Formulae for Calculation of Expected Return and Risk by crg76519

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									Lessons from Capital
   Market History


     Shad Valley



      K. Hartviksen    1
Dollar Returns
  measured in absolute dollars.
 less meaningful than percentage returns because
 they depend on the amount of the original
 investment.
 Bonds give rise to two kinds of returns:
   • Capital gains or capital losses, and
   • Interest
 Stock investments give rise to dollar returns as:
   • Capital gains or capital losses, and
   • Dividends
Holding Period Return
(For investments that yield dividend cash flow returns)



       Ending Price - Beginning Price  Dividend
 HPR 
                   Beginning Price
       P  P0  D1
 HPR  1
           P0
Holding Period Return
(For investments that yield dividend cash flow returns)


 The single period return calculation can be reformulated
 to show illustrate the two forms of investment income : capital gains/losses
 and dividends :

 HPR  capital gain yield  dividend yield
       P  P  D 
 HPR   1 0    1 
        P0   P0 
Holding Period Return
…Illustrated       (For investments that yield dividend cash flow returns)




    Price of stock at the beginning of the year  $21.25
    Price of the stock at the end of the year  $22.10
    Dividend received each quarter  $.10
            P  P  D1 ($22.10  $21.25)  (4)($0.10)
    HPR  1 0            
                 P0                    $21.25
          $0.85  $0.40
    HPR 
             $21.25
           $1.25
    HPR           0.0588  5.9%
          $21.25
The Geometric Average


Measuring Investment Returns
The Bias Inherent in the
Arithmetic Average
 Arithmetic averages can yield incorrect results
 because of the problems of bias inherent in its
 calculation.
  Example
  – Consider an investment that was purchased for $10, rose to
    $20 and then fell back to $10.
  – Let us calculate the HPR in both periods:
                     $20  $10 $10
             HPR1                  100%
                       $10      $10
                     $10  $20  $10
             HPR 2                   50%
                        $20      $20
The Bias Inherent in the
Arithmetic Average
Example Continued ...
   – Consider an investment that was purchased for $10, rose to
     $20 and then fell back to $10.
   – Let us calculate the HPR in both periods:
                            $20  $10 $10
                    HPR1                  100%
                              $10      $10
                            $10  $20  $10
                    HPR 2                   50%
                               $20      $20



   – The arithmetic average return earned on this investment was:
                          100%  50% 50%
               Average                          25%
                                 2           2
The Bias Inherent in the
Arithmetic Average
Example Continued ...
   – The answer is clearly incorrect since the investor started
     with $10 and ended with $10.
   – The correct answer may be obtained through the use of the
     geometric average:                     n
                        GeometricAverage  n          (1  r )  1
                                                     i 1
                                                             i


                                         1/ n
                           n        
                           
                          (1  ri )
                           i 1     
                                                1

                         [(1  100%)(1  (50%))]1/ 2  1
                         [(2)(.5)]1/ 2  1
                         (1)1/ 2  1  1  1  0
Geometric Versus Arithmetic
Average Returns
Consider two investments with the
following realized returns over the past                Holding Period Returns
                                                           IBM       Government
few years:                                     Year      Stock          Bonds

                                                 2000    12.0%        6.0%
                                                 2001    12.0%        6.0%
                                                 2002    12.0%        6.0%
                                                 2003    12.0%        6.0%
                                                 2004    12.0%        6.0%
                                                 2005    12.0%        6.0%




                    If the returns are equal over time, the arithmetic
                    average return will equal the geometric average
                    return.
      Geometric Versus Arithmetic
      Average Returns
                                                                           Holding Period Returns
                                                                              IBM       Government
            Arithmetic Average Return :                           Year      Stock          Bonds
            _
                   HPR1  HPR2  HPR3  HPR4  HPR5  HPR6
            R                        N
                                                                    2000
                                                                    2001
                                                                            12.0%
                                                                            12.0%
                                                                                         6.0%
                                                                                         6.0%
                  12%  12%  12%  12%  12%  12%                 2002    12.0%        6.0%
                                                                   2003    12.0%        6.0%
                                  6
                                                                    2004    12.0%        6.0%
                  72
                     12%                                          2005    12.0%        6.0%
                   6

SAME
ANSWER !   Geometric Average Return :
            _                                                                        1

