The Arbitrage Pricing Model

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The Arbitrage Pricing Model

• Lecture XXVI

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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.

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• 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

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• Computing the mean and variance of this portfolio
  yields

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

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Multifactor Models

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

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

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• Given

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

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