Lecture26-2004 - Lecture XXVI The Arbitrage Pricing Model...

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1 Lecture XXVI The Arbitrage Pricing Model I. A Single Factor Model A. Abstracting away from the specific form of the CAPM model, we posit a single factor model written as 1 K ii i k ki k za b f ε = = ++ ± ± ± 1. In this model, the random return on an investment i z ± is a linear function of some random factor i f ± and an idiosyncratic term i ± . 2. This factor specification implies () ( ) ( ) ( ) 22 2 2 0 1 iki ji kk l i k EE f E E f E f f Es S Ef εε == = = = =< = ±± ± ± ± ± ± ± B. Abstracting away from the idiosyncratic risk i i zab f =+ ± ± 1. If the s i b of two assets are the same, then the s i a must be the same for an arbitrage free model. 2. Suppose we are interested in forming a portfolio of two assets with different s i b , ij bb , 0 i b , 0 j b ( )( ) 1 j j ii j j j j j j zw ab f wa b f wa wb f a wa b f wb f wa a a wb b b f + + + −+−  =− + +− +  3. Computing the mean and variance of this portfolio yields [ ] [] {} 2 2 2 2 2 2 j j j j j j j j j j Ez wa a a Vz E wa a a a Ewa a a E wa a a Ew
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Lecture26-2004 - Lecture XXVI The Arbitrage Pricing Model...

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