           G  (1  HPR1 )(1  HPR2 )(1  HPR3 )(1  HPR4 )(1  HPR5 )(1  HPR6 )]6  1
                                                       1
                 (1.12)(1.12)(1.12)(1.12)(1.12)(1.12) 6  1
                 1.973822685.16667  1
                 12%
       Geometric Versus Arithmetic
       Average Returns                                                                      Holding Period Returns
                                                                                               IBM       Government
                                                                            Year             Stock          Bonds
      Now consider volatile returns:
                                                                                    2000      40.0%       11.0%
                  Arithmetic Average Return :                                       2001     -30.0%       4.0%
                                                                                    2002      33.0%       8.0%
                   _
                         HPR1  HPR2  HPR3  HPR4  HPR5  HPR6
                  R                         N
                                                                                    2003       5.0%       3.0%
                                                                                    2004      32.0%       6.0%
                         40%  30%  33%  5%  32%  8%
                                                                                   2005      -8.0%       4.0%
                                        6
                         72
                            12%
                         6                                      Arithmetic Average =              12.0%         6.0%
                                                                     Standard Deviation =        27.71%        3.03%
NOT THE
SAME              Geometric Average Return :
                                                                                             1
ANSWER !           _

                  G  (1  HPR1 )(1  HPR2 )(1  HPR3 )(1  HPR4 )(1  HPR5 )(1  HPR6 )]6  1
                                                             1
                        (1.40)(0.70)(1.33)(1.05)(1.32)(.92) 6  1
                        1.661991408.16667  1
                        8.84%



      Volatility of returns over time eats away at your realized returns!!!
      The greater the volatility the greater the difference between the arithmetic
      and geometric average. Arithmetic average OVERSTATES the return!!!
Measuring Returns
 When you are trying to find average returns,
 especially when those returns rise and fall, always
 remember to use the geometric average.
 The greater the volatility of returns over time, the
 greater the difference you will observe between the
 geometric and arithmetic averages.
 Of course, there are limitations inherent in the use of
 geometric averages.
Historical Returns
                    Average   Standard    Risk
                    Return    Deviation   Premium
Canadian Equities    13.20%    16.62%        7.16%
U.S. Equities        15.59%    16.86%        9.55%
Long-Term bonds       7.64%    10.57%         1.60
Treasury bills        6.04%     4.04%         0.00
Small cap stocks     14.79%    23.68%         8.75
Inflation             4.25%     3.51%        -1.79
                                                      The historical pattern of returns exhibit the classic
                                                      risk-return tradeoff


Historical Returns
                            Historical Averages - Risk and Return

                 16
                                                                             Small cap
                 14                                 Canadian
                                                    Equities
Percent Return




                 12
                 10
                  8                    Long-Term
                                         Bonds
                  6
                  4       Treasury Bills
                  2
                  0
                      0            5           10          15                 20                 25
                                            Standard Deviation
Capital Asset Pricing Model
Return
         Required return = Rf + bs [kM - Rf]
  %


 km
                          Market         Security Market
                          Premium        Line
                          for risk
  Rf
                           Real Return
                          Premium for expected inflation

                   BM=1.0            Beta Coefficient
                                                           6
Measurement of Risk in an Isolated
Asset Case
 The dispersion of returns from the mean
 return is a measure of the riskiness of an
 investment.
 This dispersion can be calculated using:
    Variance (an ‘absolute’ measure of dispersion
     expressed in units squared)
    Standard Deviation (an ‘absolute’ measure of
     dispersion expressed in the same units as the mean)
    Coefficient of Variation (this is a ‘relative’ measure
     of dispersion…it is a ratio of the standard deviation
     divided by the mean)
Ex Post and Ex Ante
Calculations

 Returns and risk can be calculated after-the-
 fact (ie. You use actual realized return data)
 This is known as an ex post calculation.
 Or you can use forecast data…this is an ex
 ante calculation.
Standard Deviation
 The formula for the standard deviation
 when analyzing sample data (realized
 returns) is:  n

                            (ki  ki )         2


                    i 1
                               n 1
   Where k is a realized return on the stock and n is the
   number of returns used in the calculation of the mean.
Standard Deviation
 The formula for the standard deviation when
 analyzing forecast data (ex ante returns) is:
              n
           (k
             i 1
                    i    k i ) Pi
                              2



 it is the square root of the sum of the squared
 deviations away from the expected value.
   Forecasting Risk and Return for
   the Individual Asset

                                     S toc k A                     D e via t io n s S q u a re d       W e ig h t e d
                   P ro b a b ilit y E x . R e t . W t d . R e t . fro m m e a n D e via t io n s S q . D e v.
R e c e s s io n       0.150            0.02              0.0030     -8 . 2 0 0 %    0.006724           0.001009
N o rm a l             0.600            0.09              0.0540     -1 . 2 0 0 %    0.000144           8 . 6 4 E -0 5
B oom                  0.250            0.18              0.0450      7.800%         0.006084           0.001521
                   E x p e c t e d R e t u rn =          10.20%                     V a ria n c e = 0 . 0 0 2 6 1 6
                                                                                    S t d . D e v. =        5.11%




                                                  K. Hartviksen
                      A Normal Probability Distribution

                      The area under the curve bounded by
Probability           -1 and +1 σ is equal to 68%



