Risk Return and Portfolio Theory

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Risk Return and Portfolio Theory Powered By Docstoc
					Portfolio Construction,
Management & Protection
Robert Strong

Chapter 5 – The Mathematics of
Diversification

     Prepared by
    Ken Hartviksen
   CHAPTER 5
The Mathematics of
  Diversification
                     Lecture Agenda

•   Learning Objectives
•   Important Terms
•   Measurement of Returns
•   Measuring Risk
•   Expected Return and Risk for Portfolios
•   The Efficient Frontier
•   Diversification
•   The New Efficient Frontier
•   The Capital Asset Pricing Model
•   The CAPM and Market Risk
•   Alternative Asset Pricing Models
•   Summary and Conclusions
                 CHAPTER 5 – The Mathematics of Diversification   8-3
               Learning Objectives


• The difference among the most important types of
  returns
• How to estimate expected returns and risk for
  individual securities
• What happens to risk and return when securities are
  combined in a portfolio
• What is meant by an “efficient frontier”
• Why diversification is so important to investors
• How to measure risk and return in portfolios



               CHAPTER 5 – The Mathematics of Diversification   8-4
               Important Chapter Terms
•   Arithmetic mean                           •    Mark to market
•   Attainable portfolios                     •    Market risk
•   Capital gain/loss                         •    Minimum variance frontier
•   Correlation coefficient                   •    Minimum variance portfolio
•   Covariance                                •    Modern portfolio theory
•   Day trader                                •    Naïve or random diversification
•   Diversification                           •    Paper losses
•   Efficient frontier                        •    Portfolio
•   Efficient portfolios                      •    Range
•   Ex ante returns                           •    Risk averse
•   Ex post returns                           •    Standard deviation
•   Expected returns                          •    Total return
•   Geometric mean                            •    Unique (or non-systematic) or
•   Income yield                                   diversifiable risk
                                              •    Variance


                    CHAPTER 5 – The Mathematics of Diversification             8-5
Introduction to Risk and Return

 Risk, Return and Portfolio Theory
            Introduction to Risk and Return
Risk and return are the two most
   important attributes of an
   investment.

Research has shown that the two
   are linked in the capital                  Return
   markets and that generally,                    %
   higher returns can only be
   achieved by taking on greater
   risk.                                                                  Risk Premium


Risk isn‟t just the potential loss of
   return, it is the potential loss of             RF
   the entire investment itself                                           Real Return

   (loss of both principal and                                            Expected Inflation Rate
   interest).
                                                                                         Risk
Consequently, taking on additional
   risk in search of higher returns
   is a decision that should not be
   taking lightly.


                         CHAPTER 5 – The Mathematics of Diversification                    8-7
     Measuring Returns

Risk, Return and Portfolio Theory
              Measuring Returns
                            Introduction


Ex Ante Returns
• Return calculations may be done „before-the-
  fact,‟ in which case, assumptions must be
  made about the future

Ex Post Returns
• Return calculations done „after-the-fact,‟ in
  order to analyze what rate of return was
  earned.


             CHAPTER 5 – The Mathematics of Diversification   8-9
                     Measuring Returns
                                  Introduction


        You know that the constant growth DDM can be decomposed into the
        two forms of income that equity investors may receive, dividends and
        capital gains.


      D1 
kc     g   Income / Dividend Yield   Capital Gain (or loss) Yield 
      P0 

     WHEREAS

        Fixed-income investors (bond investors for example) can expect to
        earn interest income as well as (depending on the movement of
        interest rates) either capital gains or capital losses.


                  CHAPTER 5 – The Mathematics of Diversification             8 - 10
                  Measuring Returns
                                Income Yield


• Income yield is the return earned in the form of
  a periodic cash flow received by investors.
• The income yield return is calculated by the
  periodic cash flow divided by the purchase
  price.
                                            CF1
          [8-1]          Income yield 
                                             P0




     Where CF1 = the expected cash flow to be received
            P0 = the purchase price

                  CHAPTER 5 – The Mathematics of Diversification   8 - 11
                 Measuring Returns
        Common Share and Long Canada Bond Yield Gap

– Table on this slide illustrates the income yield gap between stocks and bonds
  over recent decades.
– The main reason that this yield gap has varied so much over time is that the
  return to investors is not just the income yield but also the capital gain (or loss)
  yield as well.

                  Average Yield Gap between Stocks and Bonds



                    Average Yield Gap                         (%)
                    1950s                                    0.82
                    1960s                                    2.35
                    1970s                                    4.54
                    1980s                                    8.14
                    1990s                                    5.51
                    2000s                                    3.55
                    Overall                                  4.58




                   CHAPTER 5 – The Mathematics of Diversification                   8 - 12
                Measuring Returns
                              Dollar Returns


Investors in market-traded securities (bonds or stock)
receive investment returns in two different form:
     • Income yield
     • Capital gain (or loss) yield


The investor will receive dollar returns, for example:
     • $1.00 of dividends
     • Share price rise of $2.00


To be useful, dollar returns must be converted to percentage returns
as a function of the original investment. (Because a $3.00 return on a
$30 investment might be good, but a $3.00 return on a $300
investment would be unsatisfactory!)

                 CHAPTER 5 – The Mathematics of Diversification   8 - 13
                   Measuring Returns
            Converting Dollar Returns to Percentage Returns

    An investor receives the following dollar returns a stock
    investment of $25:
         • $1.00 of dividends
         • Share price rise of $2.00


    The capital gain (or loss) return component of total return is
    calculated: ending price – minus beginning price, divided by
    beginning price


                                    P  P0 $27 - $25
[8-2]   Capital gain (loss) return  1               .08  8%
                                       P0    $25


                     CHAPTER 5 – The Mathematics of Diversification   8 - 14
                   Measuring Returns
                        Total Percentage Return


• The investor‟s total return (holding period
  return) is:

            Total return  Income yield  Capital gain (or loss) yield
                            CF1  P  P0
                                   1

    [8-3]                        P0
                           CF   P  P 
                          1 1 0
                           P0   P0 
                           $1.00   $27  $25 
                                   $25   0.04  0.08  0.12  12%
                           $25               



                  CHAPTER 5 – The Mathematics of Diversification           8 - 15
                 Measuring Returns
           Total Percentage Return – General Formula


• The general formula for holding period return
  is:

           Total return  Income yield  Capital gain (or loss) yield
                            CF1  P  P0
                                   1

   [8-3]                         P0
                           CF1   P  P0 
                              
                                     1
                                           
                            P0   P0 




                CHAPTER 5 – The Mathematics of Diversification          8 - 16
        Measuring Average Returns
                         Ex Post Returns



• Measurement of historical rates of return that
  have been earned on a security or a class of
  securities allows us to identify trends or
  tendencies that may be useful in predicting the
  future.
• There are two different types of ex post mean or
  average returns used:
  – Arithmetic average
  – Geometric mean

             CHAPTER 5 – The Mathematics of Diversification   8 - 17
           Measuring Average Returns
                            Arithmetic Average



                                                            n

        [8-4]                                              r      i
                  Arithmetic Average (AM)                 i 1
                                                             n

  Where:
      ri = the individual returns
      n = the total number of observations

• Most commonly used value in statistics
• Sum of all returns divided by the total number of
  observations

                  CHAPTER 5 – The Mathematics of Diversification       8 - 18
              Measuring Average Returns
                                  Geometric Mean



                                                                         1
[8-5]
        Geometric Mean (GM)  [( 1  r1 )( 1  r2 )( 1  r3 )...(1  rn )] -1
                                                                         n




  • Measures the average or compound growth
    rate over multiple periods.




