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					  Financial Management
                 Lecture 6 (Ch.11)
Return and Risk: The Capital Asset Pricing (CAPM)
           Lecturer: Sheng-Syan Chen
             College of Management
           National Taiwan University

                                                    1
     Key Concepts and Skills
 Know   how to calculate expected returns
 Know how to calculate covariances,
  correlations, and betas
 Understand the impact of diversification
 Understand the systematic risk principle
 Understand the security market line
 Understand the risk-return tradeoff
 Be able to use the Capital Asset Pricing
  Model

                                             2
     Part 1: Individual Securities
 The characteristics of individual securities
 that are of interest are the:
    Expected Return
      This is the return that an individual expects a stock
       to earn.
      Of course, because this is only an expectation, the
       actual return may be either higher or lower.




                                                               3
   Variance and Standard Deviation
     Variance is a measure of the squared deviations of a
      security’s return from its expected return.
     Standard deviation is the square root of the variance.


 Covariance and Correlation (to another security or
  index)
 Covariance is a statistic measuring the

  interrelationship between two securities.
 Alternatively, this relationship can be restated in
  terms of the correlation between the two securities.
 Covariance and correlation are building blocks to
  an understanding of the beta coefficient.

                                                               4
Part 2 : Expected Return, Variance,
          and Covariance
Consider the  following two risky assets:
  Supertech Company and Slowpoke Company.
  There is a 1/4 chance of each state of the
  economy. The return predictions are as follows:
                                          Rate of Return
Scenario           Probability   Supertech(R At ) Slowpoke(R Bt )
Depression           25.0%           -20%              5%
Recession            25.0%           10%              20%
Normal               25.0%           30%              -12%
Boom                 25.0%           50%               9%
                                                                    5
                Variance
 Variance  can be calculated in four steps.
  An additional step is needed to calculate
  standard deviation.
 Step 1: calculate the expected return




                                               6
   Step 1: Expected Return


                                Supertech                      Slowpoke
                           Rate of   Squared              Rate of  Squared
Scenario                    Return Deviation               Return Deviation
Depression                  -20%      0.1406                 5%     0.0000
Recession                    10%      0.0056                20%     0.0210
Normal                       30%      0.0156               -12%     0.0306
Boom                         50%      0.1056                 9%     0.0012
Expected return             17.50%                          5.50%
Variance                     0.0669                         0.0132

    E (rA )  1 / 4 * (20%)  1 / 4 * (10%)  1 / 4 * (30%)  1 / 4 * (50%)
    E (rA )  17.5%

                                                                               7
Step2: Calculate the deviation of each
 possible return from the company’s
 expected return given previously




                                          8
Step   3: Square deviations




                               9
  Step  4: Calculate the average squared
  deviation (variance)

                          Supertech           Slowpoke
                     Rate of   Squared   Rate of  Squared
Scenario              Return Deviation    Return Deviation
Depression            -20%      0.1406      5%     0.0000
Recession              10%      0.0056     20%     0.0210
Normal                 30%      0.0156    -12%     0.0306
Boom                   50%      0.1056      9%     0.0012
Expected return       17.50%               5.50%
Variance               0.0669              0.0132
Standard Deviation     25.9%               11.5%




    .0669  1 / 4(.140625  .005625  .015625  .105625)
                                                             10
 Step5: Calculate standard deviation by
 taking the square root of the variance




                                           11
                          Covariance
              Rate of     R At -E(R A ) Rate of   R Bt -E(R B ) Deviations
Scenario         Return                Return
Depression       -20%      -0.3750       5%        -0.0050    0.001875
Recession         10%      -0.0750       20%        0.1450    -0.010875
Normal            30%       0.1250      -12%       -0.1750    -0.021875
Boom              50%       0.3250       9%         0.0350    0.011375
           R A  17.50%            R B  5.50%                -0.0195


Deviation compares       return in each state to the expected
 return.
Weighted takes the product of the deviations multiplied by
 the probability of that state.