                  - 1 standard                 + 1 standard
              deviation away from           deviation away from
                    the mean                     the mean




                                    13.2%                         Return on Large
                                                                  Cap Stocks
Finding the Probability of an Event
using Z-value tables

 You can find the number of
 standard deviations away
 from the mean that a point of
                                      (point of interest) - (mean)
 interest lies using the         z
 following ‘z’ value formula:             standard deviation
                                      x
                                 z
 Once you know ‘z’ then you            
 can find the areas under the
 normal curve using the z
 value table found on the
 following slide.
 Values of the Standard Normal Distribution Function
     Z        0.00     0.01    0.02    0.03     0.04    0.05    0.06     0.07    0.08    0.09


    0.0       .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359
    0.1       .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753
    0.2       .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141
    0.3       .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517
    0.4       .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879
    0.5       .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224
    0.6       .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549
    0.7       .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852
    0.8       .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133
    0.9       .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389
    1.0       .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621
    1.1       .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830
    1.2       .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015
    1.3       .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177
    1.4       .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319
    1.5       .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441
    1.6       .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545
    1.7       .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633
    1.8       .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706
    1.9       .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767
    2.0       .4773 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817
    2.1       .4821 .4826 .4830 .4834 .4838 .4842 .4846 .4850 .4854 .4857
    2.2       .4861 .4864 .4868 .4871 .4875 .4878 .4881 .4884 .4887 .4890
    2.3       .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 .4913 .4916
    2.4       .4918 .4920 .4922 .4925 .4927 .4929 .4931 .4932 .4934 .4936
    2.5       .4938 .4940 .4941 .4943 .4945 .4946 .4948 .4949 .4951 .4952
    2.6       .4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 .4963 .4964
    2.7       .4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 .4973 .4974
    2.8       .4974 .4975 .4976 .4977 .4977 .4978 .4979 .4979 .4980 .4981
    2.9       .4981 .4982 .4982 .4982 .4984 .4984 .4985 .4985 .4986 .4986
    3.0       .4987 .4987 .4987 .4988 .4988 .4989 .4989 .4989 .4990 .4990
z is the number of standard deviations from the mean. Some area tables are set up to indicate the area to the left or right of
the point of interest. In this table, we indicate the area between the mean and the point of interest.
MPT – Modern Portfolio Theory




          Shad Valley
             K. Hartviksen   1
Risk and Return - MPT
 Prior to the establishment of Modern Portfolio Theory,
 most people only focused upon investment returns…they
 ignored risk.

 With MPT, investors had a tool that they could use to
 dramatically reduce the risk of the portfolio without a
 significant reduction in the expected return of the
 portfolio.




                                                           3
Correlation
  The degree to which the returns of two
  stocks co-move is measured by the
  correlation coefficient.
  The correlation coefficient between the
  returns on two securities will lie in the
  range of +1 through - 1.
  +1 is perfect positive correlation.
  -1 is perfect negative correlation.
                                              10
    Perfect Negatively Correlated Returns
    over Time
Returns
  %
                                    A two-asset portfolio
                                    made up of equal parts
                                    of Stock A and B would
                                    be riskless. There
                                    would be no variability
                                    of the portfolios returns
  10%
                                    over time.




                                       Returns on Stock A
                                       Returns on Stock B
                                       Returns on Portfolio
          1994      1995     1996      Time          11
Ex Post Portfolio Returns
Simply the Weighted Average of Past Returns

            n
     R p   xi Ri
           i 1

     Where :
           xi  relative weight of asset i
           Ri  return on asset i

                   K. Hartviksen               5
                                              14
Ex Ante Portfolio Returns
Simply the Weighted Average of Expected Returns




              Relative       Expected Weighted
              Weight          Return   Return
 Stock X       0.400           8.0%        0.03
 Stock Y       0.350           15.0%       0.05
 Stock Z       0.250           25.0%       0.06
           Expected Portfolio Return =  14.70%



                   K. Hartviksen                   5
                                                  14
Grouping Individual Assets into
Portfolios
 The riskiness of a portfolio that is made of different risky
 assets is a function of three different factors:
 –   the riskiness of the individual assets that make up the portfolio
 –   the relative weights of the assets in the portfolio
 –   the degree of comovement of returns of the assets making up the
     portfolio
 The standard deviation of a two-asset portfolio may be
 measured using the Markowitz model:



 p   w   w  2 wA wB  A, B A B
            2
            A
                 2
                 A
                         2
                         B
                             2
                             B
       Risk of a Three-asset Portfolio
      The data requirements for a three-asset portfolio grows
      dramatically if we are using Markowitz Portfolio selection formulae.


