Risk and Return Portfolio Theory and Asset Pricing Models

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					39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls                                                                           Model
  Ch 05 Mini Case                                                                                    5/3/2003

                                                     Chapter 5. Mini Case

  Situation
  To begin, briefly review the Chapter 4 Mini Case. Then, extend your knowledge of risk and return by
  answering the following questions.

  a. Suppose Asset A has an expected return of 10 percent and a standard deviation of 20 percent. Asset B has
  an expected return of 16 percent and a standard deviation of 40 percent. If the correlation between A and B
  is 0.4, what are the expected return and standard deviation for a portfolio comprised of 30 percent Asset A
  and 70 percent Asset B?


  Expected return of a portfolio:



           ^           ^                         ^
           r p  w A r A  (1  w A ) r B


  Standard deviation of a portfolio:




           s p  WAs A  (1  WA )2 s B  2WA (1  WA )  AB s A s B
                  2 2                 2




                                 Asset A                Asset B
  Expected return, r hat          10%                    16%
  Standard deviation, s           20%                    40%


  Using the equations above, we can find the expected return and standard deviation of a
  portfolio with different percents invested in each asset.


  Correlation =             0.4
                              Proportion of
      Proportion of            Portfolio in
  Portfolio in Security A       Security B
       (Value of wA)         (Value of 1-w A)             rp              sp
                     1.00                 0.00              10.00%         20.00%
                     0.90                 0.10              10.60%         19.94%
                     0.80                 0.20              11.20%         20.55%
                     0.70                 0.30              11.80%         21.78%
                     0.60                 0.40              12.40%         23.53%
                     0.50                 0.50              13.00%         25.69%
                     0.40                 0.60              13.60%         28.17%
                     0.30                 0.70              14.20%         30.89%
                     0.20                 0.80              14.80%         33.80%
                     0.10                 0.90              15.40%         36.85%
                     0.00                 1.00              16.00%         40.00%




Michael C. Ehrhardt                                                  Page 1                                     4/24/2010
39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls                                                                                  Model
  b. Plot the attainable portfolios for a correlation of 0.4. Now plot the attainable portfolios for correlations of
  +1.0 and -1.0.



                                                 AB = +0.4: Attainable Set of
                                                  Risk/Return Combinations
                                         20%


                                         15%
                       Expected return




                                         10%


                                            5%


                                            0%
                                                 0%       10%     20%       30%      40%
                                                                 Risk, sp



  Correlation =                             1
                           Proportion of
      Proportion of         Portfolio in
  Portfolio in Security A   Security B
       (Value of wA)      (Value of 1-w A)                            rp              sp
                     1.00              0.00                             10.00%         20.00%
                     0.90              0.10                             10.60%         22.00%
                     0.80              0.20                             11.20%         24.00%
                     0.70              0.30                             11.80%         26.00%
                     0.60              0.40                             12.40%         28.00%
                     0.50              0.50                             13.00%         30.00%
                     0.40              0.60                             13.60%         32.00%
                     0.30              0.70                             14.20%         34.00%
                     0.20              0.80                             14.80%         36.00%
                     0.10              0.90                             15.40%         38.00%
                     0.00              1.00                             16.00%         40.00%


                                              AB = +1.0: Attainable Set of
                                               Risk/Return Combinations
                                            20%
                          Expected return




                                            15%


                                            10%


                                                5%


                                                0%
                                                     0%           20%                40%
                                                                Risk, sp




  Correlation =                             -1


Michael C. Ehrhardt                                                              Page 2                                4/24/2010
39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls                                                                                Model
                           Proportion of
      Proportion of         Portfolio in
  Portfolio in Security A   Security B
       (Value of wA)      (Value of 1-w A)                     rp               sp
                     1.00              0.00                      10.00%          20.00%
                     0.90              0.10                      10.60%          14.00%
                     0.80              0.20                      11.20%            8.00%
                     0.70              0.30                      11.80%            2.00%
                     0.67              0.33                      12.00%            0.00%
                     0.60              0.40                      12.40%            4.00%
                     0.50              0.50                      13.00%          10.00%
                     0.40              0.60                      13.60%          16.00%
                     0.30              0.70                      14.20%          22.00%
                     0.20              0.80                      14.80%          28.00%
                     0.10              0.90                      15.40%          34.00%
                     0.00              1.00                      16.00%          40.00%


                                           AB = -1.0: Attainable Set of
                                            Risk/Return Combinations
                                         20%
                       Expected return




                                         15%


                                         10%


                                             5%


                                             0%
                                                  0%        20%                40%
                                                          Risk, sp


  c. Suppose a risk-free asset has an expected return of 5 percent. By definition, its standard deviation is zero,
  and its correlation with any other asset is also zero. Using only Asset A and the risk-free asset, plot the
  attainable portfolios.


                                              Asset A    Risk-free Asset
  Expected return, r hat                       10%             5%
  Standard deviation, s                        20%             0%


  Using the equations above, we can find the expected return and standard deviation of a
  portfolio with different percents invested in each asset.