                      CHAPTER 5 – The Mathematics of Diversification     8 - 19
              Measuring Average Returns
                Geometric Mean versus Arithmetic Average


If all returns (values) are identical the geometric mean =
    arithmetic average.

If the return values are volatile the geometric mean <
    arithmetic average

The greater the volatility of returns, the greater the
  difference between geometric mean and arithmetic
  average.

   (Table on the following slide illustrates this principle on major asset classes 1938 – 2005)


                       CHAPTER 5 – The Mathematics of Diversification                     8 - 20
                     Measuring Average Returns
               Average Investment Returns and Standard Deviations



Average Investment Returns and Standard Deviations, 1938-2005


                                                              Annual            Annual       Standard Deviation
                                                             Arithmetic       Geometric       of Annual Returns
                                                            Average (%)        Mean (%)              (%)

 Government of Canada treasury bills                                   5.20           5.11          4.32
 Government of Canada bonds                                            6.62           6.24          9.32
 Canadian stocks                                                      11.79          10.60         16.22
 U.S. stocks                                                          13.15          11.76         17.54

 So urce: Data are fro m the Canadian Institute o f A ctuaries



                                                                 The greater the difference,
                                                                 the greater the volatility of
                                                                       annual returns.

                                    CHAPTER 5 – The Mathematics of Diversification                                8 - 21
 Measuring Expected (Ex Ante) Returns

• While past returns might be interesting,
  investor‟s are most concerned with future
  returns.
• Sometimes, historical average returns will not
  be realized in the future.
• Developing an independent estimate of ex ante
  returns usually involves use of forecasting
  discrete scenarios with outcomes and
  probabilities of occurrence.

             CHAPTER 5 – The Mathematics of Diversification   8 - 22
           Estimating Expected Returns
                 Estimating Ex Ante (Forecast) Returns


• The general formula


                                                       n
 [8-6]       Expected Return (ER)   (ri  Probi )
                                                      i 1



  Where:
         ER = the expected return on an investment
         Ri = the estimated return in scenario i
         Probi = the probability of state i occurring

                    CHAPTER 5 – The Mathematics of Diversification   8 - 23
         Estimating Expected Returns
              Estimating Ex Ante (Forecast) Returns

Example:
  This is type of forecast data that are required to
  make an ex ante estimate of expected return.


                                                             Possible
                                                            Returns on
                                     Probability of       Stock A in that
          State of the Economy        Occurrence              State
         Economic Expansion             25.0%                   30%
         Normal Economy                 50.0%                   12%
         Recession                      25.0%                  -25%




                 CHAPTER 5 – The Mathematics of Diversification             8 - 24
         Estimating Expected Returns
Estimating Ex Ante (Forecast) Returns Using a Spreadsheet Approach

Example Solution:
  Sum the products of the probabilities and possible
  returns in each state of the economy.

            (1)                     (2)           (3)      (4)=(2)×(1)
                                              Possible     Weighted
                                             Returns on      Possible
                            Probability of Stock A in that Returns on
    State of the Economy     Occurrence         State       the Stock
   Economic Expansion           25.0%            30%             7.50%
   Normal Economy               50.0%            12%             6.00%
   Recession                    25.0%           -25%            -6.25%
                            Expected Return on the Stock =        7.25%



                  CHAPTER 5 – The Mathematics of Diversification          8 - 25
         Estimating Expected Returns
  Estimating Ex Ante (Forecast) Returns Using a Formula Approach

Example Solution:
  Sum the products of the probabilities and possible
  returns in each state of the economy.


                                n
   Expected Return (ER)   (ri  Probi )
                               i 1

                             (r1  Prob1 )  (r2  Prob2 )  (r3  Prob3 )
                             (30%  0.25)  (12%  0.5)  (-25%  0.25)
                             7.25%



                 CHAPTER 5 – The Mathematics of Diversification               8 - 26
       Measuring Risk

Risk, Return and Portfolio Theory
                                  Risk


• Probability of incurring harm
• For investors, risk is the probability of earning
  an inadequate return.
   – If investors require a 10% rate of return on a given
     investment, then any return less than 10% is
     considered harmful.




               CHAPTER 5 – The Mathematics of Diversification   8 - 28
                                     Risk
                                   Illustrated


                                                        The range of total possible returns
                                                        on the stock A runs from -30% to
Probability
                                                        more than +40%. If the required
                                                        return on the stock is 10%, then
                                                        those outcomes less than 10%
              Outcomes that produce harm                represent risk to the investor.

                                                 A




         -30% -20%     -10%     0%        10%    20%     30%      40%
                                             Possible Returns on the Stock



                  CHAPTER 5 – The Mathematics of Diversification                         8 - 29
                             Range


• The difference between the maximum and
  minimum values is called the range
  – Canadian common stocks have had a range of
    annual returns of 74.36 % over the 1938-2005 period
  – Treasury bills had a range of 21.07% over the same
    period.
• As a rough measure of risk, range tells us that
  common stock is more risky than treasury bills.


             CHAPTER 5 – The Mathematics of Diversification   8 - 30
         Differences in Levels of Risk
                                   Illustrated


              Outcomes that produce harm              The wider the range of probable
                                                      outcomes the greater the risk of the
Probability
                                                      investment.
                                             B        A is a much riskier investment than B



                                                 A




         -30% -20%     -10%     0%        10%    20%     30%      40%
                                             Possible Returns on the Stock



                  CHAPTER 5 – The Mathematics of Diversification                        8 - 31
    Historical Returns on Different Asset
                   Classes

• Figure on the next slide illustrates the volatility in annual
  returns on three different assets classes from 1938 –
  2005.
• Note:
   – Treasury bills always yielded returns greater than 0%
   – Long Canadian bond returns have been less than 0% in some
     years (when prices fall because of rising interest rates), and the
     range of returns has been greater than T-bills but less than
     stocks
   – Common stock returns have experienced the greatest range of
     returns
                                                           (See Figure on the following slide)

                  CHAPTER 5 – The Mathematics of Diversification                        8 - 32
          Measuring Risk
Annual Returns by Asset Class, 1938 - 2005




     CHAPTER 5 – The Mathematics of Diversification   8 - 33
    Refining the Measurement of Risk
                       Standard Deviation (σ)




• Range measures risk based on only two
  observations (minimum and maximum value)
• Standard deviation uses all observations.
  – Standard deviation can be calculated on forecast or
    possible returns as well as historical or ex post
    returns.