      We need 3 (three) correlation coefficients between A and B; A and
      C; and B and C.
                                             A
                                     ρa,b          ρa,c
                                     B               C
                                            ρb,c



 p   A wA   B wB   C wC  2wA wB  A, B A B  2wB wC  B ,C B C  2wA wC  A,C A C
        2 2      2 2      2 2
Risk of a Four-asset Portfolio

The data requirements for a four-asset portfolio grows dramatically
if we are using Markowitz Portfolio selection formulae.


We need 6 correlation coefficients between A and B; A and C; A
and D; B and C; C and D; and B and D.

                               A
                     ρa,b             ρa,d
                               ρa,c
                    B                    D
                        ρb,d
                     ρb,c             ρc,d
                               C
Diversification Potential
  The potential of an asset to diversify a portfolio is
  dependent upon the degree of co-movement of returns of
  the asset with those other assets that make up the
  portfolio.
  In a simple, two-asset case, if the returns of the two assets
  are perfectly negatively correlated it is possible
  (depending on the relative weighting) to eliminate all
  portfolio risk.
  This is demonstrated through the following chart.
Example of Portfolio
Combinations and Correlation
                                                                    Perfect
                Expected     Standard    Correlation                Positive
     Asset       Return      Deviation   Coefficient             Correlation –
       A          5.0%        15.0%           1                        no
       B         14.0%        40.0%                              diversification

    Portfolio Components                 Portfolio Characteristics
                                         Expected       Standard
   Weight of A Weight of B                Return        Deviation
    100.00%      0.00%                     5.00%         15.0%
     90.00%      10.00%                    5.90%         17.5%
     80.00%      20.00%                    6.80%         20.0%
     70.00%      30.00%                    7.70%         22.5%
     60.00%      40.00%                    8.60%         25.0%
     50.00%      50.00%                    9.50%         27.5%
     40.00%      60.00%                   10.40%         30.0%
     30.00%      70.00%                   11.30%         32.5%
     20.00%      80.00%                   12.20%         35.0%
     10.00%      90.00%                   13.10%         37.5%
     0.00%      100.00%                   14.00%         40.0%
Example of Portfolio
Combinations and Correlation
                                                                    Positive
                Expected     Standard    Correlation             Correlation –
     Asset       Return      Deviation   Coefficient                 weak
       A          5.0%        15.0%          0.5                 diversification
       B         14.0%        40.0%                                 potential

    Portfolio Components                 Portfolio Characteristics
                                         Expected       Standard
   Weight of A Weight of B                Return        Deviation
    100.00%      0.00%                     5.00%         15.0%
     90.00%      10.00%                    5.90%         15.9%
     80.00%      20.00%                    6.80%         17.4%
     70.00%      30.00%                    7.70%         19.5%
     60.00%      40.00%                    8.60%         21.9%
     50.00%      50.00%                    9.50%         24.6%
     40.00%      60.00%                   10.40%         27.5%
     30.00%      70.00%                   11.30%         30.5%
     20.00%      80.00%                   12.20%         33.6%
     10.00%      90.00%                   13.10%         36.8%
     0.00%      100.00%                   14.00%         40.0%
Example of Portfolio
Combinations and Correlation
                                                                       No
                Expected     Standard    Correlation             Correlation –
     Asset       Return      Deviation   Coefficient                 some
       A          5.0%        15.0%           0                  diversification
       B         14.0%        40.0%                                 potential

    Portfolio Components                 Portfolio Characteristics
                                         Expected       Standard
   Weight of A Weight of B                Return        Deviation
    100.00%      0.00%                     5.00%         15.0%        Lower
     90.00%      10.00%                    5.90%         14.1%        risk than
     80.00%      20.00%                    6.80%         14.4%        asset A
     70.00%      30.00%                    7.70%         15.9%
     60.00%      40.00%                    8.60%         18.4%
     50.00%      50.00%                    9.50%         21.4%
     40.00%      60.00%                   10.40%         24.7%
     30.00%      70.00%                   11.30%         28.4%
     20.00%      80.00%                   12.20%         32.1%
     10.00%      90.00%                   13.10%         36.0%
     0.00%      100.00%                   14.00%         40.0%
Example of Portfolio
Combinations and Correlation
                                                                    Negative
                Expected     Standard    Correlation             Correlation –
     Asset       Return      Deviation   Coefficient                 greater
       A          5.0%        15.0%         -0.5                 diversification
       B         14.0%        40.0%                                 potential

