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.

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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  In this model, the random return on an investment z i is a linear function of some random factor f i and an idiosyncratic term  i.

 Abstracting away from the idiosyncratic risk  If the b i s of two assets are the same, then the a i s must be the same for an arbitrage free model.  Suppose we are interested in forming a portfolio of two assets with different b i s, b i  b j, b i  0, b j  0

 Computing the mean and variance of this portfolio yields

 Holding the variance of the portfolio equal to zero, we find

Multifactor Models:  Suppose that asset returns are generated by a two factor linear model:  A portfolio of these assets then yields

 Again to minimize systematic risk  If the portfolio is riskless, then it yields zero profit

 Given

 The matrix must be singular, or the first row must be a linear combination of the last two rows

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