  Correlation =                          0
                            Proportion of
      Proportion of       Portfolio in Risk-
  Portfolio in Security A    free Asset
       (Value of wA)       (Value of 1-w A)                    rp               sp
                     1.00              0.00                      10.00%          20.00%
                     0.90              0.10                       9.50%          18.00%
                     0.80              0.20                       9.00%          16.00%
                     0.70              0.30                       8.50%          14.00%
                     0.60              0.40                       8.00%          12.00%
                     0.50              0.50                       7.50%          10.00%


Michael C. Ehrhardt                                                        Page 3                                    4/24/2010
39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls                                                                              Model
                      0.40                             0.60              7.00%            8.00%
                      0.30                             0.70              6.50%            6.00%
                      0.20                             0.80              6.00%            4.00%
                      0.10                             0.90              5.50%            2.00%
                      0.00                             1.00              5.00%            0.00%




                                      Attainable Set of Risk/Return
                                    Combinations with Risk-Free Asset
                                         15%
                       Expected return




                                         10%




                                         5%




                                         0%
                                               0%      5%      10%         15%       20%
                                                              Risk, sp



  d. Construct a reasonable, but hypothetical, graph that shows risk, as measured by portfolio standard
  deviation, on the X axis and expected rate of return on the Y axis. Now add an illustrative feasible (or
  attainable) set of portfolios, and show what portion of the feasible set is efficient. What makes a particular
  portfolio efficient? Don't worry about specific values when constructing the graph-merely illustrate how
  things look with "reasonable" data.

  FEASIBLE AND EFFICIENT PORTFOLIOS
  The feasible set of portfolios represent all portfolios that can be constructed from a given set of stocks.
  An efficient portfolio is one that offers: the most return for a given amount of risk or the least risk for a
  given amount of return.




              Expected
              Portfolio                             Efficient Set
              Return, r p



                                                                                     Feasible Set




                                                                                          Risk, s p
       Feasible and Efficient Portfolios
          .




Michael C. Ehrhardt                                                              Page 4                            4/24/2010
39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls                                                                               Model


  e. Now add a set of indifference curves to the graph created for part b. What do these curves represent? What
  is the optimal portfolio for this investor? Finally, add a second set of indifference curves which leads to the
  selection of a different optimal portfolio. Why do the two investors choose different portfolios?

  OPTIMAL PORTFOLIOS
  An investor's optimal portfolio is defined by the tangency point between the efficient set and the investor's
  indifference curve. The inderference curve reflect an investor's attitude toward risk as reflected in his or
  her risk/return trade off function.



         Expected
                                              IB 2 I
         Return, r p                                B1




                                                                        Optimal Portfolio
                IA 2                                                        Investor B
                 IA 1

                                                                       Optimal Portfolio
                                                                          Investor A

                                                                          Risk s p
                        Optimal Portfolios
            .




  f. What is the Capital Asset Pricing Model (CAPM)? What are the assumptions that underlie the model?

  CAPM
  The Capital Asset Pricing Model is an equilibrium model that specifies the relationship between risk and
  required rate of return for assets held in well diversified portfolios.

  Assumptions
  Investors all think in terms of a single holding period.
  All investors have identical expectations.
  Investors can borrow or lend unlimited amounts at the risk free rate.
  All assets are perfectly divisible.
  There are not taxes and transaction costs.
  All investors are price takers, that is, investors buying and selling will not influence stock prices.
  Quantities of all assets are given and fixed.


  g. Now add the risk-free asset. What impact does this have on the efficient frontier?

  EFFICIENT SET WITH A RISK-FREE ASSET
  When a risk free asset is added to the feasible set, investors can create portfolios that combine this asset
  with a portfolio of risky asset. The straight line connecting krf with M, the tangency point between the line
  and the old efficiency set, becomes the new efficient frontier.




                Efficient Set with a Risk-Free Asset
Michael C. Ehrhardt                                             Page 5                                              4/24/2010
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                  Efficient Set with a Risk-Free Asset

              Expected                                     Z
              Return, r   p
                                                                     .    B

               ^
               rM
                                          M
                                              .
                                                           The Capital Market

              r   RF
                              A    .                          Line (CML):
                                                            New Efficient Set


                                              sM                                  Risk, sp
          .




  h. Write out the equation for the Capital Market Line (CML) and draw it on the graph. Interpret the CML.
  Now add a set of indifference curves, and illustrate how an investor's optimal portfolio is some combination of
  the risky portfolio and the risk-free asset. What is the composition of the risky portfolio?

  OPTIMAL PORTFOLIO WITH A RISK-FREE ASSET
  The optimal portfolio for any investor is the point of tangency between the CML and the investors indifference
  curve.



       Expected
       Return, r p
                                                                   CML
                                   I2
                                              I1



                              . .
           ^                              M
           rM
           ^                  R
           rR

                                                   R = Optimal
         r RF                                       Portfolio




                              sR         sM                               Risk,   sp
          .