                                                                             12
 Calculate the average value of the four
 states in the last column. The average is
 the covariance.




                                             13
                 Correlation
Divide the covariance by the standard
 deviations of both of the two Securities.

                         Cov(A,B)
  ρ AB  Corr(RA,R B ) 
                          σ Aσ B
             .004875
  ρ AB                  0.1639
         (.2586)(.1150)




                                             14
  Examples of Different Correlation
Coefficients-Perfect Positive Correlation




                                            15
Perfect Negative Correlation




                               16
Zero Correlation




                   17
Part 3: The Return and Risk for
           Portfolios
                          Supertech           Slowpoke
                     Rate of   Squared   Rate of  Squared
Scenario              Return Deviation    Return Deviation
Depression            -20%      0.1406      5%     0.0000
Recession              10%      0.0056     20%     0.0210
Normal                 30%      0.0156    -12%     0.0306
Boom                   50%      0.1056      9%     0.0012
Expected return       17.50%               5.50%
Variance               0.0669              0.0132
Standard Deviation     25.9%               11.5%




                                                             18
 The   expected return on a portfolio is
    simply a weighted average of the expected
    returns on the individual securities .

                             Rate of Return
Scenario             Supertech Slowpoke Portfolio    squared deviation
Depression            -20%         5%       -10.0%       0.0515
Recession              10%         20%      14.0%        0.0002
Normal                 30%        -12%      13.2%        0.0000
Boom                   50%         9%       33.6%        0.0437

Expected return      17.50%      5.50%    12.7%
Variance             0.0669     0.0132    0.0239
Standard Deviation   25.86%     11.50%    15.44%

                                                                     19
Suppose the investor with $100 invests
 $60 in Supertech and $40 in Slowpoke,
 the expected return on the portfolio:

         Returnon portfolio :
            rp  w ArA  w BrB

      Expected returnon portfolio :
        E(rp )  w A E ( rA )  w B E ( rB )

  12.7%  60%  (17.5%)  40%  (5.5%)
                                               20
           Variance of a Portfolio
   Suppose a   portfolio composes of two
    securities, A and B, the variance of the
    portfolio: (Given XA=0.6, XB=0.4)

Var (Portfolio)=

    =0.023851=0.36*0.066875+
    2*[0.6*0.4*(-0.004875)]+0.16*0.013225


                                               21
Standard Deviation of a Portfolio




                                22
       The Diversification Effect
 Itis instructive to compare the standard
  deviation of the portfolio with the
  standard deviation of the individual
  securities.
 The weighted average of the standard
  deviations of the individual securities is:




                                                23
(11.6)




         24
 Suppose ρSuper,Slow  1,
                       the highest possible
    value for correlation. The variance of the
    portfolio is:





                                            (11.9)




                                                 25
 Notethat Equations 11.9 and 11.6 are
 equal. That is, the standard deviation of a
 portfolio’s return is equal to the weighted
 average of the standard deviations of the
 individual returns when ρ = 1.
 => As long as ρ< 1, the standard
    deviation of a portfolio of two
    securities is less than the weighted
    average of the standard deviations of
    the individual securities.
 => In other words, the diversification
    effect applies as long as ρ< l.
                                               26
Part 4: The Efficient Set for Two
             Assets
   The box or “□” in the graph represents a portfolio with
    60% invested in Supertech and 40% invested in
    Slowpoke.




                                                              27
Set of Portfolios Composed of Holdings
       in Supertech and Slowpoke




                                         28
 Thereare a few important points
 concerning this graph:

    The diversification effect occurs whenever the
     correlation between the two securities is below
     1. There is no diversification effect if ρ= 1.

    The point MV represents the minimum
     variance portfolio. This is the portfolio with
     the lowest possible variance.




