The Capital Asset Pricing Model CAPM

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• The Capital Asset Pricing Model (CAPM)

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Understand the implications of capital market
theory and the CAPM to compute security risk
premiums

• What have we done this far?
• We have been concerned with how an individual or
  institution, acting on a set of estimates, could
  select an optimum portfolio.
• If investors act as we think, we should be able
  to draw on the analysis to determine how the
  aggregate of investors will behave, and more
  importantly how prices, and returns at which
  markets will clear are set
• The development of general equilibrium models
  allows us to determine the relevant measure of
  risk for any asset and the relationship between
  expected return and risk for any asset when the
  markets are in equilibrium

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Capital Asset Pricing Theory

• What is capital asset pricing theory?
• It is the theory behind the pricing of assets
  which takes into account the risk and return
  characteristics of the asset and the market
• What is the CAPM, i.e. the Capital Asset Pricing
  Model?
• It is an equilibrium model (i.e., a constant
  state model) that underlies all modern financial
  theory
• It provides a precise prediction between the
  relationship between the risk of an asset and its
  expected return when the market is in equilibrium
• With this model, we can identify mis-pricing of
  securities (in the long-run)

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CAPM (continued)

• Why is it important?
• It provides a benchmark rate of return for
  evaluating possible investments, and identifying
  potential mis-pricing of investments
• For example, an analyst might want to know
  whether the expected return she forecast is more
  or less than its fair market return.
• It helps us make an educated guess as to the
  expected return on assets that have not yet been
  traded in the marketplace
• For example, how do we price an initial public
  offering?

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CAPM (continued)

• How was it derived?
• Derived using principles of diversification with
  very simplified (i.e. somewhat unrealistic)
  assumptions
• Does it work, i.e. withstand empirical tests in
  real life?
• Not totally
• But it does offer insights that are important and
  its accuracy may be sufficient for some
  applications
• Do we use it?
• Yes, but with knowledge of its limitations

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CAPM Assumptions

• CAPM by William Sharpe (1964), John Lintner
  (1965), and Jan Mossin (1966)
• What does the model assume (some are
  unrealistic)?
• Individual investors are price takers (cannot
  affect prices)
• Single-period investment horizon (an its
  identical for all)
• Investments are limited to traded financial
  assets
• No taxes, and no transaction costs (costless
  trading)
• Information is costless and available to all
  investors
• Investors are rational mean-variance optimizers
• Investors analyze information in the same way,
  and have the same view, i.e., homogeneous
  expectations

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Resulting Equilibrium Conditions

• Based on the previous assumptions
• All investors will hold the same portfolio for
  risky assets the market portfolio (M)
• The market portfolio (M) contains all securities
  and the proportion of each security is its market
  value as a percentage of total market value. M
  will be on the efficient frontier
• The risk premium on the market depends on the
  average risk aversion of all market participants
• The risk premium on an individual security is a
  function of its covariance (correlation and ss
  sm) with the market

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The Risk Premium of the Market Portfolio

• Recall y ( a proportion allocated to the risky
  optimal portfolio M)
• y E (rp) - rf / 0.01 A ?p 2
• In the simple CAPM, net borrowing lending
  across all investors must be zero ? y 1
• with y 1, E (rp) - rf 0.01 A ?p 2
  the risk premium on the market portfolio depends
  on average degree of risk aversion and ?p 2

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Capital Market Line
E(r)
M Market portfolio rf Risk free rate E(rM) -
rf Market risk premium E(rM) - rf/sM Market
price of risk
CML
M
E(rM)
rf
The efficient frontier without lending or
borrowing
s
sm
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Expected Return and Risk of Individual Securities

• What does this imply?
• The risk premium on individual securities is a
  function of the individual securitys
  contribution to the risk of the market portfolio
• 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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CAPM Key Thoughts

• Key statements
• Portfolio risk is what matters to investors, and
  portfolio risk is what governs the risk premiums
  they demand
• Non-systematic, or diversifiable risk can be
  reduced through diversification. Investors need
  to be compensated for bearing only systematic
  risk (cannot be diversified away)
• The contribution of a security to the risk of a
  portfolio depends only on its systematic risk, as
  measured by beta. So the risk premium of the
  asset is proportional to its beta.

