The CAPM

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  {