        (The following two slides show the two different formula used for Standard
                                                                         Deviation)



                CHAPTER 5 – The Mathematics of Diversification                8 - 34
             Measuring Risk
            Ex post Standard Deviation



                                    n            _

                                   (ri  r ) 2
[8-7]     Ex post                i 1
                                          n 1


           Where :
                  the standard deviation
                 _
                r  the average return
                ri  the return in year i
                n  the number of observatio ns


        CHAPTER 5 – The Mathematics of Diversification   8 - 35
                                    Measuring Risk
              Example Using the Ex post Standard Deviation

Problem
       Estimate the standard deviation of the historical returns on investment A
       that were: 10%, 24%, -12%, 8% and 10%.
Step 1 – Calculate the Historical Average Return

                                              n

                                          r       i
                                                           10  24 - 12  8  10 40
         Arithmetic Average (AM)         i 1
                                                                                   8.0%
                                              n                      5            5


Step 2 – Calculate the Standard Deviation
                         n            _

                         (r  r )
                                i
                                          2
                                                    (10 - 8) 2  (24  8) 2  (12  8) 2  (8  8) 2  (14  8) 2
         Ex post      i 1
                                                  
                               n 1                                             5 1
                      2 2  16 2  20 2  0 2  2 2   4  256  400  0  4   664
                                                                                166  12.88%
                                    4                           4              4

                         CHAPTER 5 – The Mathematics of Diversification                                              8 - 36
                      Ex Post Risk
                   Stability of Risk Over Time

Figure on the next slide demonstrates that the relative riskiness of equities
and bonds has changed over time.

Until the 1960s, the annual returns on common shares were about four
times more variable than those on bonds.

Over the past 20 years, they have only been twice as variable.

Consequently, scenario-based estimates of risk (standard deviation) is
required when seeking to measure risk in the future. (We cannot safely
assume the future is going to be like the past!)

Scenario-based estimates of risk is done through ex ante estimates and
calculations.

              CHAPTER 5 – The Mathematics of Diversification          8 - 37
Relative Uncertainty
        Equities versus Bonds




CHAPTER 5 – The Mathematics of Diversification   8 - 38
                  Measuring Risk
                 Ex ante Standard Deviation


        A Scenario-Based Estimate of Risk



                                n
[8-8]      Ex ante           (Probi )  (ri  ERi ) 2
                               i 1




             CHAPTER 5 – The Mathematics of Diversification   8 - 39
 Scenario-based Estimate of Risk
Example Using the Ex ante Standard Deviation – Raw Data

     GIVEN INFORMATION INCLUDES:
     - Possible returns on the investment for different
       discrete states
     - Associated probabilities for those possible returns



                                                  Possible
           State of the                          Returns on
            Economy           Probability        Security A

        Recession                25.0%              -22.0%
        Normal                   50.0%               14.0%
        Economic Boom            25.0%               35.0%



             CHAPTER 5 – The Mathematics of Diversification   8 - 40
     Scenario-based Estimate of Risk
      Ex ante Standard Deviation – Spreadsheet Approach




• The following two slides illustrate an approach
  to solving for standard deviation using a
  spreadsheet model.




               CHAPTER 5 – The Mathematics of Diversification   8 - 41
Scenario-based Estimate of Risk
       First Step – Calculate the Expected Return

                 Determined by multiplying
                  the probability times the
                      possible return.

                                       Possible              Weighted
  State of the                        Returns on             Possible
   Economy         Probability        Security A              Returns

Recession             25.0%           -22.0%                  -5.5%
Normal                50.0%            14.0%                   7.0%
Economic Boom         25.0%            35.0%                   8.8%
                               Expected Return =              10.3%



  Expected return equals the sum of
    the weighted possible returns.

            CHAPTER 5 – The Mathematics of Diversification              8 - 42
          Scenario-based Estimate of Risk
      Second Step – Measure the Weighted and Squared Deviations

                                Now multiply the square deviations by
       First calculate the deviation of
                                    their probability of occurrence.
     possible returns from the expected.

                                                        Deviation of                  Weighted
                             Possible     Weighted       Possible                        and
 State of the               Returns on    Possible      Return from       Squared      Squared
  Economy       Probability Security A     Returns       Expected        Deviations   Deviations

Recession         25.0%      -22.0%          -5.5%         -32.3%        0.10401       0.02600
Normal            50.0%       14.0%           7.0%           3.8%        0.00141       0.00070
Economic Boom     25.0%       35.0%           8.8%          24.8%        0.06126       0.01531
                     Expected Return =       10.3%                      Variance =     0.0420
                                                              Standard Deviation =     20.50%



                  Second, square those deviations
         The sum of the weighted and square deviations
                            from deviation
                   The standardthe mean. is the square root
                       of the variance squared terms.
            is the variance in percent (in percent terms).
                        CHAPTER 5 – The Mathematics of Diversification                     8 - 43
             Scenario-based Estimate of Risk
            Example Using the Ex ante Standard Deviation Formula

                                 Possible      Weighted
  State of the                  Returns on     Possible
   Economy       Probability    Security A      Returns

Recession          25.0%          -22.0%        -5.5%
Normal             50.0%           14.0%         7.0%
Economic Boom      25.0%           35.0%         8.8%
                           Expected Return =    10.3%


                                                               n
                                               Ex ante       (Prob )  (r  ER )
                                                              i 1
                                                                        i     i      i
                                                                                         2




                                                           P (r1  ER1 ) 2  P2 (r2  ER2 ) 2  P (r3  ER3 ) 2
                                                             1                                    1

                                                           .25(22  10.3) 2  .5(14  10.3) 2  .25(35  10.3) 2
                                                           .25(32.3) 2  .5(3.8) 2  .25(24.8) 2
                                                           .25(.10401)  .5(.00141)  .25(.06126)
                                                           .0420
                                                           .205  20.5%

                               CHAPTER 5 – The Mathematics of Diversification                                      8 - 44
  Modern Portfolio Theory

Risk, Return and Portfolio Theory
                              Portfolios

• A portfolio is a collection of different securities such as stocks
  and bonds, that are combined and considered a single asset

• The risk-return characteristics of the portfolio is demonstrably
  different than the characteristics of the assets that make up
  that portfolio, especially with regard to risk.

• Combining different securities into portfolios is done to
  achieve diversification.




                  CHAPTER 5 – The Mathematics of Diversification   8 - 46
                          Diversification

Diversification has two faces:

   1. Diversification results in an overall reduction in portfolio risk
      (return volatility over time) with little sacrifice in returns, and
   2. Diversification helps to immunize the portfolio from potentially
      catastrophic events such as the outright failure of one of the
      constituent investments.

       (If only one investment is held, and the issuing firm goes
       bankrupt, the entire portfolio value and returns are lost. If a
       portfolio is made up of many different investments, the outright
       failure of one is more than likely to be offset by gains on others,
       helping to make the portfolio immune to such events.)

                   CHAPTER 5 – The Mathematics of Diversification      8 - 47
   Expected Return of a Portfolio
                   Modern Portfolio Theory

The Expected Return on a Portfolio is simply the weighted
average of the returns of the individual assets that make up the
portfolio:


                                    n
    [8-9]              ER p   ( wi  ERi )
                                   i 1




The portfolio weight of a particular security is the percentage of
the portfolio‟s total value that is invested in that security.

             CHAPTER 5 – The Mathematics of Diversification     8 - 48
   Expected Return of a Portfolio
                              Example

Portfolio value = $2,000 + $5,000 = $7,000
           rA = 14%, rB = 6%,
wA = weight of security A = $2,000 / $7,000 = 28.6%
wB = weight of security B = $5,000 / $7,000 = (1-28.6%)= 71.4%



                n
       ER p   ( wi  ERi )  (.286  14%)  (.714  6% )
               i 1

             4.004%  4.284%  8.288%




            CHAPTER 5 – The Mathematics of Diversification   8 - 49
Range of Returns in a Two Asset Portfolio


   In a two asset portfolio, simply by changing the weight of the
   constituent assets, different portfolio returns can be achieved.

