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

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Unformatted text preview: The Arbitrage Pricing Model Lecture XXVI A Single Factor Model Abstracting away from the specific form of the CAPM model, we posit a single factor model written as K % % % z = a + b f + i i k =1 ik k i In this model, the random return on an investment zi is a linear function of some random factor fi and an idiosyncratic term i. % % % % %% % % E ( i ) = E f k = E ( i j ) = E i f k = E f k fl = 0 %i2 ) = si2 < Si2 E( %2 =1 E fk ( ) ( ) ( ) ( ) Abstracting away from the idiosyncratic risk % % zi = ai + bi f i If the bis of two assets are the same, then the ais must be the same for an arbitrage free model. Suppose we are interested in forming a portfolio of two assets with different bis, bi bj , bi 0, bj 0 z = w ( ai + bi f ) + ( 1 - w ) ( a j + b j f ) = wai + wbi f + a j - wa j + b j f - wb j f = ( ai - a j ) + a j ( bi - b j ) + b j w + w f Computing the mean and variance of this portfolio yields E [ z ] = w ( ai - a j ) + a j V [ z ] = E ( ai - a j ) + a j ( bi - b j ) + b j w + w f - w ( ai - a j ) + a j i j j 2 { } 2 { } = E { w ( a - a ) + a } + E { ( a - a ) + a w ( b - b ) + b } + w f E { ( b - b ) + b } -{ w( a - a ) + a } w f 2 i j j i j j 2 2 i j j i j j = ( bi - b j ) + b j w 2 Holding the variance of the portfolio equal to zero, we ( bi - b j ) + b j = 0 w 2 find w ( bi - b j ) + b j = 0 w = * bj b j - bi bj bj z= ( ( b - b j ) + b j f ai - a j ) + a j + j - bi i b j - bi b = i b j ( ai - a j ) b j - bi j + a j = R the riskfree rate j (a -a ) = R-a ( b -b ) b j i j R - ai = bi zi (a -a ) b = R+ ( b -b ) i j j i (a -a ) b R-z = ( b -b ) (a -a ) b = z R- ( b -b ) i j i j i i j j j i R - zi ( ai - a j ) = bj ( b j - bi ) j i j 1 (a -a ) = ( b -b ) i j j i zi = R + 1bi zi = 0 + 1bi Multifactor Models: Suppose that asset returns are generated by a two factor linear model: % % % zi = ai + bi f1 + ci f 2 A portfolio of these assets then yields % wi zi = i ai + i bi f1 + i ci f 2 w w % w % i i i i Again to minimize systematic risk w w b = c i i i i i i =0 If the portfolio is riskless, then it yields zero profit R= i wi ai wi ai - R = 0 i Given w i i =1 iR = R w i i a 0 1 - R a2 - R a3 - R w1 wi ( ai - R ) = 0 b1 b2 b3 w2 = 0 c c2 c3 w3 0 1 The matrix a 1 - R a2 - R a3 - R b2 b3 b1 c c2 c3 1 must be singular, or the first row must be a linear combination of the last two rows a 1 - R a2 - R a3 - R b2 b3 R1 + 1 R2 + 2 R3 = 0 b1 c c2 c3 1 zi - R = ai - R = 1bi + 2 ci i or zi = 0 + 1bi + 2 ci i ...
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This note was uploaded on 07/15/2011 for the course AEB 6182 taught by Professor Weldon during the Fall '08 term at University of Florida.

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