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Expected Return Beta Relationship

• Expected return - beta relationship of CAPM
• E(rM) - rf E(rs)
  - rf
• 1.0
  bs
• In other words, the expected rate of return of an
  asset exceeds the risk-free rate by a risk
  premium equal to the assets systematic risk (its
  beta) times the risk premium of the market
  portfolio. This leads to the familiar
  re-arrangement of terms to give (memorize this)
• E(rs) rf bs E(rM) - rf

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The CAPM

• The expected return-beta relationship
• E(ri) rf ?i E(rM) - rf , where ?i cov(ri,
  rM) /?M2.
• ?M 1 ?i gt 1 aggressive, ?i lt 1
  defensive
• The beta of a stock measures the stocks
  contribution to the variance of the market
  portfolio measures only systematic risk
• CAPM the securitys risk premium is directly
  proportional to both the beta the risk premium
  of the mkt port.

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The Security Market Line

• Notice that instead of using standard deviation,
  the SML uses Beta
• SML Relationships
• b COV(ri,rm) / sm2
• Slope SML E(rm) rf market risk
  
• premium

E(r)
SML
E(rM)
rf
SML rf bE(rm) - rf
ß
ß
1.0
M
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CML and SML

• Beta and Standard Deviation
• Total risk of a share Market risk of the share
• Specific Risk
• Total risk of a portfolio Market risk of the
  portfolio
• Specific Risk(negligible)

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Example SML Calculations

• Put the following data on the SML. Are they in
  equilibrium?
• Market data E(rm) - rf .08 rf .03
• Asset data bx 1.25 by .60
• Calculations
• bx 1.25 so E(r) on x
• E(rx) .03 1.25(.08) .13 or 13
• by .60 so E(r) on y
• E(ry) .03 .6(.08) .078 or 7.8

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Graph of Sample Calculations
E(r)
SML
Rx13
.08
Rm11
Ry7.8
They are in equilibrium
3
ß
1.0
1.25
.6
ß
ß
ß
m
y
x
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Disequilibrium Example

• Suppose a security with a beta of 1.25 is
  offering expected return of 15
• According to SML, it should be 13
• Under priced offering too high of a rate of
  return for its level of risk. Investors
  therefore would
• Buy the security, which would increase demand,
  which would increase the price, which would
  decrease the return until it came back into line.
• fairly priced assets plot on the SML
• under-priced assets plot above the SML
• over-priced assets plot below the SML

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Disequilibrium Example
E(r)
The return is above the SML, so you would buy it
SML
15
As more people bought the security, it would push
the price up, which would bring the return down
to the line.
Rm11
rf3
ß
1.0
1.25
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CAPM and Index Models

• CAPM Problems
• It relies on a theoretical market portfolio which
  includes all assets
• It deals with expected returns
• To get away from these problems and make it
  testable, we change it and use an Index model
  which
• Uses an actual index, i.e. the SP 500 for
  measurement
• Uses realized, not expected returns
• Now the Index model is testable

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The Index Model

• With the Index model, we can
• Specify a way to measure the factor that affects
  returns (the return of the Index)
• Separate the rate of return on a security into
  its macro (systematic) and micro (firm-specific)
  components
• Components
• ? excess return if market factor is zero
• ßiRm component of returns due to movements in
  the overall market
• ei component attributable to company specific
  events
• Ri a i ßiRm ei
• (Notice the similarity to the Single Index model
  discussed earlier)

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Alpha

• The difference between the fair and actually
  expected rates of return on a stock.
• SML E(ri) rf ?i E(rM) - rf
• E(ri) - rf ?i E(rM) - rf
• Ex. ? 1.2 rf 5 E(rM) - rf 8 E(ri)
  16
• SML E(ri) 5 1.2 (8) 14.6 fair return
  
• ? 16 - 14.6 1.4 gt 0.

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Does the CAPM hold?

• There is much evidence that supports the CAPM
• There is also evidence that does not support the
  CAPM
• Is the CAPM useful?
• Yes. Return and risk are linearly related for
  securities and portfolios over long periods of
  time
• Yes. Investors are compensated for taking on
  added market risk, but not diversifiable risk
• Perhaps instead of determining whether the CAPM
  is true or not, we might ask Are there better
  models?