   Because the expected return on the portfolio is a simple
   weighted average of the individual returns of the assets, you can
   achieve portfolio returns bounded by the highest and the lowest
   individual asset returns.




               CHAPTER 5 – The Mathematics of Diversification         8 - 50
Range of Returns in a Two Asset Portfolio


   Example 1:

    Assume ERA = 8% and ERB = 10%



                                    (See the following 6 slides based on Figure 8-4)




                CHAPTER 5 – The Mathematics of Diversification                 8 - 51
                                    Expected Portfolio Return
Affect on Portfolio Return of Changing Relative Weights in A and B




                        10.50


                        10.00                                                          ERB= 10%
    Expected Return %




                         9.50


                         9.00


                         8.50


                         8.00        ERA=8%

                         7.50


                         7.00
                                0        0.2      0.4       0.6        0.8       1.0         1.2
                                                              Portfolio Weight



                                        CHAPTER 5 – The Mathematics of Diversification             8 - 52
                                    Expected Portfolio Return
Affect on Portfolio Return of Changing Relative Weights in A and B



                                    A portfolio manager can select the relative weights of the two
                                    assets in the portfolio to get a desired return between 8% (100%
                                    invested in A) and 10% (100% invested in B)
                        10.50


                        10.00                                                           ERB= 10%
    Expected Return %




                         9.50


                         9.00


                         8.50


                         8.00        ERA=8%

                         7.50


                         7.00
                                0         0.2       0.4      0.6        0.8       1.0         1.2
                                                               Portfolio Weight



                                         CHAPTER 5 – The Mathematics of Diversification                8 - 53
                                    Expected Portfolio Return
Affect on Portfolio Return of Changing Relative Weights in A and B




                        10.50

                                                                                       ERB= 10%
                        10.00
    Expected Return %




                         9.50                                                             The potential returns of
                                                                                          the portfolio are
                                                                                          bounded by the highest
                         9.00                                                             and lowest returns of
                                                                                          the individual assets
                         8.50                                                             that make up the
                                                                                          portfolio.

                         8.00
                                     ERA=8%
                         7.50


                         7.00
                                0        0.2      0.4       0.6        0.8       1.0          1.2
                                                              Portfolio Weight



                                        CHAPTER 5 – The Mathematics of Diversification                               8 - 54
                                    Expected Portfolio Return
Affect on Portfolio Return of Changing Relative Weights in A and B




                        10.50

                                                                                                   ERB= 10%
                        10.00
    Expected Return %




                         9.50


                         9.00
                                                                  The expected return on
                                                                  the portfolio if 100% is
                         8.50                                     invested in Asset A is
                                                                  8%.
                         8.00
                                                    ERp  wA ERA  wB ERB  (1.0)(8%)  (0)(10%)  8%
                                    ERA=8%
                         7.50


                         7.00
                                0            0.2        0.4        0.6           0.8         1.0         1.2
                                                                     Portfolio Weight



                                             CHAPTER 5 – The Mathematics of Diversification                    8 - 55
                                    Expected Portfolio Return
Affect on Portfolio Return of Changing Relative Weights in A and B




                        10.50                                The expected return on
                                                             the portfolio if 100% is
                                                             invested in Asset B is             ERB= 10%
                        10.00                                10%.
    Expected Return %




                         9.50


                         9.00


                         8.50
                                                                     ERp  wA ERA  wB ERB  (0)(8%)  (1.0)(10%)  10%
                         8.00

                                    ERA=8%
                         7.50


                         7.00
                                0            0.2       0.4         0.6            0.8     1.0         1.2
                                                                     Portfolio Weight



                                             CHAPTER 5 – The Mathematics of Diversification                               8 - 56
                                    Expected Portfolio Return
Affect on Portfolio Return of Changing Relative Weights in A and B




                        10.50                                The expected return on
                                                             the portfolio if 50% is
                                                             invested in Asset A and               ERB= 10%
                        10.00                                50% in B is 9%.
    Expected Return %




                         9.50
                                                                                ER p  wA ERA  wB ERB
                         9.00
                                                                                        (0.5)(8%)  (0.5)(10%)
                         8.50                                                           4 %  5 %  9%
                         8.00

                                    ERA=8%
                         7.50


                         7.00
                                0            0.2       0.4         0.6           0.8         1.0         1.2
                                                                     Portfolio Weight



                                             CHAPTER 5 – The Mathematics of Diversification                       8 - 57
Range of Returns in a Two Asset Portfolio


   Example 1:

    Assume ERA = 14% and ERB = 6%



                                                            (See the following 2 slides )




                CHAPTER 5 – The Mathematics of Diversification                      8 - 58
Range of Returns in a Two Asset Portfolio
                    E(r)A= 14%, E(r)B= 6%


         Expected return on Asset A =                14.0%
         Expected return on Asset B =                 6.0%
                                            Expected
            Weight of     Weight of       Return on the
             Asset A       Asset B          Portfolio
              0.0%         100.0%              6.0%
             10.0%          90.0%              6.8%
             20.0%          80.0%              7.6%
             30.0%          70.0%              8.4%
             40.0%          60.0%              9.2%          A graph of this
             50.0%          50.0%             10.0%          relationship is
             60.0%          40.0%             10.8%          found on the
             70.0%          30.0%             11.6%
                                                             following slide.
             80.0%          20.0%             12.4%
             90.0%          10.0%             13.2%
             100.0%         0.0%              14.0%


            CHAPTER 5 – The Mathematics of Diversification              8 - 59
Range of Returns in a Two Asset Portfolio
                                                       E(r)A= 14%, E(r)B= 6%


                                                  Range of Portfolio Returns
Expected Return on Two




                         16.00%
                         14.00%
    Asset Portfolio




                         12.00%
                         10.00%
                          8.00%
                          6.00%
                          4.00%
                          2.00%
                          0.00%




                                                                                                                   0%
                                         %


                                                   %


                                                           %


                                                                   %


                                                                           %


                                                                                   %


                                                                                           %


                                                                                                   %


                                                                                                           %
                                0%


                                       .0


                                                 .0


                                                         .0


                                                                 .0


                                                                         .0


                                                                                 .0


                                                                                         .0


                                                                                                 .0


                                                                                                         .0


                                                                                                                 0.
                              0.


                                     10


                                               20


                                                       30


                                                               40


                                                                       50


                                                                               60


                                                                                       70


                                                                                               80


                                                                                                       90


                                                                                                               10
                                                               Weight Invested in Asset A


                                             CHAPTER 5 – The Mathematics of Diversification                             8 - 60
    Expected Portfolio Returns
          Example of a Three Asset Portfolio




              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%



          CHAPTER 5 – The Mathematics of Diversification
                          K. Hartviksen                    8 - 61
      Risk in Portfolios

Risk, Return and Portfolio Theory
         Modern Portfolio Theory - MPT


•   Prior to the establishment of Modern Portfolio Theory (MPT),
    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.




                  CHAPTER 5 – The Mathematics of Diversification   8 - 63
   Expected Return and Risk For Portfolios
         Standard Deviation of a Two-Asset Portfolio using Covariance




[8-11]      p  ( wA ) 2 ( A ) 2  ( wB ) 2 ( B ) 2  2( wA )( wB )(COV A, B )



     Risk of Asset A                                 Risk of Asset B          Factor to take into
   adjusted for weight                              adjusted for weight    account comovement
     in the portfolio                                 in the portfolio     of returns. This factor
                                                                               can be negative.




