The CAPM

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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)

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A. Idea of equilibrium in the capital market

• E(ri) rF ?iE(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)

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Idea of equilibrium(contd)

• 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?

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Idea of equilibrium(contd)

• 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

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Idea of equilibriumCapital Market Line

• In equilibrium, the CAL becomes the CML the
  Capital Market Line

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Idea of equilibrium(contd)

• Slope of the CML
• E(rM) rF Market risk premium
• What is the market risk premium equal to in
  equilibrium?

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Idea of equilibriummarket 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?

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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

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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

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Assumptions (contd)

• 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 ?

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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

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Resulting equilibrium conditions (contd)

• 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

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

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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

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Graph of Numerical Example
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Example of underpricing
?