CAPM and APT

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					    BM410: Investments




Capital Asset Pricing
  Theory and APT
           or
 How do you value stocks?
                 Objectives

A. Review and solve problems using the CAL,
    MPT, and the Single Index model
B. Understand the implications of capital asset
    pricing theory and the CAPM to compute
    security risk premiums
C. Understand the arbitrage pricing theory and
    how it works
A. Solve problems using the CAL, CML,
      MPT and Single Index Models
 Capital Market Line Review
 •   You estimate that a passive portfolio invested to
     mimic the S&P 500 (an index fund) has an expected
     return of 13% with a standard deviation of 25%.
     Your portfolio has an expected return of 17% with a
     standard deviation of 27%. With the risk-free rate at
     7%, draw the CML and your fund’s CAL on an
     expected return-standard deviation diagram.
        A. What is the slope of the CML? Your CAL?
        B. Characterize in one short paragraph the
            advantage of your fund over the passive fund.
                           Answer

          Slope of the CML = (13-7)/25 = .24
          Slope of your CAL = (17-7)/27 = .37
                                                    Your Fund

17%
13%
                                                       Index Fund

7%
                    10%           20%                30%
      b. Your fund allows an investor a higher mean for any
          given standard deviation than the passive strategy.
                 MPT Review

   Suppose that for some reason you are required to
    invest 50% of your portfolio in bonds (sb = 12%,
    E(rb) = 10%) and 50% in stocks (ss = 25%, E(rs) =
    17%).
       A. If the standard deviation of your portfolio is
       15%, what must be the correlation coefficient
       between stock and bond returns?
       B. What is the expected rate of return on your
       portfolio?
       C. Now suppose that the correlation between stock
       and bond returns is 0.22 but that you are free to
       choose whatever portfolio proportions you desire.
       Are you likely to be better or worse off that you
       were in part a?
                          Answer

     A.         sp2 = w12s12 + w22s22 + 2W1W2 (r1,2s1s2 )
    (.15) 2 =[(.512.1212) +(.522.2522) + 2(.51.52)*(.121.252 )] * r1,2
    r1,2 = .2183 or 21.8% (take my word for this)
•     B. E(rp) = (.5 * .10) + (.5 * .17) = 13.5%
•     C. While the current correlation is slightly lower than 22%,
      this implies slightly greater benefits from diversification.
      However, the 50% bond constraint represents a cost since
      you cannot choose your optimal risk-return tradeoff for
      your risk level. Unless you would choose to have 50%
      bonds anyway, you are better off with the slightly higher
      correlation and the ability to choose your own portfolio
      weights.
               Factor Review

 Investors expect the market rate of return to be
  10%. The expected rate of return on the stock
  with a beta of 1.2 is currently 12%.
   If the market return this year turns out to be
      8%, how would you review/change your
      expectations of the rate of return on the
      stock?
                   Answer

 The expected return on the stock would be
  your beta (1.2) times the market return or:
      1.2 * 8% = 9.6%
 Likewise, you could also determine how much
  the return would decrease by multiplying the
  beta times the change in the market return or:
      1.2 * (8%-10%) = -2.4% + 12% =
       9.6%
                 Questions

Any questions of Capital Allocation Lines,
 Modern Portfolio Theory, or Single Index
 Models?
  B. Implications of Capital Market
          Theory and CAPM
 What have we done this far?
  • We have been concerned with how an individual or
    institution would select an optimum portfolio.
      • If investors act as we think, we should be able to
        determine how investors will behave, and how
        prices at which markets will clear are set
  • This market clearing of prices and returns has
    resulted in the development of so-called general
    equilibrium models
      • These models allow us to determine the risk for
        any asset and the relationship between expected
        return and risk for any asset when the markets
        are in equilibrium, i.e. balance or constant state
       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 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)
                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?
                   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
                 CAPM Assumptions

 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
   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
   • 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
          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


                      sm                            s
       Expected Return and Risk
        of Individual Securities
 What does this imply?
  • The risk premium on individual securities is
    a function of the individual security’s
    contribution to the risk of the market
    portfolio
  • Individual security’s risk premium is a
    function of the covariance of returns with
    the assets that make up the market portfolio
          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
    non-systematic risk (risk that 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. (ß = [COV(ri,rm)] / sm2)
 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
   asset’s 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 ]
        The Security Market Line

           Notice that instead of using standard
   E(r)     deviation, the Security Market Line uses Beta
           SML Relationships
            ß = [COV(ri,rm)] / sm2        SML
            Slope SML = E(rm) – rf = market risk
              premium
E(rM)
  rf                         SML = rf + ß[E(rm) - rf]



                ß M = 1.0                       ß
Differences Between the SML and CML

 What are the differences?
  • The CML graphs risk premiums of efficient
    portfolios , i.e. complete portfolios made up of the
    risk portfolio and risk-free asset, as a function of
    standard deviation
  • The SML graphs individual asset risk premiums as
    a function of asset risk.
      • The relevant measure of risk for individual
        assets is not standard deviation; rather, it is beta
      • The SML is also valid for portfolios
     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%
      Graph of Sample Calculations

      E(r)
                                   SML
Rx=13%                                .08
Rm=11%
Ry=7.8%                 They are in equilibrium
     3%
                                         ß
             .6 1.0 1.25
             ßy ßm ßx
        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.
          Disequilibrium Example

         E(r)   The return is above the
                SML, so you would buy it     SML
   15%
                                    As more people bought
Rm=11%                              the security, it would
                                    push the price up,
                                    which would bring the
  rf=3%                             return down to the line.