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Boeings Historical Beta
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Problem CAPM

• Suppose the risk premium on the market portfolio
  is 9, and we estimate the beta of Dell as bs
  1.3. The risk premium predicted for the stock is
  therefore 1.3 times the market risk premium of 9
  or 11.7. The expected return on Dell is the
  risk-free rate plus the risk premium. For
  example, if the T-bill rate were 5m the expected
  return of Dell would be 51.39 16.7.
• a. If the estimate of the beta of Dell were only
  1.2, what would be Dells required risk premium?
• b. If the market risk premium were only 8 and
  Dells beta was 1.3, what would be Dells risk
  premium?

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Answer

• a. If Dells beta was 1.2 the required risk
  premium would be (remember the risk premium is
  the expected return less the risk-free rate)
• E(rs) rf bs E(rM) - rf or the expected
  return on Dell 5 1.2 (9) 15.8
• Dells risk premium (over the risk free rate)
  
• 15.8 - 5 10.8
• b. If the market risk premium was 8
• E(rs) rf bs E(rM) - rf
• E(r) of Dell 5 1.3 (8) 15.4
• Dells new risk premium is 15.4 5 10.4

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Using the Markowitz model to do Active Portfolio
Management

• The Markowitz portfolio selection model requires
• E(ri) N
• var(ri) N ? need (N2
  3N)/2 estimates
• cov(ri, rj) (N2 - N)/2
• Ex. If N 100, we need 10,300/2 5,150
  estimates.
• 500,
  125,750
• Another difficulty in applying the Markowitz
  model to portfolio optimization is that errors in
  the assessment or estimation of correlation
  coefficients can lead to nonsensical results.

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Using the Markowitz model to do Active Portfolio
Management

• Moreover, using seemingly reasonable estimates of
  expected returns and covariances can lead to
  unreasonable portfolio weights.
• Small errors in mean or covariance estimates lead
  to unreasonable weights
• A remedy for both of these problems is to use (1)
  a single factor model to calculate asset
  covariances, (2) and use the CAPM to determine
  what market expectations must be, and then
  combine your views with the CAPM-derived
  estimates to get portfolio weights.
• The key input we will need for both of these is
  the set of asset ßs. So, first, we must consider
  the problem of estimating ßs.

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The Single Factor Model

• By the properties of regression, the residuals
  have a mean of zero, and are uncorrelated with
  the returns on the SP 500.
• Thus, the return on Microsoft is decomposed into
• A constant term.
• Movements in the SP 500 index return.
• Movements in a component unrelated to market
  movements.

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Specification of a Single Index Model

• Is this a good model?
• A model is of little use unless we can test a way
  to measure the factor we say affects security
  returns. It is not testable in its above form.
  To make it testable, we
• Use the rate of return on a broad index of
  securities, such as the SP 500, for a proxy. So
  now ßiM becomes ßiRm, or M is equal to the return
  on the market
• The expected holding period return E(Ri) becomes
  ai, because that is the stocks excess return
  assuming the markets excess return is zero
• So the new testable model becomes
• RiaißiRmei or substituting (ri-rf)
  aißi(rm-rf) ei

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Active Portfolio Management

• An alternative approach is to accept market
  opinion and simply hold the market portfolio.
• Calculate ßs for the securities we plan to hold
• Using these ßs, calculate E(ri)s and ?i,js by
  using the single index model (and the CAPM)
• Using these estimates and the Markowitz portfolio
  optimization tools, determine our optimal
  portfolio weighs

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A single Index Model

• Use the mkt index return to proxy for F F ?
  rM
• ri - rf ?i ?i rM - rf ei, or
• Ri ?i ?i RM ei, where Ri ri - rf,
  RM rM rf
• Each security has two sources of risk market
  risk and firm-specific risk
• For the index model
• E(ri) N
• ?i N
• var(ei) N ? need (3N 1) estimates
• ?M2 1
• Ex. If N 100, we need 301 estimates (vs.
  5,150)

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A single Index Model

• var(Ri) ?i2 ?i2 ?M2 ?2(ei),
• Because cov(RM, ei) 0,
• cov(Ri, Rj) cov (?i RM, ?jRM) ?i ?j ?M2.
• Because cov( ei, ej) 0.
• Advantages
• Reduce the number of inputs for diversification
• Easier for security analysts to specialize
• The index model suggests a simple way to compute
  covariance.
• However, the single index model oversimplifies
  sources of real-world uncertainty and misses some
  important sources of dependence in stock returns.