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The CAPM

The CAPM. An extension of the Markowitz portfolio selection model Idea of equilibrium in the capital market Assumptions of the CAPM Resulting equilibrium conditions The Security Market Line (SML). A. Idea of equilibrium in the capital market. E(r i ) = r F +  i [E(r M ) - r F ]

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The CAPM

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  1. The CAPM An extension of the Markowitz portfolio selection model • Idea of equilibrium in the capital market • Assumptions of the CAPM • Resulting equilibrium conditions • The Security Market Line (SML)

  2. A. Idea of equilibrium in the capital market E(ri) = rF + i[E(rM) - rF] • The CAPM relationship is an equilibrium condition, i.e., it holds when the capital market is in equilibrium • Parallel work by Sharpe (1964) and Lintner (1965)

  3. Idea of equilibrium(cont’d) • Equilibrium: supply = demand • Claim: in equilibrium, the tangency/optimal risky portfolio is the “market” portfolio • What does “market” mean? • Why are all risky assets included in the tangency portfolio in equilibrium?

  4. Idea of equilibrium(cont’d) • What if one is not included? Think about an important implication of the Markowitz model • Imagine a world in which investors all face the same universe of assets • Each investor holds the tangency portfolio on the efficient frontier

  5. E(r) CML M E(rM) rF  M Idea of equilibrium:Capital Market Line • In equilibrium, the CAL becomes the CML – the Capital Market Line

  6. Idea of equilibrium(cont’d) = Slope of the CML E(rM) – rF = Market risk premium • What is the market risk premium equal to in equilibrium?

  7. Idea of equilibrium:market risk premium • Recall that: • In equilibrium, P is replaced by M • What is the average holding (among investors) of the risk-free asset, F? • What is the average holding of the market portfolio, M? That is, what is the average y?

  8. B. Assumptions of the CAPM • One of the most important theories in modern finance • Derived using principle of diversification • Extension of the Markowitz model • Think of as an implication of the Markowitz model in capital market equilibrium

  9. Seven assumptions • Individual investors are price takers • Market structure: perfect competition • Single-period investment horizon • Multi-period: same relationship • Investments are limited to traded, perfectly divisible assets • No taxes, no transaction costs

  10. Assumptions (cont’d) • Information is costless and available to all investors (no information cost) • Investors are rational mean-variance optimizers • Investors’ expectations are homogeneous • For each asset, all investors expect the same E(r) and 

  11. C. Resulting equilibrium conditions • All investors hold the same portfolio of risky assets • Result from the Markowitz model • Use as starting point here for the CAPM • In equilibrium, this tangency portfolio is called the “market” portfolio (M) , as it contains all assets in the universe • The proportion/weight of each asset in M is its market value as a percentage of total market value

  12. Resulting equilibrium conditions (cont’d) • Risk premium on M depends on the average risk aversion of all market participant, and the variance of M • Risk premium on an individual security is a function of its covariance with the market • Key insight • Risk of a security  its standard deviation (total risk) because of diversification

  13. D. Security Market Line E(r) SML E(rM) rF ß ß = 1.0 M

  14. SML Relationships SML equation: E(ri) = rF + [E(rM) - rF] i Hence, slope of the SML = E(rM) - rF = market risk premium Since i = Cov(ri,rM) / M2 therefore M = Cov (rM,rM) / sM2 = sM2 / sM2 = 1

  15. SML: Numerical Example E(rM) - rF = .08 rF = .03 a) x = 1.25 E(rx) = .03 + 1.25(.08) = .13 or 13% b) y = .6 E(ry) = .03 + .6(.08) = .078 or 7.8%

  16. E(r) SML Rx=13% .08 RM=11% Ry=7.8% 3% ß .6 1.0 1.25 ß ß ß y m x Graph of Numerical Example

  17. E(r) SML 15% Rm=11% rf=3% ß 1.25 1.0 Example of underpricing  {

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