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Capital Asset Pricing Model and the Arbitrage Pricing Theorem

Capital Asset Pricing Model and the Arbitrage Pricing Theorem. Lecture XXV. Deriving the Capital Asset Pricing Model. Using the results from the Expected Value-Standard Deviation frontier, we can derive the security market line (SML). Security Market Line (SML).

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Capital Asset Pricing Model and the Arbitrage Pricing Theorem

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  1. Capital Asset Pricing Model and the Arbitrage Pricing Theorem Lecture XXV

  2. Deriving the Capital Asset Pricing Model

  3. Using the results from the Expected Value-Standard Deviation frontier, we can derive the security market line (SML)

  4. Security Market Line (SML)

  5. Which is consistent with the standard CAPM relationship • Starting with a two-asset portfolio, we construct a portfolio using investment and asset .

  6. Next, we examine the risk/return relationship based on changes in the share of asset .

  7. Consider what happens as the share held in asset becomes small

  8. The risk/return relationship as the share in asset becomes small is then computed as

  9. This relationship then yields

  10. Empirical Tests of CAPM • The typical estimation procedure for empirically testing the CAPM is a two step model. • First, the annual returns are estimated as a function of the returns on the market portfolio:

  11. Using the estimated results from the first estimation, the SML is estimated across equations

  12. The “testable” implications of the CAPM are: • The intercept term 0 should be equal to zero. • Beta should be the only factor that explains the rate of return on a risky asset. • The relationship in beta should be linear. • The coefficient 1 should be equal to Rmt-Rft.

  13. Uses of CAPM • Risk Adjusted Discount Rate

  14. Certainty Equivalent Approach

  15. Arbitrage Pricing Theorem • Again, the concept is to show under what conditions a riskless, wealthless trade provides no expected rate of return. The basic construct for this argument is the arbitrage portfolio w. The arbitrage portfolio is defined as that set of purchases and sales that leaves wealth unchanged:

  16. The vector form of the factor model is • where R is the vector of returns on assets • E is the vector of expected returns on the assets • F is a matrix of factor loadings relating changes in the common factors with the fluctuations in asset returns •  is a vector of idiosyncratic risks .

  17. No cost portfolio No profit No Risk

  18. The algebraic consequence of this statement is that or the expected return on an asset is a linear function of the factor loadings. This result is identical to the CAPM results if there is a single factor.

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