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The Multifactor APT
Consider a two factor, two security case
Suppose that returns are generated by the following process
i
i
i
F
F
r
E
r
ε
β
~
~
~
)
(
~
2
2
1
1
+
+
+
=
One can form
"factor portfolios"
defined as portfolios with a beta (sensitivity) equal to 1 on
one factor (say factor 1) and zero to all other factors.
Let us denote a factor portfolio with a
beta of 1 with respect to factor 1 as FP
1
.
Similarly, refer to the other factor portfolio with a
beta of 1 with respect to the second factor as FP
2
.
Let the expected rates of return on these
factor portfolios be E(r
1
) and E(r
2
), respectively.
Expected return  beta relationship
Consider a welldiversified portfolio 'P' with a beta equal to
β
1
on the first factor and a beta
equal to
β
2
on the other factor.
What is the expected rate of return on such a portfolio under the
APT?
Answer:
To answer this question let us form a mimicking portfolio 'A' which has the same
factor sensitivities to the two factors in the economy as portfolio 'P'.
To do this, invest a
proportion '
β
1
' in factor portfolio FP
1
, '
β
2
' in FP
2
and a proportion (1
β
1

β
2
)
in the riskfree
asset.
Note that the expected return on portfolio 'A' is:
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This note was uploaded on 09/27/2010 for the course BUSINESS 6F:111 taught by Professor Tongyao during the Spring '09 term at University of Iowa.
 Spring '09
 TongYao
 Management

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