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Lecture26-2004

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

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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 i i ik k i k z a 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 ε ± . 2. This factor specification implies ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 0 1 i k i j i k k l i i i k E E f E E f E f f E s S E f ε ε ε ε ε = = = = = = < = ± ± ± ± ± ± ± ± ± ± B. Abstracting away from the idiosyncratic risk i i i i z a b 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 , i j b b , 0 i b , 0 j b ( ) ( ) ( ) ( ) ( ) 1 i i j j i i j j j j i j j i j j z w a b f w a b f wa wb f a wa b f wb f w a a a w b b b f = + + + = + + + = + + + 3. Computing the mean and variance of this portfolio yields [ ] ( ) [ ] ( ) ( ) { } ( ) { } ( ) { } ( ) ( ) { } ( ) { } ( ) { } ( ) 2 2 2 2 2 2 i j

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