    Portfolio Components                 Portfolio Characteristics
                                         Expected       Standard
   Weight of A Weight of B                Return        Deviation
    100.00%      0.00%                     5.00%         15.0%
     90.00%      10.00%                    5.90%         12.0%
     80.00%      20.00%                    6.80%         10.6%
     70.00%      30.00%                    7.70%         11.3%
     60.00%      40.00%                    8.60%         13.9%
     50.00%      50.00%                    9.50%         17.5%
     40.00%      60.00%                   10.40%         21.6%
     30.00%      70.00%                   11.30%         26.0%
     20.00%      80.00%                   12.20%         30.6%
     10.00%      90.00%                   13.10%         35.3%
     0.00%      100.00%                   14.00%         40.0%
Example of Portfolio
Combinations and Correlation                                         Perfect
                                                                    Negative
                Expected     Standard    Correlation             Correlation –
     Asset       Return      Deviation   Coefficient                greatest
       A          5.0%        15.0%          -1                  diversification
       B         14.0%        40.0%                                 potential

    Portfolio Components                 Portfolio Characteristics
                                         Expected       Standard
   Weight of A Weight of B                Return        Deviation
    100.00%      0.00%                     5.00%         15.0%
     90.00%      10.00%                    5.90%          9.5%         Risk of the
     80.00%      20.00%                    6.80%          4.0%         portfolio is
                                                                       almost
     70.00%      30.00%                    7.70%          1.5%
                                                                       eliminated at
     60.00%      40.00%                    8.60%          7.0%
                                                                       70% asset A
     50.00%      50.00%                    9.50%         12.5%
     40.00%      60.00%                   10.40%         18.0%
     30.00%      70.00%                   11.30%         23.5%
     20.00%      80.00%                   12.20%         29.0%
     10.00%      90.00%                   13.10%         34.5%
     0.00%      100.00%                   14.00%         40.0%
       Diversification of a Two Asset Portfolio Demonstrated Graphically


                                         The Effect of Correlation on Portfolio Risk:
                                                    The Two-Asset Case




Expected Return                                                                                  B



                                        AB = -0.5
         12%
                       AB = -1




          8%
                                                                                      AB = 0

                                                                  AB= +1



                                                     A
          4%




          0%

                  0%              10%                    20%                30%                      40%

                                                                            Standard Deviation
An Exercise using T-bills, Stocks and Bonds
                Base Data:                        Stocks    T-bills  Bonds
                          Expected Return         12.73383 6.151702 7.007872
                          Standard Deviation         0.168     0.042   0.102

                           Correlation Coefficient Matrix:
                                       Stocks              1      -0.216   0.048
                                       T-bills        -0.216       1.000   0.380
                                       Bonds           0.048       0.380   1.000

                Portfolio Combinations:

                            Weights                            Portfolio
                                                 Expected          Standard
     Combination Stocks   T-bills    Bonds       Return   Variance Deviation
         1         100.0%       0.0%     0.0%      12.7    0.0283   16.8%
         2          90.0%     10.0%      0.0%      12.1    0.0226   15.0%
         3          80.0%     20.0%      0.0%      11.4    0.0177   13.3%
         4          70.0%     30.0%      0.0%      10.8    0.0134   11.6%
         5          60.0%     40.0%      0.0%      10.1    0.0097    9.9%
         6          50.0%     50.0%      0.0%       9.4    0.0067    8.2%
         7          40.0%     60.0%      0.0%       8.8    0.0044    6.6%
         8          30.0%     70.0%      0.0%       8.1    0.0028    5.3%
         9          20.0%     80.0%      0.0%       7.5    0.0018    4.2%
        10          10.0%     90.0%      0.0%       6.8    0.0014    3.8%
        11           0.0%    100.0%      0.0%       6.2    0.0017    4.2%
Results Using only Three Asset Classes

                                        Attainable Portfolio Combinations
                                              and Efficient Set of Portfolio Combinations


                                     14.0
                                                                    Efficient Set
     Portfolio Expected Return (%)




                                     12.0
                                                  Minimum
                                                  Variance
                                     10.0
                                                  Portfolio
                                      8.0
                                      6.0
                                      4.0
                                      2.0
                                      0.0
                                            0.0               5.0             10.0   15.0   20.0
                                                  Standard Deviation of the Portfolio (%)
                           Plotting Achievable Portfolio Combinations




Expected Return on
the Portfolio



         12%




          8%




          4%




          0%

                0%   10%               20%                 30%                  40%

                                                          Standard Deviation of the Portfolio
                           The Efficient Frontier




Expected Return on
the Portfolio



         12%




          8%




          4%




          0%

                0%   10%     20%                    30%                   40%

                                                    Standard Deviation of the Portfolio
                           The Capital Market Line
                                                                                    Capital
                                                                                  Market Line

Expected Return on
the Portfolio



         12%




          8%




          4%

 Risk-free
 rate
          0%

                0%   10%      20%                30%                   40%

                                                 Standard Deviation of the Portfolio
                                  The Capital Market Line and Iso Utility Curves


                      Highly
                                                           A risk-
                       Risk
                                                           taker
Expected Return on    Averse
the Portfolio
                     Investor

