Econ173A_topic6

Econ173A_topic6 - Ec 173A FINANCIAL MARKETS Foster UCSD LECTURE NOTES 21-Jul-08 TOPIC 6 ASSET EQUILIBRIUM PRICING MODELS A The Capital Asset

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Ec 173A – FINANCIAL MARKETS LECTURE NOTES Foster, UCSD 21-Jul-08 TOPIC 6 – ASSET EQUILIBRIUM PRICING MODELS A. The Capital Asset Pricing Model 1. Introduction: a) One question from MPT may bother you. 1) Asset X in the CML diagram is com- pletely dominated by market portfolio M and by other single assets. [Fig. 1] 2) Yet, by definition, X is included in M, so somebody owns it! Why? b) The reason asset X is in the market, yet in the interior of the opportunity set of efficient portfolios, is that the RELEVANT risk of a single asset is NOT σ(r x ). What we said earlier about asset dominance as an equilibra- ting process must now be revised. 1) We know that investors diversify portfolios between a riskless asset and a market portfolio M of risky assets. As E(U) maximizers, they are concerned only with over- all portfolio risk and return μ(r p ) and σ(r p ). And furthermore, since they choose indi- vidual portfolios along the CML, they are only concerned with μ(r m ) and σ(r m ). (This is what we called the “Separation Theorem.”) 2) Hence, when an investor is contemplating purchase of asset X for addition to an already diversified portfolio, the only thing that matters is what X will do to overall portfolio risk and return. This is only weakly related to σ(r x ). But if σ(r x ) isn't the relevant risk of asset X, then what is? c) The CAPM provides an answer. It was developed by William F. Sharpe in 1964 (Nobel Prize 1990). It adds 3 assumptions to those made by MPT: #6 There are no taxes, brokerage fees, or other market imperfections. #7 Total asset quantities are fixed, and all assets are marketable. Hence, we are discus- sing secondary market transactions, not new issues of securities. #8 Perfect competition -- all investors are small and take security prices as given -- they are price takers. (Not strictly true!) 2. Derivation of the CAPM: a) Starting with market portfolio M, create enlarged portfolio M+ with proportion δ in asset X and 1−δ in M. For the new portfolio: σ(r m ) σ(r x ) σ(r p ) M • Y ●X μ(r m ) μ(r x ) r f μ(r p ) Fig. 1 ) , ( ) 1 ( 2 ) ( ) 1 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( 2 2 2 2 2 m x m x m m s m r r r r r r r r σ δ μ δμ - + - + = - + = + +
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Ec 173A ASSET PRICING MODELS p. 2 b) To find the change in return and risk from adding X to M, take the following derivatives: c) Now the trick. 1) We know that in general equilibrium, supply = demand in all markets and excess demand = 0. In financial market equilibrium, excess demand for each asset = 0. 2) Now all of X WAS ALREADY INCLUDED in market portfolio M, so M M+. Trying to add δ% of X to M is excess demand for X. So we must evaluate the derivatives above at δ = 0, which yields the following: d) The extra return needed on the market portfolio M to compensate for the extra risk from having δ% of X already in it is: e) But all investors diversify by choosing a portfolio along the CML, where their portfolio risk premium = slope of CML = [μ(r
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This note was uploaded on 06/30/2009 for the course ECON 173A taught by Professor Foster during the Spring '09 term at UCSD.

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Econ173A_topic6 - Ec 173A FINANCIAL MARKETS Foster UCSD LECTURE NOTES 21-Jul-08 TOPIC 6 ASSET EQUILIBRIUM PRICING MODELS A The Capital Asset

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