                                                      29
   An individual contemplating an investment in
    a portfolio of Slowpoke and Supertech faces
    an opportunity set or feasible set.
     He can achieve any point on the curve by selecting
      the appropriate mix between the two securities.
     However, He cannot achieve any point above the
      curve because he cannot increase the return on the
      individual securities, decrease the standard
      deviations of the securities, or decrease the
      correlation between the two securities.




                                                           30
   Note that the curve is backward bending
    between the Slowpoke point and MV.
     This indicates that, for a portion of the feasible set,
      standard deviation actually decreases as we increase
      expected return.
     Actually, backward bending always occurs if ρ  0.


   No investor would want to hold a portfolio
    with an expected return below that of the
    minimum variance portfolio.




                                                          31
 E.g.no investor would choose portfolio 1, because
  this portfolio has less expected return but more
  standard deviation than the minimum variance
  portfolio has.

 Though the entire curve from Slowpoke to
  Supertech is called the feasible set, investors
  consider only the curve from MV to SuPertech.

 The curve from MV to Supertech is called the
  efficient set or the efficient frontier.


                                                      32
   Figure 11.4 (opportunity sets composed of holdings
    in supertech and slowpoke




The lower the   correlation , the more bend there is
 in the curve.
This indicates that the diversification effect rises
 as ρdeclines.                                           33
Portfolios with Various Correlations
return

                                    Supertech
          = -1.0


                                    = 1.0
                    Slowpoke
                                = -0.1639

                                                
          Relationship depends    on correlation coefficient
                             -1.0 <  < +1.0
          If  = +1.0, no risk reduction is possible
          If  = –1.0, complete risk reduction is possible
                                                                34
Part 5: The Efficient Set for
      Many Securities
 Figure11.6 (The feasible set of portfolios
 constructed from many securities)




                                               35
 Figure 11.6  represents the opportunity set
  or feasible set when many securities are
  considered.
 The shaded area represents all the possible
  combinations of expected return and
  standard deviation for a portfolio.
 The upper edge between MV and X is
  called the efficient set.




                                            36
 Any  point below the efficient set would
 receive less expected return and the same
 standard deviation as a point on the
 efficient set.
  For example, consider R on the efficient set
   and W directly below it.
  If W contains the risk level you desire, you

   should choose R instead to receive a higher
   expected return.



                                                  37
 Diversification and Portfolio Risk
 Diversification can   substantially reduce
  the variability of returns without an
  equivalent reduction in expected returns.
 This reduction in risk arises because
  worse than expected returns from one
  asset are offset by better than expected
  returns from another.
 However, there is a minimum level of risk
  that cannot be diversified away, and that is
  the systematic portion.
                                             38
    Portfolio Risk and Number of
                Stocks


               Diversifiable Risk;
               Nonsystematic Risk;
               Firm Specific Risk;
               Unique Risk
                                   Portfolio risk
               Nondiversifiable risk;
               Systematic Risk;
               Market Risk
                                n

                                                    39
           Systematic Risk
 Risk  factors that affect a large number of
  assets
 Also known as non-diversifiable risk or
  market risk
 Includes such things as changes in GDP,
  inflation, interest rates, etc.




                                                40
Unsystematic (Diversifiable) Risk
   Risk factors that affect a limited number of assets
   Also known as unique risk and asset-specific risk
   Includes such things as labor strikes, part shortages,
    etc.
   The risk that can be eliminated by combining assets
    into a portfolio
   If we hold only one asset, or assets in the same
    industry, then we are exposing ourselves to risk that
    we could diversify away.


