The Capital Asset Pricing Model

1
The Capital Asset Pricing Model

• This chapter has one of the most important models
  in investment modeling. It addresses the
  question of what is a reasonable price for an
  asset. The same model also gives some very good
  investment advice. The results of the chapter
  build upon the Markowitz mean-variance portfolio
  theory of Chapter 6.

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• Assumptions
• Everyone is a mean-variance optimizer, as in
  Chapter 6.
• Everyone assigns to returns of all available
  market assets the same mean values, variances,
  and covariances (available from Morning Star, for
  example).
• There is a unique risk-free rate of borrowing and
  lending available to all, with no transaction
  costs.

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• Under the above assumptions, everyone would rely
  on the one-fund theorem. They would buy a single
  (efficient) fund of risky assets, and then also
  borrow or lend at the risk-free rate. The mix of
  the fund and the risk-free asset would depend on
  a investors attitude towards risk.
• Basic Question What is the one fund (shared by
  all investors) ?

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• Example of the Market Fund, after Table 7.1.
• Suppose there are only three risky assets in the
  market, as follows

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• Notes
• The table illustrates the definitions of
  capitalization (shares ? price) and
  capitalization weights.
• The weights are NOT the same as the relative
  shares in the market.
• Each weight is the ratio of the capital value of
  the asset to the total market capital value.

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• If you invested in this market fund, you would
  use the weights in the last column a 1,000
  investment would give you
• (3/20) ? 1000/6 25 shares of Mahler Inc
• (3/10) ? 1000/4 75 shares of Mozart Inc.
• (11/20) ? 1000/5.5 100 shares of Verdi, Inc.

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• The actual market fund would be comprised of
  every investment asset available. Just as in the
  example, each investment weight would be the
  ratio of the capital value of the asset to the
  total market capital value.
• Funds similar to the market fund actually exist,
  and are called index funds. Of course, they do
  not include every available asset, but they may
  well include 500 or more. Index funds typically
  out-perform most actively managed funds.
  Vanguard Securities offers many such funds (e.g.
  Index Trust 500)

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• Index funds have been somewhat resisted by the
  financial world, including many private
  investors. The first reaction is usually that
  surely an actively managed fund will give better
  performance. The facts, however, are otherwise.
  About 90 of actively managed funds perform worse
  than the SP 500. You can also check performance
  with the Morning Star services.

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• Equilibrium Arguments
• If everyone buys just one fund, and their
  purchases add up to the market, then that one
  fund must be the market as well.
• The fund must contain shares of every stock in
  proportion to that stocks representation in the
  entire market.
• If everyone else solved the one-fund problem, we
  would not need to.
• Suppose everyone else solves the mean-variance
  portfolio problem with their common estimates,
  and places orders in the market to acquire their
  portfolios. This solution is efficient, because
  it minimizes the total variance of the return.

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• Equilibrium Arguments (Contd)
• If the orders placed do not match what is
  available, the prices must change. Prices of
  assets under heavy demand will go up, prices of
  assets under light demand will go down. The
  price changes will affect the estimates of asset
  returns directly. Therefore, investors will
  recalculate their optimal portfolios. The
  process continues until demand exactly matches
  supply that is, until there is an equilibrium.

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Flow Chart Visualization
Solve mean-variance portfolio problem for
efficient
portfolio.
Place orders.
Supply Demand? Yes Equilibrium
No
Prices adjustments occur.
Asset return changes cause portfolio data changes.
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• Extra Comments
• Everyone would buy just one portfolio in this
  idealized world, and it would be the market
  portfolio.
• Prices adjust to drive the market to efficiency.
• After the adjustments, the portfolio will be
  efficient, so we would not need to make the
  calculations.
• This argument is most plausible when applied to
  assets traded repeatedly over time, which is
  certainly the case with the stock market.
• The fact that index funds perform well provides
  some verification that the equilibrium argument
  is plausible.

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• Final Word If you have better information than
  your rivals, you will want your portfolio to
  include relatively large investments in the
  stocks you think are undervalued. In a
  competitive market you are unlikely to have a
  monopoly of good ideas. In that case, there is
  no reason to hold a different portfolio of common
  stocks from anybody else. In other words, you
  might just as well hold the market portfolio.
  That is why many professional investors invest in
  a market-index portfolio, and why most others
  hold well-diversified portfolios.

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• The Capital Market Line
• M is the market portfolio
• M (?M, ErM)
• ErM - rf
• Er rf -------------- ?
• ?M

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• The line shows the efficient set, starting at
  the risk-free point, and passing through the
  market portfolio.
• The CML shows the relation between the expected
  rate of return and the risk of return (as
  measured by the standard deviation), for
  efficient assets or portfolios of assets.
• The CML is also called the pricing line. Prices
  should adjust so that efficient assets fall on
  this line.equations w.r.t.

