The Capital Asset Pricing Model

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The Capital Asset Pricing Model

• Chapter 9

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Capital Asset Pricing Model (CAPM)

• It is the equilibrium model that underlies all
  modern financial theory.
• Derived using principles of diversification with
  simplified assumptions.
• Markowitz, Sharpe, Lintner and Mossin are
  researchers credited with its development.
• William Sharpe won the Nobel Price in economics
  for his work on CAPM.

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Assumptions

• Individual investors are price takers.
• Single-period investment horizon.
• Investments are limited to traded financial
  assets.
• No taxes and transaction costs.
• Information is costless and available to all
  investors.
• Investors are rational mean-variance optimizers.
• There are homogeneous expectations.

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Resulting Equilibrium Conditions

• All investors will hold the same portfolio for
  risky assets market portfolio (recall from
  Chapter 7).
• Market portfolio contains all securities and the
  proportion of each security is its market value
  as a percentage of total market value.

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Resulting Equilibrium Conditions (contd)

• Risk premium on the market depends on the average
  risk aversion of all market participants.
• Risk premium on an individual security is a
  function of its covariance with the market.

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The Premium of the Market Portfolio

• Lets assume that there are 3 investors, with
  risk aversion parameters A1, A2, and A3. Each
  has 1 to invest. Recall that in the capital
  allocation analysis, the optimal weight each
  investor assigns to the risky market portfolio
  should be

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The Premium of the Market Portfolio (contd)

• The total money invested in the market portfolio
  therefore is

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The Premium of the Market Portfolio (contd)

• What should be the total investment in the market
  portfolio?
• Lets take a look at a simple example. Let
  A11.5, A22, A33, and E(rm)-rf 9, sm20.

• Can this be an equilibrium?
• Risk free asset
• Market portfolio

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The Premium of the Market Portfolio (contd)

• What if E(rm)-rf 6, sm20?

• Can this be an equilibrium?
• Risk free asset
• Market portfolio

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The Premium of the Market Portfolio (contd)

• What if E(rm)-rf 8, sm20.

• Is this an equilibrium?
• What is the total investment in the market
  portfolio?

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The Premium of the Market Portfolio (contd)

• Therefore, total money invested in the market
  portfolio has to be 3.

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The Premium of the Market Portfolio (contd)

• If

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The Premium of the Market Portfolio and Risk
Aversion-In-class Exercise

• Historical market risk premium (proxied by the
  SP 500 index) is 8.2, and the standard
  deviation of the market portfolio is 20.6.
  Based on these statistics, what is the average
  coefficient of risk aversion?

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The Premium of the Market Portfolio (contd)

• Risk premium (E(rM) rf ) on the market depends
  on the (harmonic) average risk aversion of all
  market participants.

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Capital Market Line
E(r)
CML
M
E(rM)
rf
?
?m
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Slope and Market Risk Premium

• M Market portfolio rf Risk free
  rate E(rM) - rf Market risk premium E(rM) -
  rf
• Market price of risk
• Slope of the CML

?
M
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Return and Risk For Individual Securities

• The risk premium on individual securities is a
  function of the individual securitys
  contribution to the risk of the market portfolio.
• An individual securitys risk premium is a
  function of the covariance of returns with the
  assets that make up the market portfolio.

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Return and Risk For Individual Securities
Simplified Derivation

• For simplicity, lets assume that there are only
  three assets in the market, then

• Therefore, the marginal contribution of asset 1
  to the expected risk premium of the market
  portfolio is

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Return and Risk For Individual Securities
Simplified Derivation (Contd)

• Now, lets look at the variance of the market
  portfolio

• The marginal contribution of asset 1 to the risk
  (variance) of the market portfolio is

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Return and Risk For Individual Securities
Simplified Derivation (Contd)

• The reward-to-risk ratio for asset 1 therefore is

• Now, recall that the market reward to risk ratio
  is

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Return and Risk For Individual Securities
Simplified Derivation (Contd)

• In equilibrium, the reward to risk ratio should
  be the same for all the assets.
• Why?
• Therefore,

• or,

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Return and Risk For Individual Securities
Simplified Derivation (Contd)

• We call the following ratio as the beta for asset
  1.

• The equation for the expected rate of return can
  be simplified as

• We did it! This is the CAPM model you saw and
  used in other finance classes.

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Return and Risk For Individual Securities

• More generally,

• The equation for the expected rate of return for
  individual asset i can be simplified as

• The graph that describes this equation is called
  the Security Market Line (SML).

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Return and Risk For Individual Securities

• Beta of security i measures how the return of i
  moves with the return of the market. In other
  words, it is a measure of the systematic risk.
• Only systematic risk matters in determining the
  equilibrium expected return.
• Unsystematic risk affects only a single security
  or a limited number of securities.
• Systematic risk affects the entire market.

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Security Market Line
E(r)
SML
E(rM)
rf
b
bM 1.0
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SML Relationships

• ???????????????????? Cov(ri,rm) / ?M2
• Slope SML E(rM) - rf
• market risk premium
• Betam Cov (rM,rM) / sM2
• sM2 / sM2 1

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Examples of SML

• E(rm) - rf .08 rf .03
• What is the expected return for a security with a
  beta of 0?
• What is the expected return for a security with a
  beta of 0.6?
• What is the expected return for a security with a
  beta of 1.25?

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Graph of Sample Calculations
E(r)
SML
Rx13
Slope0.08
Rm11
Ry7.8
3
b
1.0
1.25 bx
.6 by
0 bf
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Alpha and Disequilibrium

• The difference between the actual expected rate
  of return and that dictated by the SML is often
  called as alpha.

• What should alpha be if a security is fairly
  priced according to CAPM?

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Disequilibrium Example

• E(rm) - rf .08 rf .03
• Suppose a security with a ? of 1.25 is offering
  expected return of 15.
• According to SML, it should be 13.
• What is the alpha for this security?
• Is this security overpriced or underpriced, why?

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Disequilibrium Example
E(r)
SML
15
Rm11
rf3
b
1.25
1.0
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Alpha and Security Price

• What would happen if alpha is positive?
• What would happen if alpha is negative?
• When are securities overpriced or underpriced?

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Extensions of CAPM

• Blacks Zero Beta Model
• CAPM and Liquidity

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Extensions of CAPM

• Blacks Zero Beta Model
• Absence of a risk-free asset
• Combinations of portfolios on the efficient
  frontier are efficient.
• All frontier portfolios have companion portfolios
  that are uncorrelated.
• Returns on individual assets can be expressed as
  linear combinations of efficient portfolios.

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Zero Beta Market Model
CAPM with E(rz (m)) replacing rf
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CAPM Liquidity

• Liquidity
• Illiquidity Premium
• If there are two assets with identical expected
  rate of returns and beta, but one costs more to
  trade, which asset do you prefer?
• Research supports a premium for illiquidity.
• Amihud and Mendelson

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CAPM with a Liquidity Premium
f (ci) liquidity premium for security i f (ci)
increases at a decreasing rate
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Liquidity and Average Returns
Average monthly return()
Bid-ask spread ()
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Assignments

• Chapter 9 problems
• 4, 6-12, 16-17, 21-27, 30