  Capital Market Line
  The capital market line is all linear combinations of the risk free asset and portfolio M.


              rhat=                     rrf           +            (rm-rrf)/sm          x              sp



                                  Intercept                          Slope                       Risk Measure



Michael C. Ehrhardt                                              Page 6                                             4/24/2010
39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls                                                                                                  Model
  The CML gives the risk and return relationship for efficient portfolios
  The SML , also part of CAPM, gives the risk and return relationship for individual stocks.

          SML =                                  ri                +            (RPm)                x                  b


  i. What is a characteristic line? How is this line used to estimate a stock's beta coefficient? Write out and
  explain the formula that relates total risk, market risk, and diversifiable risk.



  Beta Calculation
  Run a regression line of past returns on Stock I versus returns on the market. The regression line is the characteristic line.


                                               Year                rm                ri
                                                1                 15%            18%
                                                2                 -5%            -10%
                                                3                 12%            16%



                                                                  Beta Calculation

                                                      25%
                                                      20%
                                                      15%
                        Stock Return




                                                      10%                                                beta calculation

                                                      5%
                                                                                                         Linear (beta
                                                      0%                                                 calculation)
                                       -10%           -5% 0%              10%             20%
                                                  -10%                                          y = 1.4441x - 0.0259
                                                                                                     R² = 0.9943
                                                  -15%
                                                            Market Return


                                                                            R2 measures the percent of a stock's variance as explained by the market.

  Relationship between stand alone, market, and diversifiable risk
           s2j =                              b2j *s2m                +          s2ej

              s2j =                     stand alone risk of stock J
          2       2
        b j *s     m=                   market risk of stock J
              s2ej =                    diversifiable risk of stock J


  j. What are two potential tests that can be conducted to verify the CAPM? What are the results of such tests?
  What is Roll's critique of CAPM tests?

  Test to verify CAPM
  Beta stability test and tests based on the slope of the SML.

  Test of the SML indicate a more-or-less linear relationship between realized return and market risk.
  Slope is less than predicted
  Irrelevance of diversifiable risk specified in the CAPM model can be questioned.
  Betas of individual securities are not good estimators of future risk.

Michael C. Ehrhardt                                                         Page 7                                                    4/24/2010
39fef436-2dd7-4b0e-94c1-d7acc274fe8f.xls                                                                                                Model
  Betas of ten or more randomly selected stocks are reasonably stable.
  Past betas are good estimates of future portfolio volitility.

  Conclusions regarding CAPM
  It is impossible to verify.
  Recent studies have questioned its validity.
  Investors seemed to be concerned with both market and stand alone risk. Therefore, the SML may not produce the correct estimate of rj.
  CAPM/SML concepts are based on expectations, yeta betas are calculated using historical data.

  k. Briefly explain the difference between the CAPM and the Arbitrage Pricing Theory (APT).

  CAPM and the Arbitrage Pricing Theory
  The CAPM is a single factor model. The APT proposes that the relationship between risk and return is more complex and may be due
  to multiple factos such as GDP, growth, expected inflation, tax rate changes, and dividend yield.



  l. Suppose you are given the following information. The beta of company, bi, is 0.9, the risk-free rate, rRF, is 6.8 percent, and
  the expected market premium, rM - rRF, is 6.3 percent. Because your company is larger than average and more successful than
  average (that is, it has a lower book-to-market ratio), you think the Fama-French three-factor model might be more
  appropriate than the CAPM. You estimate the additional coefficients from the Fama-French three-factor model: The
  coefficient for the size effect, ci, is -0.5, and the coefficient for the book-to-market effect, di, is -0.3. If the expected value of the
  size factor is 4 percent and the expected value of the book-to-market factor is 5 percent, what is the required return using the
  Fama-French three-factor model? (Assume that ai = 0.0.) What is the required return using CAPM?

  Required Return for stock I under the Fama-French-3-Factor Model
  Fama and French propose three factors:
  The excess market return, rm-rrf.
  The return on, S, a portfolio of small firms minus the return on B, a portfolio of big firms. This return is called
  rsmb, for S minus B.
  The return on, H, a portfolio of firms with high book-to-market ratios minus the return on L, a portfolio of firms
  with low book-to-market ratios. This return is called rhml, for H minus L.

  Required return for Stock I

            ri =                    rrf                 +              (rm-rrf) b             +              (rsmb) c              +


                        b= Sensitivity of stock I to the market
                        c= Sensitivity of stock I to the size factor
                        c= Sensitivity of stock I to the book-to-market factor

  b=                                         0.9
  rrf =                                   6.8%
  RPm =                                   6.3%
  c=                                        -0.5
  value for size factor =                 4.0%
  d=                                        -0.3
  book-to-market factor=                  5.0%

            ri =                          8.97%

         CAPM =                       12.47%




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                             Excess Returns
                              on Wal-Mart,
                                 kS-kRF
                             120%

                             110%

                             100%

                              90%

                              80%

                              70%

                              60%

                              50%

                              40%                          Excess Returns
                                                    on the Market, kM-kRF
                              30%

                              20%

                              10%

                               0%
      -30%    -20%    -10%          0%        10%   20%         30%




Michael C. Ehrhardt                                         Page 19         4/24/2010