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• Example 7.1
• rf 6, ErM 12, ?M 15.
• John Eager wants to retire in 10 years. For
  this he needs 1,000,000. He currently has
  1,000. At the market rate, it would take about
  60 years for 1,000 to grow into 1,000,000. If
  he can get 100 return each year he concludes he
  will grow 1,000 into 1,000,000 in 10 years.

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• Example 7.1 (Contd)

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Example 7.2.
The Capital Market Line
Er

M
r
f
Oil Drilling


s

s
r
0.10, M (
, Er
) (0.12, 0.17), OD (0.4,0.14).
f
M
M
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The Pricing Model

• The CML relates the expected rr of any efficient
  portfolio to its standard deviation. Another
  step beyond the CML is to show how the expected
  rate of return of any individual asset relates to
  its individual risk. That is what the capital
  asset pricing model does.

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• Example (We use t instead of ?).
• The equation for the standard deviation of the
  rate of the return combining the market portfolio
  with any asset i is
• x f(t) t2 ?i2 2 t (1-t) ?iM (1-t)2 ?M2
  ½
• The expected return for this combination becomes
• y g(t) t Eri (1-t) ErM
• Note f(0) ?M, g(0) ErM .

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• Example (Contd)
• We thus have
• dy/dt g?(t) Eri - ErM
• dx/dt f?(t) t ?i2 (1-2 t) ?iM (t-1)
  ?M2/f(t)
• Note, when t 0,
• f?(t)t0 (?iM - ?M2)/f(0) (?iM - ?M2)/ ?M

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• Example (Contd)

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• CAPM Fundamental Result
• If the market portfolio M is efficient, then the
  expected return Eri of any asset i satisfies
• Eri rf ?i (ErM rf)
• where
• ?i ?iM/?M2.

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• Proof of CAPM.
• For any t, consider the portfolio consisting of a
  portion t invested in asset i and a portion 1-t
  invested in the Market Portfolio M. (t corresponds to selling short the asset.)
• The expected rate of return of this portfolio is
• y g(t) t Eri (1-t) ErM
• The standard deviation of the rate of return is
• x f(t) t2 ?i2 2 t (1-t) ?iM (1-t)2 ?M2 ½

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• As t varies, the values (f(t), g(t)) trace out a
  curve in the expected return-sd diagram, as shown
  below.
• In particular, the point on the curve (f(0),g(0))
  for
• t 0 corresponds to the market portfolio M.

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• This curve cannot cross the capital market line.
  If it did, the portfolio corresponding to a point
  above the capital market line would violate the
  definition of the capital market line as being
  the efficient boundary of the feasible set.
  Hence as t passes through zero, the curve must be
  tangent to the capital market line at M. This
  tangency is the condition that we exploit to
  derive the formula.
• The tangency condition can be translated into the
  condition that the slope of the curve (f(t),
  g(t)) is equal to the slope of the capital market
  line at the point M, where t 0.

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• If we solve this equation for Eri we get
• Eri rf (ErM rf)/?M2 ?iM
• rf ?i (ErM
  rf).
• where ?i ?iM/?M2 .
• Equivalently, we have Eri - rf ?i (ErM
  rf).
• ?i is called the beta of asset i. Sometimes the
  subscript is omitted.

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• The Morning Star service estimates betas.
• Since Eri - rf ?i (ErM rf), Eri - rf is
  called the expected excess rate of return of
  asset i. It is the amount by which the rate of
  return is expected to exceed the risk-free rate.
• (ErM rf) is called the expected excess rate
  of return of the market portfolio.
• THE CAPM says the expected excess rate of return
  of an asset is proportional to the expected
  excess rate of return of the market portfolio.
  The constant of proportionality factor is ?.

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• Because ?i ?iM/?M2, it is a normalized version
  of the covariance of an asset with the market
  portfolio. The excess rate of return for the
  asset is directly proportional to its covariance
  with the market.
• Generally speaking, we expect aggressive
  assets/companies or highly leveraged companies to
  have high betas. Conservative companies whose
  performance is unrelated to the general market
  behavior are expected to have low betas. We
  expect that companies in the same business will
  have similar beta values.

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• Bottom Line
• The CAPM changes our concept of the risk of
  an asset from ? to ?. It is still true that,
  overall, we measure the risk of a portfolio in
  terms of ?. But this does not translate into a
  concern for the ?s of individual assets. For
  those, the proper measure is their ?s.

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• After Table 7.2. Betas and Sigmas for Some U.S.
  Companies (1979)

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• The concept of beta is well-accepted.
• Various financial service organizations (e.g.,
  Morning Star) provide beta and other estimates.
• Estimates may be based on 6 to 18 months of
  weekly values.
• Companies in the same business should have
  similar betas compare, for instance, JC Penny
  with Sears Roebuck, and Exxon with Standard Oil
  of California.