                          CHAPTER 5 – The Mathematics of Diversification                     8 - 64
   Expected Return and Risk For Portfolios
         Standard Deviation of a Two-Asset Portfolio using Correlation
                                  Coefficient




[8-15]       p  ( wA ) 2 ( A ) 2  ( wB ) 2 ( B ) 2  2( wA )( wB )(  A, B )( A )( B )


                                           Factor that takes into
                                          account the degree of
                                         comovement of returns.
                                          It can have a negative
                                           value if correlation is
                                                 negative.




                          CHAPTER 5 – The Mathematics of Diversification                        8 - 65
    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




                     CHAPTER 5 – The Mathematics of Diversification          8 - 66
            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




                         CHAPTER 5 – The Mathematics of Diversification                   8 - 67
     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


              CHAPTER 5 – The Mathematics of Diversification          8 - 68
                      Covariance

• A statistical measure of the correlation of the
  fluctuations of the annual rates of return of
  different investments.


                          n                      _           _
  [8-12]   COVAB   Probi ( x A,i  xi )( xB ,i - xB )
                         i 1




            CHAPTER 5 – The Mathematics of Diversification       8 - 69
                         Correlation

• The degree to which the returns of two stocks
  co-move is measured by the correlation
  coefficient (ρ).
• The correlation coefficient (ρ) 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
                                    COV AB
          [8-13]            AB 
                                     A B


               CHAPTER 5 – The Mathematics of Diversification   8 - 70
  Covariance and Correlation Coefficient

• Solving for covariance given the correlation
  coefficient and standard deviation of the two
  assets:


            [8-14]       COVAB   AB A B




              CHAPTER 5 – The Mathematics of Diversification   8 - 71
         Importance of Correlation


• Correlation is important because it affects the
  degree to which diversification can be achieved
  using various assets.
• Theoretically, if two assets returns are perfectly
  positively correlated, it is possible to build a
  riskless portfolio with a return that is greater
  than the risk-free rate.




             CHAPTER 5 – The Mathematics of Diversification   8 - 72
   Affect of Perfectly Negatively Correlated Returns
                    Elimination of Portfolio Risk

Returns
                                                          If returns of A and B are
  %
          20%                                             perfectly negatively correlated,
                                                          a two-asset portfolio made up
                                                          of equal parts of Stock A and B
                                                          would be riskless. There would
          15%                                             be no variability
                                                          of the portfolios returns over
                                                          time.

          10%

                                                             Returns on Stock A
                                                             Returns on Stock B
          5%
                                                                 Returns on Portfolio


      Time 0               1                        2
                CHAPTER 5 – The Mathematics of Diversification                          8 - 73
 Example of Perfectly Positively Correlated Returns
                No Diversification of Portfolio Risk

Returns
                                                          If returns of A and B are
  %
          20%                                             perfectly positively correlated,
                                                          a two-asset portfolio made up
                                                          of equal parts of Stock A and B
                                                          would be risky. There would be
          15%                                             no diversification (reduction of
                                                          portfolio risk).


          10%
                                                             Returns on Stock A
                                                                 Returns on Stock B
          5%
                                                                 Returns on Portfolio


      Time 0               1                        2
                CHAPTER 5 – The Mathematics of Diversification                          8 - 74
   Affect of Perfectly Negatively Correlated Returns
                    Elimination of Portfolio Risk

Returns
                                                          If returns of A and B are
  %
          20%                                             perfectly negatively correlated,
                                                          a two-asset portfolio made up
                                                          of equal parts of Stock A and B
                                                          would be riskless. There would
          15%                                             be no variability
                                                          of the portfolios returns over
                                                          time.

          10%

                                                             Returns on Stock A
                                                             Returns on Stock B
          5%
                                                                 Returns on Portfolio


      Time 0               1                        2
                CHAPTER 5 – The Mathematics of Diversification                          8 - 75
Affect of Perfectly Negatively Correlated Returns
                             Numerical Example



            Weight of Asset A =                 50.0%
            Weight of Asset B =                 50.0%
                                                                            n

                                       Expected                     ER p   ( wi  ERi )  (.5  5%)  (.5  15% )
                                                                           i 1
       Return on    Return on        Return on the                        2.5%  7.5%  10%
Year    Asset A      Asset B           Portfolio
xx07      5.0%        15.0%              10.0%
xx08     10.0%        10.0%              10.0%
xx09     15.0%        5.0%               10.0%
                                                                            n
                                                                    ER p   ( wi  ERi )  (.5 15%)  (.5  5% )
                                                                           i 1

                                                                          7.5%  2.5%  10%
        Perfectly Negatively
        Correlated Returns
        over time


                   CHAPTER 5 – The Mathematics of Diversification                                    8 - 76
               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 series of
    spreadsheets, and then summarized in graph format.




                  CHAPTER 5 – The Mathematics of Diversification   8 - 77
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          Both
     Weight of A Weight of B                 Return        Deviation         portfolio
      100.00%      0.00%                      5.00%         15.0%            returns and
       90.00%      10.00%                     5.90%         17.5%            risk are
       80.00%      20.00%                     6.80%         20.0%            bounded by
       70.00%      30.00%                     7.70%         22.5%            the range set
       60.00%      40.00%                     8.60%         25.0%            by the
       50.00%      50.00%                     9.50%         27.5%            constituent
       40.00%      60.00%                    10.40%         30.0%            assets when
       30.00%      70.00%                    11.30%         32.5%            ρ=+1
       20.00%      80.00%                    12.20%         35.0%
       10.00%      90.00%                    13.10%         37.5%
       0.00%      100.00%                    14.00%         40.0%
            CHAPTER 5 – The Mathematics of Diversification                        8 - 78
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
                                                                             When ρ=+0.5
     Weight of A Weight of B                 Return        Deviation
                                                                             these portfolio
      100.00%      0.00%                      5.00%         15.0%
                                                                             combinations
       90.00%      10.00%                     5.90%         15.9%
                                                                             have lower
       80.00%      20.00%                     6.80%         17.4%
                                                                             risk –
       70.00%      30.00%                     7.70%         19.5%
                                                                             expected
       60.00%      40.00%                     8.60%         21.9%
                                                                             portfolio return
       50.00%      50.00%                     9.50%         24.6%
                                                                             is unaffected.
       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%
            CHAPTER 5 – The Mathematics of Diversification                        8 - 79
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         Portfolio
      100.00%      0.00%                      5.00%         15.0%            risk is
       90.00%      10.00%                     5.90%         14.1%            lower than
       80.00%      20.00%                     6.80%         14.4%            the risk of
       70.00%      30.00%                     7.70%         15.9%            either
       60.00%      40.00%                     8.60%         18.4%            asset A or
       50.00%      50.00%                     9.50%         21.4%            B.
       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%
            CHAPTER 5 – The Mathematics of Diversification                        8 - 80
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       Portfolio risk
                                                                           for more
      100.00%      0.00%                      5.00%         15.0%
                                                                           combinations
       90.00%      10.00%                     5.90%         12.0%
                                                                           is lower than
       80.00%      20.00%                     6.80%         10.6%
                                                                           the risk of
       70.00%      30.00%                     7.70%         11.3%
                                                                           either asset
       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%
            CHAPTER 5 – The Mathematics of Diversification                        8 - 81
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% invested in
        50.00%      50.00%                     9.50%         12.5%           asset A
        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%
             CHAPTER 5 – The Mathematics of Diversification                        8 - 82
     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


                           CHAPTER 5 – The Mathematics of Diversification                                  8 - 83
   Impact of the Correlation Coefficient

• Figure on the next slide illustrates the
  relationship between portfolio risk (σ) and the
  correlation coefficient
  – The slope is not linear a significant amount of
    diversification is possible with assets with no
    correlation (it is not necessary, nor is it possible to
    find, perfectly negatively correlated securities in the
    real world)
  – With perfect negative correlation, the variability of
    portfolio returns is reduced to nearly zero.