                                                             41
               Total Risk
 Total risk = systematic risk +
  unsystematic risk
 The standard deviation of returns is a
  measure of total risk.
 For well-diversified portfolios,
  unsystematic risk is very small.
 Consequently, the total risk for a
  diversified portfolio is essentially
  equivalent to the systematic risk.
                                           42
Optimal Portfolio with a Risk-Free Asset

      return
                           100%
                           stocks



      rf
                   100%
                   bonds


                                    
In  addition to stocks and bonds, consider a
  world that also has risk-free securities like
  T-bills.                                        43
 Part 6: Riskless Borrowing and
             Lending
   return
                           100%
                           stocks
             Balanced
             fund


  rf
                   100%
                   bonds
                                    

Now  investors can allocate their money
 across the T-bills and a balanced mutual
 fund.
                                            44
Riskless Borrowing and Lending

     return




      rf

                                       P


 With a risk-free asset available and the efficient
 frontier identified, we choose the capital
 allocation line with the steepest slope.
                                                 45
                    Example
   Ms. Bagwell is considering investing in the
    common stock of Merville Enterprises. In addition,
    Ms. Bagwell will either borrow or lend at the risk-
    free rate. The relevant parameters are these :




 Suppose     Ms. Bagwell chooses to invest a total
    of $l,000, $ 350 of which is to be invested in
    Merville Enterprises and $650 placed in the
    risk-free asset. The expected return on her total
    investment is simply a weighted average of the
    two returns : 0.35*0.14+0.65*0.1=0.114
                                                          46
 Variance of portfolio composed of one
 riskless and one risky asset.




 Thestandard deviation of portfolio
 composed of one riskless and one risky
 asset.



                                          47
Relationship between Expected Return and Risk for a
Portfolio of One Risky Asset and One Riskless Asset




                                                      48
   Suppose that, alternatively, Ms. Bagwell borrows
    $ 200 at the risk-free rate. Combining this with her
    original sum of $l,000, she invests a total of $ l,200
    in Merville. Her expected return would be :
          1.2*0.14+(-0.2*0.1)=14.8%

   The standard deviation is:


    ( is greater than 0.20, the standard deviation of the
    Merville investment, because borrowing increases
    the variability of the investment.)


                                                             49
        The Optimal Portfolio
 The  previous section concerned a portfolio
  formed between one riskless asset and one
  risky asset.
 In reality, an investor is likely to combine an
  investment in the riskless asset with a
  portfolio of risky assets.
 E.g. Figure 11.9, Point Q is in the interior of
  the feasible set of risky securities, and the
  point represents a portfolio of 30 percent in
  AT&T, 45 percent in General Motors (GM),
  and 25 percent in IBM.

                                                    50
51
point 1 on the line represents a portfolio of 70 percent in the
 riskless asset and 30 percent in stocks represented by Q.
An investor with $ 100 choosing point 1 as his portfolio would
 put $70 in the risk-free asset and $ 30 in Q.
This can be restated as $ 70 in the riskless asset, $9 (=
 0.3×$30) in AT&T, $13.50(= 0.45× $30) in GM, and $7.50
 (= 0.25×$30) in IBM.
Point 3 is obtained by borrowing to invest in Q. An investor
 with $100 of her own would borrow $40 from the bank or
 broker to invest $140 in Q. This can be stated as borrowing
 $40 and contributing $100 of her money to invest $42 = 0.3×
 $140 ) in AT&T, $63 (= 0.45×$140 ) in GM, and $35(=0.25×
 $140 ) in IBM.
                                                                   52
 Though any investor can obtain any point on
  line 1, no point on the line is optimal.
 Consider line II, Line II represents portfolios

  formed by combinations of the risk-free asset
  and the securities in A, and point A represents
  a portfolio of risky securities.
 In fact, because line II is tangent to the

  efficient set of risky assets, it provides the
  investor with the best possible opportunities.
 => In other words, line II can be viewed as the
     efficient set of all assets, both risky and
     riskless.


                                                    53
       Part 7: Market Equilibrium

        Expected return
                          M


                   rf


                                      P
 Ina world with homogeneous expectations, M is the
 same for all investors. With the capital allocation line
 identified, all investors choose a point along the
 line—some combination of the risk-free asset and
 the market portfolio M.                               54
      Expected return
                        M


                 rf


                                     P
Where the investor chooses along the Capital Market
Line (CML) depends on his risk tolerance. The big
point is that all investors have the same CML.