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• We never know the beta and sigma values we only
  have estimates of them.
• Generally speaking, we expect aggressive
  companies or highly leveraged companies to have
  high betas, whereas conservative companies whose
  performance is unrelated to the general market
  behavior are expected to have low betas. Also we
  expect that companies in the same business will
  have similar, but not identical, beta values.

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• Facts about covariances
• covU V, Z covU,Z covV,Z
• cova U b V, Z cova U,Z covb V,Z
• a covU,Z b
  covV,Z

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• Beta of a Portfolio
• Suppose a portfolio P has 2 assets with returns
  r1, r2 and weights w1, w2. Let rM denote the
  market return. We know the portfolio return is r
  w1 r1 w2 r2. Let ?P denote the beta of the
  portfolio (ratio of the portfolio covariance with
  the market and ?M2 ).
• ?P covr, rM /?M2 cov w1 r1 w2 r2, rM
  /?M2
• cov w1 r1, rM /?M2 covw2 r2, rM/?M2
• w1 covr1, rM /?M2 w2 covr2, rM/?M2
• w1 ?1 w2 ?2

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• Beta of a Portfolio (Contd)
• The two dimension formula generalizes to n
  assets
• ?P covr, rM /?M2 w1 ?1 w2 ?2 ? wn
  ?n.
• The portfolio beta is just the weighted average
  of the betas of the individual assets in the
  portfolio. The weights are those that define the
  portfolio.
• Risk neutral portfolio ?P 0 .

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The Security Market Line (SML)
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• Under the equilibrium conditions assumed by the
  CAPM, any asset should fall on the SML.
• The SML expresses the reward-risk structure of
  assets according to the CAPM, and emphasizes that
  the risk of an asset is a function of its
  covariance with the market or, equivalently, a
  function of its beta.

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• Systematic Risk
• There are several types of risk with an
  investment
• - systematic risk
• - nonsystematic, idiosyncratic, or specific
  risk.
• The systematic risk is risk associated with the
  market as a whole. The second type of risk is
  uncorrelated with the market, and can be reduced
  by diversification.
• We can use the CAPM to quantify these two types
  of risks.

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• Consider the equation
• ri rf ?i (rM - rf) ?i ()
• To begin with, we view this equation simply as a
  definition of the random variable ?i. Namely,
• ?i ri rf ?i (rM - rf)
• The CAPM provides some information on ?i. Note
  first that

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• Let us show that
• ?i2 ? var(ri) ?i2 ?M2 var(?i)
• We can write
• ri ?i rM ?i k,
• where k is a constant.

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• Implications. For asset i, its risk is the sum
  of
• (1) ?i2 ?M2, the systematic risk, and
• (2) var(?i), the nonsystematic or specific risk.
• The systematic risk is the risk associated with
  the market as a whole. It cannot be reduced by
  diversification, because every asset with nonzero
  beta contains this risk.

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• The specific risk is uncorrelated with the market
  and can be reduced by diversification.
• It is the systematic risk, measured by beta,
  that is most important. It directly combines
  with the systematic risk of other assets.
• There is a limit to how much diversification can
  achieve in reducing risk.

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• The Capital Market Line
• M is the market portfolio
• M (?M, ErM)
• ErM - rf
• Er rf -------------- ?
• ?M

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• For asset i, its risk is the sum of
• (1) ?i2 ?M2, the systematic risk, and
• (2) var(?i), the nonsystematic or specific
  risk.
• If it has only systematic risk, then its standard
  deviation is
• ?i ?i ?M.

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• Now consider other funds with the same ? value,
  ?i, as asset i. The CAPM implies all these funds
  have an expected return of
• rf ?i (ErM - rf)
• But this is the expected return of asset i,
  Eri. Suppose these other assets have
  nonsystematic risk. Then each will have a
  variance, for some ?, of
• ?i2 ?M2 var(?) ?i2 ?M2 ?i2.

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• Assets with the same beta as asset i, but which
  also have systematic risk, have the same expected
  return as asset i but do not fall on the CML.
• Bottom Line. The horizontal distance of an asset
  point from the CML is a measure of the
  nonsystematic risk of the asset.

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• CAPM Investment Implications
• A CAPM purist is one who completely believes the
  CAPM theory as applied to publicly traded
  securities. A purist would just purchase the
  market fund and some risk-free securities (e.g.,
  U.S. Treasury bills), adjusting the relative
  investment in the two according to her/his
  tolerance for risk.
• Individual investors cannot easily purchase the
  market fund. They can, however, purchase an
  index fund. These funds allocate their
  investments in order to duplicate the portfolio
  of a major stock market index, such as the SP
  500 or the Wiltshire 2,000.