               CHAPTER 5 – The Mathematics of Diversification   8 - 84
                     Expected Portfolio Return
                              Impact of the Correlation Coefficient




                                                  15
Standard Deviation (%)
of Portfolio Returns




                                                  10




                                                   5




                                                   0
                         -1        -0.5                0                0.5     1
                                          Correlation Coefficient (ρ)




                               CHAPTER 5 – The Mathematics of Diversification       8 - 85
                     Zero Risk Portfolio

• We can calculate the portfolio that removes all risk.
• When ρ = -1, then

 [8-15]    p  ( wA ) 2 ( A ) 2  ( wB ) 2 ( B ) 2  2( wA )( wB )(  A, B )( A )( B )



• Becomes:

           [8-16]            p  w A  (1  w) B


                    CHAPTER 5 – The Mathematics of Diversification                            8 - 86
An Exercise to Produce the Efficient
   Frontier Using Three Assets

    Risk, Return and Portfolio Theory
An Exercise using T-bills, Stocks and Bonds

               Base Data:                         Stocks    T-bills   Bonds               Historical
                         Expected Return(%)       12.73383 6.151702 7.0078723           averages for
                       Standard Deviation (%)        0.168     0.042    0.102
                                                                                     returns and risk for
                          Correlation Coefficient Matrix:
                                                                                         three asset
                                    Stocks                1      -0.216   0.048
                                                                                  Each achievable
                                                                                           classes
                                    T-bills          -0.216           1   0.380   portfolio
                                    Bonds             0.048       0.380       1
                                                                                  combination is
                                                                                           Historical
               Portfolio Combinations:                                                     on
                                                                                  plotted correlation
                                                                                  expected return,
                                                                                          coefficients
                          Weights                             Portfolio
                                                 Expected          Standard       risk between the asset
                                                                                       (σ) space,
                                                                                            classes
 Combination    Stocks      T-bills   Bonds      Return   Variance Deviation      found on the
     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%
                                                                                  following slide.
     3            80.0%       20.0%     0.0%        11.4   0.0177    13.3%           Portfolio
     4            70.0%       30.0%     0.0%        10.8   0.0134    11.6%           characteristics for
     5            60.0%       40.0%     0.0%        10.1   0.0097     9.9%           each combination
     6            50.0%       50.0%     0.0%        9.4    0.0067     8.2%           of securities
     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%

                            CHAPTER 5 – The Mathematics of Diversification                      8 - 88
                                                    Achievable Portfolios
                                                 Results Using only Three Asset Classes



                                   Attainable Portfolio Combinations                                     The efficient set is that set of
                                         and Efficient Set of Portfolio Combinations                          achievable portfolio
                                                                                                          combinations that offer the
                                                                                                           highest rate of return for a
                                14.0
                                                            Efficient Set                                given level of risk. The solid
Portfolio Expected Return (%)




                                12.0                                                                    blue line indicates the efficient
                                         Minimum Variance
                                              Portfolio
                                                                                                                       set.
                                10.0
                                 8.0
                                                                                                            The plotted points are
                                 6.0                                                                         attainable portfolio
                                 4.0                                                                            combinations.

                                 2.0
                                 0.0
                                       0.0            5.0             10.0     15.0         20.0
                                             Standard Deviation of the Portfolio (%)


                                                       CHAPTER 5 – The Mathematics of Diversification                            8 - 89
                         Achievable Two-Security Portfolios
                                          Modern Portfolio Theory



                                                                                   This line
                                                                                   represents
                    13                                                             the set of
                    12
                                                                                   portfolio
                                                                                   combinations
Expected Return %




                    11
                                                                                   that are
                    10                                                             achievable by
                     9
                                                                                   varying
                                                                                   relative
                     8
                                                                                   weights and
                     7                                                             using two
                                                                                   non-
                     6
                         0   10   20         30        40       50      60
                                                                                   correlated
                                       Standard Deviation (%)                      securities.

                                  CHAPTER 5 – The Mathematics of Diversification          8 - 90
                       Dominance


• It is assumed that investors are rational, wealth-
  maximizing and risk averse.
• If so, then some investment choices dominate
  others.




             CHAPTER 5 – The Mathematics of Diversification   8 - 91
                     Investment Choices
                 The Concept of Dominance Illustrated


Return                                           A dominates B
  %                                              because it offers
                     A                     B     the same return
 10%                                             but for less risk.
                                                 A dominates C
                     C                           because it offers a
  5%                                             higher return but
                                                 for the same risk.


                     5%                  20%       Risk
 To the risk-averse wealth maximizer, the choices are clear, A dominates B,
 A dominates C.

                CHAPTER 5 – The Mathematics of Diversification                8 - 92
                                 Efficient Frontier
                        The Two-Asset Portfolio Combinations



                                                                             A is not attainable
                                                                             B,E lie on the
                                                                             efficient frontier and
                                                                             are attainable
                        A              B                                     E is the minimum
Expected Return %




                                                                             variance portfolio
                                                 C                           (lowest risk
                                                                             combination)

                                                                             C, D are
                    E                                                        attainable but are
                             D                                               dominated by
                                                                             superior portfolios
                                                                             that line on the line
                                 Standard Deviation (%)                      above E



                            CHAPTER 5 – The Mathematics of Diversification               8 - 93
                                 Efficient Frontier
                        The Two-Asset Portfolio Combinations



                                                                             Rational, risk
                                                                             averse
                                                                             investors will
                                                                             only want to
                        A              B                                     hold
Expected Return %




                                                                             portfolios
                                                 C                           such as B.


                    E                                                        The actual
                             D                                               choice will
                                                                             depend on
                                                                             her/his risk
                                 Standard Deviation (%)
                                                                             preferences.

                            CHAPTER 5 – The Mathematics of Diversification           8 - 94
        Diversification

Risk, Return and Portfolio Theory
                           Diversification

• We have demonstrated that risk of a portfolio can be
  reduced by spreading the value of the portfolio across,
  two, three, four or more assets.
• The key to efficient diversification is to choose assets
  whose returns are less than perfectly positively
  correlated.
• Even with random or naïve diversification, risk of the
  portfolio can be reduced.
   – This is illustrated in the Figure and Table found on the following
     slides.
       • As the portfolio is divided across more and more securities, the risk
         of the portfolio falls rapidly at first, until a point is reached where,
         further division of the portfolio does not result in a reduction in risk.
       • Going beyond this point is known as superfluous diversification.