                                                      55
Risk When Holding the Market Portfolio
   Researchers have     shown that the best
    measure of the risk of a security in a large
    portfolio is the beta (b)of the security.
   Beta measures the responsiveness of a
    security to movements in the market
    portfolio (i.e., systematic risk).

              Cov( Ri , RM )
       bi 
                 ( RM )
                  2

                                                   56
Estimating b with Regression


    Security Returns

                                Slope = bi
                                     Return on
                                     market %




                       Ri = a i + biRm + ei
                                                 57
The Formula for Beta
        Cov( Ri , RM )
 bi 
           ( RM )
            2




                         58
 One  useful property is that the average beta
  across all securities, when weighted by the
  proportion of each security’s market value to
  that of the market portfolio, is 1.
 That is:




 where X i is the proportion of security i’s
 market value to that of the entire market and
 N is the number of securities in the market.

                                                  59
Part 8: Relationship between Risk
 and Expected Return (CAPM)
 Expected     Return on the Market:

    R M  RF  Market Risk Premium
   Expected return on an individual security:

           Ri  RF  βi  ( R M  RF )

                         Market Risk Premium

 This applies to individual securities held within well-
 diversified portfolios.                                   60
             Expected Return on a Security
 This    formula is called the Capital Asset
     Pricing Model (CAPM):
             Ri  RF  βi  ( R M  RF )
Expected
                    Risk-     Beta of the   Market risk
return on     =             +             ×
                  free rate    security      premium
a security


    Assume bi = 0, then the expected return is RF.
    Assume bi = 1, then Ri  R M

                                                          61
 Equation 11.16   can be represented graphically by
  the upward-sloping line in Figure 11.11.
 Note that the line begins at R F and rises to RM
  when beta is l. This line is frequently called the
  security market line (SML).
                                                       62
         Relationship Between Risk & Return
                   (Equation 11.16)
Expected return


                       Ri  RF  βi  ( R M  RF )

                  RM

                  RF

                             1.0             b


                                                     63
     Relationship Between Risk & Return



           15%
Expected
return




            3%

                                 1.5     b

           β i  1.5   RF  3%    R M  11%
            R i  3%  1.5  (11%  3%)  15%
                                                64
               Example
 The stock of Aardvark Enterprises has a
 beta of 1.5 and that of Zebra Enterprises
 has a beta of 0.7. The risk-free rate is
 assumed to be 3 percent, and the
 difference between the expected return on
 the market and the risk-free rate is
 assumed to be 8.0 percent.




                                             65
 The   expected returns on the two securities
 are:




                                                 66
 Three
     additional points concerning the
 CAPM should be mentioned:

    Linearity: The relationship between expected
     return and beta corresponds to a straight line.
      Figure 11.11, Securities (S,T) lying below the SML
       are overpriced.
      Securities lying above the SML are underpriced.




                                                            67
   Portfolio as well as securities :
     Example, Aardvark and Zebra, the expected return
      on the portfolio is:

                                                       (11.18)


     The beta of the portfolio is simply a weighted
      average of the betas of the two securities:


                                                        (11.19)



     Because the expected return in Equation 11.18 is the
      same as the expected return in Equation 11.19, the
      example shows that the CAPM holds for portfolios
      as well as for individual securities.
                                                             68
   A potential confusion:
     Line II in Figure 11.9 traces the efficient set of
      portfolios formed from both risky assets and the
      riskless asset. Each point on the line represents an
      entire portfolio.
     Under homogeneous expectations, point A in Figure
      11.9 becomes the market portfolio. In this situation ,
      line II is referred to as the capital market line (CML).
     SML differs from line II in at least two ways:
       First , beta appears in the horizontal axis of Figure 11.11,
        but standard deviation appears in the horizontal axis of
        Figure 11.9.
       Second, the SML in Figure 11.11 holds both for all
        individual securities and for all possible portfolios, whereas
        line II in Figure 11.9 holds only for efficient portfolios.

                                                                         69

				
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