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# Apt - The Multifactor APT Consider a two factor two...

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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 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 well-diversified 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 risk-free asset.

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