                    CHAPTER 5 – The Mathematics of Diversification             8 - 96
                                          Diversification
                                          Domestic Diversification



                                               Average Portfolio Risk
                                           January 1985 to December 1997
                         14


                         12


                         10


                          8
Standard Deviation (%)




                          6


                          4


                          2


                          0
                              0      50       100        150        200        250   300

                                               Number of Stocks in Portfolio



                                  CHAPTER 5 – The Mathematics of Diversification           8 - 97
                                          Diversification
                                         Domestic Diversification

Monthly Canadian Stock Portfolio Returns, January 1985 to December 1997

   Number of              Average           Standard Deviation                  Ratio of Portfolio                    Percentage of
    Stocks in             Monthly               of Average                    Standard Deviation to                  Total Achievable
    Portfolio             Portfolio          Monthly Portfolio               Standard Deviation of a                  Risk Reduction
                         Return (%)             Return (%)                        Single Stock
          1                   1.51                     13.47                                 1.00                               0.00
          2                   1.51                     10.99                                 0.82                              27.50
          3                   1.52                      9.91                                 0.74                              39.56
          4                   1.53                      9.30                                 0.69                              46.37
          5                   1.52                      8.67                                 0.64                              53.31
          6                   1.52                      8.30                                 0.62                              57.50
          7                   1.51                      7.95                                 0.59                              61.35
          8                   1.52                      7.71                                 0.57                              64.02
          9                   1.52                      7.52                                 0.56                              66.17
         10                   1.51                      7.33                                 0.54                              68.30
        14                    1.51                      6.80                                 0.50                             74.19
        40                    1.52                      5.62                                 0.42                             87.24
        50                    1.52                      5.41                                 0.40                             89.64
        100                   1.51                      4.86                                 0.36                             95.70
        200                   1.51                      4.51                                 0.34                             99.58
        222                   1.51                      4.48                                 0.33                             100.00
 So urce: Cleary, S. and Co pp D. "Diversificatio n with Canadian Sto cks: Ho w M uch is Eno ugh?" Canadian Investment Review (Fall 1999), Table 1.



                              CHAPTER 5 – The Mathematics of Diversification                                                                          8 - 98
                             Total Risk of an Individual Asset
                                  Equals the Sum of Market and Unique Risk

                                  Average Portfolio Risk
                                                                                   •      This graph illustrates
                                                                                          that total risk of a
                                                                                          stock is made up of
                                                                                          market risk (that
    Standard Deviation (%)




                                                                                          cannot be diversified
                                                             Diversifiable                away because it is a
                                                             (unique) risk
                                                                                          function of the
[8-19]                                                                                    economic „system‟)
                                                                                          and unique, company-
                                                             Nondiversifiable             specific risk that is
                                                             (systematic) risk            eliminated from the
                                                                                          portfolio through
                                 Number of Stocks in Portfolio
                                                                                          diversification.



[8-19]                       Total risk  Market (systematic) risk  Unique (non - systematic) risk



                                         CHAPTER 5 – The Mathematics of Diversification                     8 - 99
         International Diversification

• Clearly, diversification adds value to a portfolio
  by reducing risk while not reducing the return
  on the portfolio significantly.
• Most of the benefits of diversification can be
  achieved by investing in 40 – 50 different
  „positions‟ (investments)
• However, if the investment universe is
  expanded to include investments beyond the
  domestic capital markets, additional risk
  reduction is possible.
                                                           (See the following slide.)
              CHAPTER 5 – The Mathematics of Diversification                   8 - 100
                                        Diversification
                                    International Diversification




               100


                80
Percent risk




                60


                40
                                                     U.S. stocks
                20
                                                  International stocks
                         11.7
                 0
                     0             10        20           30         40    50    60
                                             Number of Stocks




                                CHAPTER 5 – The Mathematics of Diversification        8 - 101
Achievable Portfolio Combinations

   The Capital Asset Pricing Model
               (CAPM)
     Achievable Portfolio Combinations
                           The Two-Asset Case

• It is possible to construct a series of portfolios with
  different risk/return characteristics just by varying the
  weights of the two assets in the portfolio.
• Assets A and B are assumed to have a correlation
  coefficient of -0.379 and the following individual
  return/risk characteristics

                         Expected Return                   Standard Deviation
       Asset A                 8%                              8.72%
       Asset B               10%                              22.69%

   The following table shows the portfolio characteristics for 100
     different weighting schemes for just these two securities:


                  CHAPTER 5 – The Mathematics of Diversification                8 - 103
       Example of Portfolio Combinations and
                   Correlation
You repeat this
    procedure
                                Expected     Standard    Correlation
down until you
                     Asset       Return      Deviation   Coefficient
have determine
                       A          8.0%         8.7%        -0.379
  the portfolio
                       B         10.0%        22.7%
characteristics
    The first
    for second
  The all 100       Portfolio Components                  Portfolio Characteristics
combination
    portfolios.
     portfolio                                            Expected       Standard
     simply
 assumes 99%       Weight of A Weight of B                 Return        Deviation
  Next plot the
    A and 1%
inassumesin          100%         0%                       8.00%           8.7%
  returns on a
 B. Notice the        99%         1%                       8.02%           8.5%
  you invest
   increase the
graph (see in         98%         2%                       8.04%           8.4%
    solely in
   next slide)
 return and the       97%         3%                       8.06%           8.2%
    Asset A
   decrease in        96%         4%                       8.08%           8.1%
 portfolio risk!      95%         5%                       8.10%           7.9%
                      94%         6%                       8.12%           7.8%
                      93%         7%                       8.14%           7.7%
                      92%         8%                       8.16%           7.5%
                      91%         9%                       8.18%           7.4%
                      90%         10%                      8.20%           7.3%
                      89%         11%                      8.22%           7.2%


                          CHAPTER 5 – The Mathematics of Diversification              8 - 104
Example of Portfolio Combinations and
    Attainable Portfolio Combinations for a
              Two Asset Portfolio
                Correlation
                           12.00%
  Expected Return of the


                           10.00%

                           8.00%
         Portfolio




                           6.00%

                           4.00%

                           2.00%

                           0.00%
                                0.0%       5.0%        10.0%       15.0%         20.0%   25.0%
                                          Standard Deviation of Returns


                                CHAPTER 5 – The Mathematics of Diversification                   8 - 105
        Two Asset Efficient Frontier


• Figure on the next slide describes five different
  portfolios (A,B,C,D and E in reference to the
  attainable set of portfolio combinations of this
  two asset portfolio.