         12%



                                                                                                       Capital
          8%
                                                                                                     Market Line


          4%

 Risk-free
 rate
          0%

                0%          10%                 20%                  30%                   40%

                                                                     Standard Deviation of the Portfolio
                                 The Capital Market Line and Iso Utility Curves



                      The risk-                           A risk-taker’s
                        taker’s                            utility curve
Expected Return on
the Portfolio          optimal
                       portfolio
         12%
                     combination


                                                                                                     Capital
          8%
                                                                                                   Market Line


          4%

 Risk-free
 rate
          0%

                0%         10%                 20%                 30%                   40%

                                                                   Standard Deviation of the Portfolio
CML versus SML
 Please notice that the CML is used to
 illustrate all of the efficient portfolio
 combinations available to investors.
 It differs significantly from the SML that is
 used to predict the required return that
 investors should demand given the riskiness
 (beta) of the investment.
Data Limitations
 Because of the need for so much data, MPT
 was a theoretical idea for many years.
 Later, a student of Markowitz, named
 William Sharpe worked out a way around
 that…creating the Beta Coefficient as a
 measure of volatility and then later
 developing the CAPM.
CAPM

The Capital Asset Pricing Model was the
work of William Sharpe, a student of Harry
Markowitz at the University of Chicago.
CAPM is an hypothesis …
Capital Asset Pricing Model
Return
         Required return = Rf + bs [kM - Rf]
  %


 km
                          Market         Security Market
                          Premium        Line
                          for risk
  Rf
                           Real Return
                          Premium for expected inflation

                   BM=1.0            Beta Coefficient
                                                           6
CAPM
 This model is an equilibrium based model.
 It is called a single-factor model because the slope of the SML is
 caused by a single measure of risk … the beta.
 Although this model is a simplification of reality…it is robust (it
 explains much of what we see happening out there) and it enjoys
 widespread use in a great variety of applications.
 Although it is called a ‘pricing model’ there are not prices on that
 graph….only risk and return.
 It is called a pricing model because it can be used to help us
 determine appropriate prices for securities in the market.
Risk
 Risk is the chance of harm or loss; danger.
 We know that various asset classes have
 yielded very different returns in the past:
Historical Returns and Standard Deviations
1948 - 941



                            Average Return   Standard Deviation
Canadian common stock           12.73%            16.81%
U.S. common stock (Cdn $)       14.09             16.60
Long term bonds                  7.01             10.20
Small cap stocks                15.67             24.40
Inflation                        4.52              3.54
Treasury bills                   6.15              4.17

___________________
1The Alexander Group
Risk and Return
 The foregoing data point out that those asset
 classes that have offered the highest rates of
 return, have also offered the highest risk levels as
 measured by the standard deviation of returns.
 The CAPM suggests that investors demand
 compensation for risks that they are exposed
 to…and these returns are built into the decision-
 making process to invest or not.
Capital Asset Pricing Model
Return
         Required return = Rf + bs [kM - Rf]
  %


 km
                          Market         Security Market
                          Premium        Line
                          for risk
  Rf
                           Real Return
                          Premium for expected inflation

                   BM=1.0            Beta Coefficient
                                                           6
CAPM
 The foregoing graph shows that investors:
 –   demand compensation for expected inflation
 –   demand a real rate of return over and above expected inflation
 –   demand compensation over and above the risk-free rate of return for any
     additional risk undertaken.

 We will make the case that investors don’t need
 compensation for all of the risk of an investment
 because some of that risk can be diversified away.
 Investors require compensation for risk they can’t
 diversify away!
Beta Coefficient
The beta is a measure of systematic risk of an investment.
Systematic risk is the only relevant risk to a diversified investor
according to the CAPM since all other risk may be diversified away.
Total risk of an investment is measured by the securities’ standard
deviation of returns.
According to the CAPM total risk may be broken into two
parts…systematic (non-diversifiable) and unsystematic (diversifiable)

TOTAL RISK = SYSTEMATIC RISK + UNSYSTEMATIC RISK

The beta can be determined by regressing the holding period returns
(HPRs) of the security over 30 periods against the returns on the
overall market.
                                                                      7
Measuring Risk of the Individual
Security
 Risk is the possibility that the actual return that will be
 realized, will turn out to be different than what we expect
 (or have forecast).
 This can be measured using standard statistical measures
 of dispersion for probability distributions. They include:
 –   variance
 –   standard deviation
 –   coefficient of variation
Standard Deviation
 The formula for the standard deviation
 when analyzing population data (realized
 returns) is:

                 n

                      (ki  ki )   2


              i 1
                        n 1
Standard Deviation
 The formula for the standard deviation when
 analyzing forecast data (ex ante returns) is:

             n
          (k
            i 1
                   i    k i ) Pi
                             2


 it is the square root of the sum of the squared
 deviations away from the expected value.
Using Forecasts to Estimate Beta
The formula for the beta coefficient for a stock ‘s’ is:

                          Cov (k s k M )
                   Bs 
                        Variance (k M )
Obviously, the calculate a beta for a stock, you must first
  calculate the variance of the returns on the market
  portfolio as well as the covariance of the returns on the
  stock with the returns on the market.
Systematic Risk
 The returns on most assets in our economy are influenced by the
 health of the ‘system’
 Some companies are more sensitive to systematic changes in the
 economy. For example durable goods manufacturers.
 Some companies do better when the economy is doing poorly (bill
 collection agencies).
 The beta coefficient measures the systematic risk that the security
 possesses.
 Since non-systematic risk can be diversified away, it is irrelevant to
 the diversified investor.
Systematic Risk
 We know that the economy goes through
 economic cycles of expansion and
 contraction as indicated in the following:
Canada’s Business cycles from 1873-1992


Trough to ExpansionPeak to Contraction
(months from trough to peak)(months from peak to trough)

                                                 Nov 1873    66
May 1879                 38                      July 1882   32
Mar 1885                 23                      Feb 1887    12
Feb 1888                 29                      July 1890   9
Mar 1891                 23                      Apr 1893    13
Mar 1894                 17                      Aug 1895    12

Aug 1896                 44                      Apr 1900    10
Feb 1901                 22                      Dec 1902    18
June 1904                30                      Dec 1906    19
July 1908                20                      Mar 1910    16
July 1911                16                      Nov 1912    25

Jan 1915                 36(WWI)                 Jan 1918    15
Apr 1919                 14                      June 1920   15
Sep 1921                 21                      June 1923   14
Aug 1924                 56                      Apr 1929    47 (Depression)
Mar 1933                 52                      July 1937   15 (Depression)

Oct 1938                 80(WWII)                June 1945   8
Feb 1946                 33                      Oct 1948    11
Sep 1949                 44(Korean War)          May 1953    14
July 1954                31                      Feb 1957    12
Feb 1958                 26                      Apr 1960    10

Feb 1961                 160                     June 1974   10
Apr 1975                 58                      Feb 1980    6
July 1980                12                      July 1981   6
Nov 1982                 89                      Apr 1990    22
Feb 1992
Companies and Industries
 Some industries (and by implication the companies that
 make up the industry) move in concert with the expansion
 and contraction of the economy.
 Some lead the overall economy. (stock market)
 Some lag the overall economy. (ie. automotive industry)
Amount of Systematic Risk
 Some industries may find that their fortunes are positively
 correlated with the ebb and flow of the overall
 economy…but that this relationship is very insignificant.
 An example might be Imperial Tobacco. This firm does
 have a positive beta coefficient, but very little of the
 returns of this company can be explained by the beta.
 Instead, most of the variability of returns on this stock is
 from diversifiable sources.
 A Characteristic line for Imperial Tobacco would show a
 very wide dispersion of points around the line. The R2
 would be very low (.05 = 5% or lower).
Characteristic Line for Imperial
Tobacco
                       Characteristic
     Returns on
                       Line for Imperial
     Imperial
                       Tobacco
     Tobacco %




                            Returns on the
                            Market %
                            (TSE 300)
High R2
 An R2 that approaches 1.00 (or 100%) indicates that the
 characteristic (regression) line explains virtually all of the
 variability in the dependent variable.
 This means that virtually of the risk of the security is
 ‘systematic’.
 This also means that the regression model has a strong
 predictive ability. … if you can predict what the market
 will do…then you can predict the returns on the stock
 itself with a great deal of accuracy.
Characteristic Line General
Motors
                      Characteristic
    Returns on
                      Line for GM
    General
    Motors %          (high R2)




                           Returns on the
                           Market %
                           (TSE 300)
Diversifiable Risk
(non-systematic risk)

   Examples of this type of risk include:
   –   a single company strike
   –   a spectacular innovation discovered through the company’s R&D
       program
   –   equipment failure for that one company
   –   management competence or management incompetence for that
       particular firm
   –   a jet carrying the senior management team of the firm crashes
   –   the patented formula for a new drug discovered by the firm.
   Obviously, diversifiable risk is that unique factor that
   influences only the one firm.
Partitioning Risk under the
CAPM
    Remember that the CAPM assumes that total risk (variability of a security’s
    returns) can be separated into two distinct components:

Total risk = systematic risk + unsystematic risk
 100% = 40% + 60%                                                      (GM)
or
100% = 5% + 95%                                          (Imperial Tobacco)

