The Capital Asset Pricing Model CAPM and its Extensions

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Lecture 4

• The Capital Asset Pricing Model (CAPM) and its
  Extensions

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Contribution of a Stock to the Portfolio Risk

• A stocks contribution to the risk of a
  diversified portfolio does not depend on its own
  variance.
• Its contribution to portfolio risk is through the
  covariance of the stock with the other stocks in
  the portfolio.
• Decomposition of the market returns variance
  into the weighted average of covariance of
  individual assets return with the markets
  return
• Covariance is a linear operator

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What is the Beta?

• Beta is a scaled covariance of an assets
  return with the market portfolios return.
• The average of betas for all assets is 1.
• Beta can also be linked with the co-efficient of
  correlation
• Beta standard deviation of the assets return /
  standard deviation of market return, multiplied
  by the coefficient of correlation between the
  two.

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Intuitive Explanation of the Capital Asset
Pricing Model (CAPM)

• Investors will only want to hold a security in
  their portfolio if it provides a reasonable
  amount of reward (i.e., the marginal excess
  return) in return for the risk it adds to the
  portfolio.
• This risk is characterized by the beta.
• Well term this risk the systematic risk or
  market risk.
• The ratio of marginal excess return to marginal
  risk must be the same for all assets in
  equilibrium.

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An Example How does the market achieve the
equilibrium?

• Asset A has an expected return of 12 and a beta
  of 1.40. Asset B has an expected return of 8
  and a beta of 0.80. Are these two assets valued
  correctly relative to each other if the risk-free
  rate is 5?
• NO! The reward-to-systematic risk ratio is 0.05
  for A, 0.0375 for B.
• Please note this reward-to-systematic risk
  ratio is different from the Sharpe ratio.
• Asset B offers insufficient return for its level
  of systematic risk, relative to A. Bs price is
  too high, therefore it is overvalued (or A is
  undervalued).
• What should investors do?

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Capital Asset Pricing Model

• It has to be
• Expected return of stock Risk-free rate Beta
  x Risk premium of market portfolio.

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Assumptions needed for the CAPM

• 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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Security Market Line (SML)

• SML is the line depicting the relationship
  between the assets expected return and its beta.
  
• The slope of the SML, E(rm) rf, measures the
  price of risk.
• If you graph the actual average return-beta
  combination of assets, most of them deviate from
  the SML. The difference is called alpha.

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Security Market Line
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Disequilibrium Example

• Suppose a security with a ? of 1.25 is offering
  expected return of 15
• According to SML, it should be 13
• The (Jensens) alpha of this security is 2.
  Thus, this security is under-priced offering
  too high of an expected rate of return for its
  level of risk

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Disequilibrium Example
a 2
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The Usefulness of Alpha

• An asset with a negative alpha calculated based
  on past performance means this asset
  underperformed in the market.
• Stocks with a forecasted positive alpha in the
  near future should be bought, if you trust the
  forecast.
• On average, an assets return/beta relationship
  should follow the SML in the long term.
• This may imply mean-reverting observations.

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Portfolios Beta

• A portfolios beta is simply the weighted-average
  of its assets betas.
• This is because covariance is a linear operator.

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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 with them
• Returns on individual assets can be expressed as
  linear combinations of efficient portfolios

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Blacks Zero Beta Model Formulation

• P and Q are minimum-variance portfolios. For an
  arbitrary asset i,

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Illustration of Efficient Portfolios and Zero
Companions
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Zero Beta Market Model

• Choose M as P, Z(M) as Q, then
• This is the CAPM with E(rz (m)) replacing rf.
• Portfolio Z(m) has a beta of zero.

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Zero Beta Model Applications

• (No riskfree borrowing) The mutual funds
  theorem will no longer be valid. Less risk
  averse people will invest in risk free asset and
  another efficient portfolio which is not the
  tangent portfolio. The CAPM should be changed
  accordingly.
• (Differential borrowing and lending rates) We
  will have a kinked CAL.
• Implications Not everyone holds a combination of
  riskfree and the market portfolio anymore.

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Liquidity A Recap

• Liquidity how ease to trade a security. If
  transaction costs are high, then liquidity will
  be low.
• Trading costs
• Bid-ask spread ? This is considered the major
  part of trading costs which will vary with the
  size of the firm. Stocks of smaller firms have
  bigger bid-ask spreads
• Commissions, taxes
• Implicit market impact costs

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CAPM and Liquidity

• Illiquidity Premium If a security is illiquid,
  then investors demand a higher expected rate of
  return for this security.
• Liquidity is related to the length of holding
  period.
• Liquidity and the size effect.
• 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, a concave
function. (f gt 0, f lt 0). ci is the liquidation
cost, which is partly determined by the bid-ask
spread of the security.
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Index Model and Factor Model

• We are not able to construct a true market
  portfolio in reality. Instead, we use a stock
  market index as a substitute for the market
  portfolio.
• Or, we could say that all securities returns are
  affected by one single macro factor.
• Employing the index model makes it easier for
  security analysts to do the analysis
• Reduce drastically the number of parameters
  needed to be estimated

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Liquidity and Average Returns
Average monthly return()
Bid-ask spread ()
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Single Factor Model

• Assumption a broad market index is the common
  factor
• ri E(ri) ßiF ei
• ßi sensitivity of an assets return to the
  factor
• F some macro factor in this case F is
  unanticipated movement
• F is commonly related to security returns
• Multi-factor models will be addressed in the
  future.

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Single Index Model
Risk Prem
Market Risk Prem
or Index Risk Prem
the stocks expected return if the markets
excess return is zero
ai is zero under the CAPM.
a
i
(rm - rf) 0
ßi(rm - rf) the component of return due to
movements in the market index b has the same
expression as in the CAPM.
ei firm specific component, not due to market
movements. This is an important assumption.
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Using the Text Example from Table 10-1
Excess Mkt. Ret.
Excess GM Ret.
Jan. Feb. . . Dec Mean Std Dev
5.41 -3.44 . . 2.43 -.60 4.97
7.24 .93 . . 3.90 1.75 3.32
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Regression Results
a
rGM - rf ß(rm - rf)
a
ß
-2.590 (1.547)
1.1357 (0.309)
Estimated coefficient Std error of
estimate Variance of residuals 12.601 Std dev
of residuals 3.550 R2 0.575
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Beta Coefficients for Selected Australian
Companies (as of 2/8/2005, FinAnalysis)
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Components of Risk

• Market or systematic risk risk related to the
  market portfolio
• However, it is not possible to get the beta under
  this definition, so the beta is obtained under
  the index model.
• Unsystematic or firm specific risk risk not
  related to the market portfolio or market index
• Total risk Systematic Unsystematic

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Measuring Components of Risk

• ?i2 ?i2 ?m2 ?2(ei)
• where
• ?i2 total variance
• ?i2 ?m2 systematic variance
• ?2(ei) unsystematic variance

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Examining Percentage of Variance

• Running a market model regression first, then we
  can decompose the total variance of a stocks
  return.
• Total Risk Systematic Risk Unsystematic Risk
• Systematic Risk/Total Risk ?2 R2
• ßi2 ? m2 / ?2 ?2 ? of systematic
  risk
• ?i2 ?m2 /( ?i2 ?m2 ?2(ei)) ?2

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Risk Reduction with Diversification
St. Deviation
Unique Risk s2(eP)s2(e) / n
bP2sM2
Market Risk
Number of Securities