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The Arbitrage Pricing Model

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

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The Arbitrage Pricing Model

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  1. The Arbitrage Pricing Model Lecture XXVI

  2. 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 zi is a linear function of some random factor fi and an idiosyncratic term i.

  3. Abstracting away from the idiosyncratic risk • 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

  4. Computing the mean and variance of this portfolio yields

  5. Holding the variance of the portfolio equal to zero, we find

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

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

  8. Given

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

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