the assets in the portfolio weighted
by the share of the security in the portfolio holdings. Also, note that by
model assumptions the specific returns of the securities are uncorrelated
with each other. If the returns are uncorrelated with each other, then the
overall variance is a weighted sum of the individual specific variances. In the
preceding two-security example, the specific variance is given as

(3.16)

Writing this out in matrix form, we have

(3.17)

or

(3.18)

Adding it all together we are now able to put down an expression for the
total risk in a portfolio based on the portfolio holdings and the parameters
of the APT model. The risk/portfolio variance is therefore given as

σspecific^2 =hh∆T

σspecific^2121

2

0

0

=[]

()

()

































hh

r

r

h

h

eA

eB

var

var

,

,

σspecific^2 =hr hr 12 var()eA,,+^22 var()eB

σcf^2 =hXVX hTT

σcf^2 =eVep pT

ehhpABAA

BB

=[]

















ββ

ββ

,,

,,

12

12

46 BACKGROUND MATERIAL