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• The CAPM requires the assumption that everyone
  has identical information about the expected
  returns and variance of returns of all assets.
  The assumption is certainly open to criticism.
• Therefore, a key question for an investor is the
  following
• Do I possess superior information to that
  required by the Markowitz model and the CAPM?
• With superior information, it is likely that one
  can do better.

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• Few people quarrel with the idea that investors
  require some extra return for taking on risk ....
• Investors do appear to be concerned principally
  with those risks that they cannot eliminate by
  diversification.
• The CAPM captures these ideas in a simple way.
  That is why many financial managers find it the
  most convenient tool for coming to grips with the
  slippery notion of risk.

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• Tests of the CAPM
• There are two problems with the CAPM.
• - First, it is concerned with expected returns,
  whereas we can observe only actual returns.
  Stock returns reflect expectations, but they also
  embody lots of noise the steady flow of
  surprises that gives many stocks standard
  deviations of 30 or 40 percent per year.
• - Second, the market portfolio should comprise
  all risky investments .... Most market indexes
  contain only a sample of common stocks.

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• A classic paper by Fama and MacBeth avoids the
  main pitfalls that come from having to work with
  actual rather than expected returns. Fama and
  MacBeth (Risk, Return and Equilibrium Empirical
  Tests J. of Political Economy, 81, 607-636 ,May,
  1973) grouped all New York Stock Exchange stocks
  into 20 portfolios. They then plotted the
  estimated beta of each portfolio in one 5-year
  period against the portfolios average return
  over a subsequent 5-year period. 1 Figures 8-10
  show what they found. You can see that the
  estimated beta of each portfolio told investors
  quite a lot about its future return.

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• If the CAPM is correct, investors would not have
  expected any of these portfolios to perform
  better or worse than a comparable package of
  Treasury bills and the market portfolio.
  Therefore, the expected return on each portfolio,
  given the market return, should plot along the
  sloping lines in Figures 8 10. Notice that the
  actual returns do plot roughly along those lines.

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• 1 Fama and MacBeth first estimated the beta of
  each stock during one period and then formed (the
  20) portfolios on the basis of these estimated
  betas. Next they reestimated the beta of each
  portfolio by using the returns in the subsequent
  period. This ensured that the estimated betas
  for each portfolio were largely unbiased and free
  from error. Finally, these portfolio betas were
  plotted against returns in an even later period.

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Performance Evaluation

• Many institutional portfolios (pension funds,
  mutual funds) now have their performance
  evaluated using the CAPM framework.
• The following example illustrates the evaluation
  ideas and the use the CAPM.

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• Example, ABC fund

12.39

9.43

0.47

Std. Dev.

12.34

11.63

7.60

Geom. Mn


Cov(ABC,SP)

107


1.20375

1

Beta

0.104

0

Jensen

Sharpe

0.43577368

0.46669


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• According to the CAPM, the value of J should be
  zero when true expected returns are used. Hence
  J measures, approximately, how much the
  performance of ABC has deviated from the
  theoretical value of zero. A positive value of J
  presumably implies that the fund did better than
  the CAPM prediction (but of course we recognize
  that approximations are introduced by the use of
  a finite amount of data to estimate the important
  quantities.

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• The Jensen index can be indicated on the security
  market line (Figure 7.5 a).

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• Sharpe Index (Note. Figure 7.5 in the text has
  mistakes)
• The slope of the heavy dashed line is the
• Sharpe Index (Ratio)

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We compare S for ABC (0.43577) with S for the
market (0.46669). The conclusion is that ABC is
not efficient.
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• CAPM as a Pricing Formula

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• Example 7.5 (Contd)

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• Certainty Equivalent Form of the CAPM

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• Certainty Equivalent Form of the CAPM (Contd)

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• Practical Implication. If we want the CAPM for
  two assets, and have the CAPM for each, we can
  get the CAPM for the two in total by adding the
  certainty equivalent forms for each, or
  equivalently, using the latter equation.
•  
• The reason for linearity can be traced back to
  the principle of no arbitrage .... This linearity
  of pricing is therefore a fundamental tenet of
  financial theory (in the context of perfect
  markets) ....

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• Example 7.7 (Certainty Equivalent version of
  Example 7.5)
•  
• A fund 
• invests 10 of its money at rf 0.07
• invests 90 of its money at the market rate,
• ErM 0.15.
•  
• Its expected return in a year is 0.1 ? 0.07 0.9
  ? 0.15 0.142
•  
• The epected value of a 100 share in a year will
  be 100 ?1.142 114.20.
•  
• The ? value is 0.10 ? 0 0.9 ? 1.0 0.9 and
  ?P90
•  
• Certainty Equivalent Version 
• P (1 rf)-1EQ - (cov(Q,rM)/?M2 )(ErM
  rf) 
• (1 rf)-1EQ - ?P)(ErM rf) 
• (1.07) 1114.20 - 90 ? 0.08 100