Obviously, if you were to add Imperial Tobacco to your portfolio, you could
    diversify away much of the risk of your portfolio. (Not to mention the fact
    that Imperial has realized some very high rates of return in addition to
    possessing little systematic risk!)
Using the CAPM to Price Stock
 The CAPM is a ‘fundamental’ analyst’s tool to estimate
 the ‘intrinsic’ value of a stock.
 The analyst needs to measure the beta risk of the firm by
 using either historical or forecast risk and returns.
 The analyst will then need a forecast for the risk-free rate
 as well as the expected return on the market.
 These three estimates will allow the analyst to calculate
 the required return that ‘rational’ investors should expect
 on such an investment given the other benchmark returns
 available in the economy.
Required Return
 The return that a rational investor should demand is
 therefore based on market rates and the beta risk of the
 investment.
 To find this, you solve for the required return in the
 CAPM:



     R(k )  R f  b s [k M  R f ]
 This is a formula for the straight line that is the SML.
Security Market Line
 This line can easily be plotted.
 Draw Cartesian coordinates.
 Plot the yield on 91-day Government of Canada Treasury Bills as the
 risk-free rate of return on the vertical axis.
 On the horizontal axis set a scale that includes Beta=1 (this is the beta
 of the market)
 Plot the point in risk-return space that represents your expected return
 on the market portfolio at beta =1
 Draw a straight line to connect the two points.
 Plot the required and expected returns for the stock at it’s beta.
    Plot the Risk-Free Rate
Return
%




    Rf




                  1.0   Beta Coefficient
    Plot Expected Return on the
    Market Portfolio
Return
%

km =12%




Rf = 4%




                  1.0   Beta Coefficient
    Draw the Security Market Line
Return                        SML
%

km =12%




Rf = 4%




                  1.0   Beta Coefficient
        Plot Required Return
        (Determined by the formula = Rf + bs[kM - Rf]


   Return                                                    SML
   %
R(k) = 13.6%

   km =12%                                              R(k) = 4% + 1.2[8%] = 13.6%




   Rf = 4%




                                        1.0     1.2 Beta Coefficient
        Plot Expected Return
        E(k) = weighted average of possible returns


   Return                                                    SML
   %
R(k) = 13.6%
                                                        R(k) = 4% + 1.2[8%] = 13.6%
   km =12%

                                                      E(k)

   Rf = 4%




                                        1.0     1.2 Beta Coefficient
        If Expected = Required Return
        The stock is properly (fairly) priced in the market. It is in
        EQUILIBRIUM.

   Return                                                        SML
   %
R(k) = 13.6%
                                                           R(k) = 4% + 1.2[8%] = 13.6%
   km =12%
                                                          E(k)



   Rf = 4%




                                           1.0      1.2 Beta Coefficient
        If E(k) < R(k)
        The stock is over-priced. The analyst would issue a sell recommendation in anticipation of
        the market becoming ‘efficient’ to this fact. Investors may ‘short’ the stock to take advantage
        of the anticipated price decline.

   Return                                                                   SML
   %
R(k) = 13.6%
                                                                      R(k) = 4% + 1.2[8%] = 13.6%
   km =12%


E(k) = 9%                                                         E(k)

   Rf = 4%




                                                   1.0       1.2 Beta Coefficient
Let’s Look at the Pricing
Implications
In this example:
– E(k) = 9%
– R(k) = 13.6%
If the market expects the company to pay a dividend of $1.00 next year, and
the stock is currently offering an expected return of 9%, then it should be
priced at:
                          d1
                 P0 
                        E (k s )
                        $1.00
                 P0           $11.11
                         .09
But, given the other rates in the economy and our judgement about the
riskiness of this investment we think that this stock should be worth:
                                           $1.00
                                    P0           $7.35
                                           .136
Practical Use of the CAPM

 Regulated utilities justify rate increases using the model to demonstrate that
 their shareholders require an appropriate return on their investment.
 Used to price initial public offerings (IPOs)
 Used to identify over and under value securities
 Used to measure the riskiness of securities/companies
 Used to measure the company’s cost of capital. (The cost of capital is then
 used to evaluate capital expansion proposals).
 The model helps us understand the variables that can affect stock prices…and
 this guides managerial decisions.
Rf rises
                                SML2
Return
  %                             SML1

 ks2

 ks1
                    Rising interest rates will
                    cause all required rates of
  Rf2               return to increase and this
  Rf1               will force down stock and
                    bond prices.



           Bs=1.2   Beta Coefficient
The Slope of The SML rises
(indicates growing pessimism about the future of the economy)
                                                       SML2
Return
  %                                                     SML1

 ks2
                                                Growing pessimism
                                                will cause investors to
 ks1
                                                demand greater
                                                compensation for
                                                taking on risk…this
                                                will mean prices on
  Rf1
                                                high beta stocks will
                                                fall more than low
                                                beta stocks.

                               Bs=1.2        Beta Coefficient

								
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