                                                   ß
                     1.0 1.25
       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 S&P 500 for
     measurement
   • Uses realized, not expected returns
 Now the Index model is testable
                    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
   ά = 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)
       Security Characteristic Line
 Excess Returns (i)                                   SCL

                       . .. .
            Plot of a company’s excess return as a
              . .. .
          function of the excess return of the market
            .              . .
          .       . .. .
   . . .. . .
       .        . . . Excess returns
. .. .. .
      . .      . . ..       on market index

    . . . .. . .       .. . .
 .                R = a + ßR + e
                         i       i     i   m      i
            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?
               Questions

Any questions on capital asset pricing
 and the Capital Asset Pricing Model?
                  CAPM Problem

 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
   5%, the expected return of Dell would be 5% +(1.3 *
   9%) = 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
      Dell’s beta was 1.3, what would be Dell’s risk
      premium?
                      Answer

  a. If Dell’s 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%
   Dell’s 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%
    Dell’s new risk premium is 15.4 – 5% = 10.4%
C. Understand Arbitrage Pricing Theory
    (APT) and How it Works

 What is arbitrage?
    • The exploitation of security mis-pricing to earn
      risk-free economic profits
        • It rises if an investor can construct a zero
          investment portfolio (with a zero net investment
          position netting out buys and sells) with a sure
          profit
  Should arbitrage exist?
    • In efficient markets (and in CAPM theory),
      profitable arbitrage opportunities will quickly
      disappear as more investors try to take advantage of
      them
Arbitrage Pricing Theory (APT) (continued)

 What is APT based on?
   • It is a variant of the CAPM, and is an attempt to
     move away from the mean-variance efficient
     portfolios (the calculation problem)
   • Ross instead calculated relationships among
     expected returns that would rule out riskless profits
     by any investor in a well-functioning capital market
 What is it?
   • It is a another theory of risk and return similar to the
     CAPM.
   • It is based on the law of one price: two items that
     are the same can’t sell at different prices
                 APT (continued)

 In its simplest form, it is:
   Ri = a i + ßiRm + ei the same as CAPM
   The only value for a which rules out arbitrage
     opportunities is zero. So set a to zero and you get:
   Ri = ßiRm Subtract the risk-free rate and you get the
     well-known equation:
    E(rs) = rf + bs [E(rM) - rf ] from CAPM
        APT and CAPM Compared

 Differences:
   • APT applies to well diversified portfolios and not
     necessarily to individual stocks
   • With APT it is possible for some individual stocks
     to be mispriced – to not lie on the SML
   • APT is more general in that it gets to an expected
     return and beta relationship without the assumption
     of the market portfolio
   • APT can be extended to multifactor models, such
     as:
       Ri = a i + ß1R1 + ß2R2 + ß3R3 + ßnRn + ei
     APT and Investment Decisions

 Roll and Ross argue that APT offers an approach to
  strategic portfolio planning
   • Investors need to recognize that a few systematic
      factors affect long-term average returns
        • Investors should understand those factors and set
          up their portfolios to take those factors into
          account
   • Key Factors:
        • Changes in expected inflation
        • Unanticipated changes in inflation
        • Unanticipated changes in industrial production
        • Unanticipated changes in default-risk premium
        • Unanticipated changes in the term structure of
          interest rates
                 Questions

 Any questions on Arbitrage Pricing Theory
  and how it differs from CAPM?
                   Problem

 Suppose two factors are identified for the U.S.
  economy: the growth rate of industrial
  production (IP) and the inflation rate (IR). IP
  is expected to be 4% and IR 6% this year. A
  stock with a beta of 1.0 on IP and 0.4 on IR
  currently is expected to provide a rate of return
  of 14%. If industrial production actually
  grows by 5% while the inflation rate turns out
  to be 7%, what is your best guess on the rate
  of return on the stock?
                   Answer

 The revised estimate on the rate of return on
  the stock would be:
   • Before
      • 14% = a +[4%*1] + [6%*.4]
            a = 7.6%
   • With the changes:
      • 7.6% + [5%*1] + [7%*.4]
           The new rate of return is 15.4%
          Review of Objectives

 A. Can you solve problems using the CAL,
  MPT, and the Single Index model?
 B. Do you understand the implications of
  capital asset pricing theory and the CAPM to
  compute security risk premiums?
 C. Do you understand arbitrage pricing theory
  and how it works?

				
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posted:1/14/2012
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