                                             (See Figure on the following slide)




             CHAPTER 5 – The Mathematics of Diversification                8 - 106
                                 Efficient Frontier
                        The Two-Asset Portfolio Combinations



                                                                             A is not attainable
                                                                             B,E lie on the
                                                                             efficient frontier and
                                                                             are attainable
                        A              B                                     E is the minimum
Expected Return %




                                                                             variance portfolio
                                                 C                           (lowest risk
                                                                             combination)

                                                                             C, D are
                    E                                                        attainable but are
                             D                                               dominated by
                                                                             superior portfolios
                                                                             that line on the line
                                 Standard Deviation (%)                      above E



                            CHAPTER 5 – The Mathematics of Diversification              8 - 107
Achievable Set of Portfolio Combinations
                    Getting to the „n‟ Asset Case

• In a real world investment universe with all of the
  investment alternatives (stocks, bonds, money
  market securities, hybrid instruments, gold real
  estate, etc.) it is possible to construct many
  different alternative portfolios out of risky
  securities.
• Each portfolio will have its own unique expected
  return and risk.
• Whenever you construct a portfolio, you can
  measure two fundamental characteristics of the
  portfolio:
   – Portfolio expected return (ERp)
   – Portfolio risk (σp)
                 CHAPTER 5 – The Mathematics of Diversification   8 - 108
      The Achievable Set of Portfolio
             Combinations

• You could start by randomly assembling ten
  risky portfolios.
• The results (in terms of ER p and σp )might look
  like the graph on the following page:




             CHAPTER 5 – The Mathematics of Diversification   8 - 109
Achievable Portfolio Combinations
        The First Ten Combinations Created



  ERp


                                                     10 Achievable
                                                     Risky Portfolio
                                                     Combinations




                                         Portfolio Risk (σp)



         CHAPTER 5 – The Mathematics of Diversification                8 - 110
      The Achievable Set of Portfolio
             Combinations

• You could continue randomly assembling more
  portfolios.
• Thirty risky portfolios might look like the graph
  on the following slide:




             CHAPTER 5 – The Mathematics of Diversification   8 - 111
Achievable Portfolio Combinations
       Thirty Combinations Naively Created



 ERp




                                                    30 Risky Portfolio
                                                    Combinations




                                      Portfolio Risk (σp)



        CHAPTER 5 – The Mathematics of Diversification                   8 - 112
Achievable Set of Portfolio Combinations
    All Securities – Many Hundreds of Different Combinations



• When you construct many hundreds of different
  portfolios naively varying the weight of the
  individual assets and the number of types of
  assets themselves, you get a set of achievable
  portfolio combinations as indicated on the
  following slide:




               CHAPTER 5 – The Mathematics of Diversification   8 - 113
      Achievable Portfolio Combinations
                   More Possible Combinations Created


                                                                         The highlighted
                                                                         portfolios are
ERp                                                                      ‘efficient’ in that
                                                                         they offer the
                                                                         highest rate of
       E is the                                                          return for a given
       minimum                                                           level of risk.
       variance                                                          Rationale investors
       portfolio                                     Achievable Set of   will choose only
                                                     Risky Portfolio     from this efficient
                                                     Combinations        set.

           E



                                       Portfolio Risk (σp)


                    CHAPTER 5 – The Mathematics of Diversification               8 - 114
   The Efficient Frontier

The Capital Asset Pricing Model
            (CAPM)
      Achievable Portfolio Combinations
                     Efficient Frontier (Set)


                                                                  Efficient
ERp                                                               frontier is the
                                                                  set of
                                                                  achievable
                                                                  portfolio
                                                                  combinations
                                              Achievable Set of   that offer the
                                              Risky Portfolio
                                              Combinations
                                                                  highest rate
                                                                  of return for a
                                                                  given level of
         E                                                        risk.


                                Portfolio Risk (σp)


             CHAPTER 5 – The Mathematics of Diversification              8 - 116
     The New Efficient Frontier
                     Efficient Portfolios


                                                           Figure 9 – 1
                                                           illustrates
                             Efficient Frontier            three
ER
                                                           achievable
                                                           portfolio
                                                           combinations
                             B
                                                           that are
      A                                                    „efficient‟ (no
                                                           other
                                                           achievable
      MVP                                                  portfolio that
                                                           offers the
                                                           same risk,
                    Risk                                   offers a higher
                                                           return.)

          CHAPTER 5 – The Mathematics of Diversification           8 - 117
               Underlying Assumption
               Investors are Rational and Risk-Averse



• We assume investors are risk-averse wealth maximizers.
• This means they will not willingly undertake fair gamble.
   – A risk-averse investor prefers the risk-free situation.
   – The corollary of this is that the investor needs a risk premium to
     be induced into a risky situation.
   – Evidence of this is the willingness of investors to pay insurance
     premiums to get out of risky situations.
• The implication of this, is that investors will only choose
  portfolios that are members of the efficient set (frontier).



                  CHAPTER 5 – The Mathematics of Diversification    8 - 118
The New Efficient Frontier and
    Separation Theorem

 The Capital Asset Pricing Model
             (CAPM)
              Risk-free Investing


• When we introduce the presence of a risk-free
  investment, a whole new set of portfolio
  combinations becomes possible.
• We can estimate the return on a portfolio made
  up of RF asset and a risky asset A letting the
  weight w invested in the risky asset and the
  weight invested in RF as (1 – w)




             CHAPTER 5 – The Mathematics of Diversification   8 - 120
          The New Efficient Frontier
                          Risk-Free Investing



– Expected return on a two asset portfolio made up of
  risky asset A and RF:


                       ER p  RF  w (ER A - RF)




  The possible combinations of A and RF are found graphed on the following slide.


                CHAPTER 5 – The Mathematics of Diversification             8 - 121
     The New Efficient Frontier
         Attainable Portfolios Using RF and A



                                                                This means
                                                                you can 9 – 2
                                                                Equation
                                                                Rearranging 9
ER                                                              achieve any
                                                                illustrates w=σ
                                                                -2 where
                                                                portfolio can
                                                                p / σA and
                                                                what you
                                                                combination
                                                                see…portfolio
                                                                substituting in
                                     [9-2]  E(RpA - w A
                                                ) RF           along the blue
                                                                risk increases
                    [9-3]        ER P  RF              P
                                                                Equation 1 we
     A                                         A             coloured line
                                                                in direct
                                                                get an
                                                                simply by to
                                                                proportion a
                                                                equation for
                                                                changing the
                                                                the amount
RF                                                              straight line
                                                                relative weight
                                                                invested in the
                                                                with a
                                                                risky and A
                                                                of RFasset. in
                                                                constant
                                                                the two asset
                                                                slope.
                                                                portfolio.
                    Risk



          CHAPTER 5 – The Mathematics of Diversification               8 - 122
          The New Efficient Frontier
     Attainable Portfolios using the RF and A, and RF and T



                                                                   Which risky
                                                                   portfolio
ER                                                                 would a
                                                                   rational risk-
                    T
                                                                   averse
                                                                   investor
              A                                                    choose in the
                                                                   presence of a
                                                                   RF
RF                                                                 investment?
                                                                   Portfolio A?
                                                                   Tangent
                                 Risk                              Portfolio T?

                  CHAPTER 5 – The Mathematics of Diversification          8 - 123
      The New Efficient Frontier
     Efficient Portfolios using the Tangent Portfolio T

                                                               Clearly RF with
                                                               T (the tangent
                                                               portfolio) offers
ER                                                             a series of
                                                               portfolio
                                                               combinations
                T
                                                               that dominate
          A                                                    those produced
                                                               by RF and A.
                                                               Further, they
RF
                                                               dominate all but
                                                               one portfolio on
                                                               the efficient
                             Risk
                                                               frontier!


              CHAPTER 5 – The Mathematics of Diversification           8 - 124
         The New Efficient Frontier
                             Lending Portfolios

                                                                   Portfolios
                                                                   between RF
                                                                   and T are
ER
     Lending Portfolios                                            „lending‟
                                                                   portfolios,
                                                                   because they
                    T
                                                                   are achieved by
              A                                                    investing in the
                                                                   Tangent
                                                                   Portfolio and
RF                                                                 lending funds to
                                                                   the government
                                                                   (purchasing a
                                                                   T-bill, the RF).
                                 Risk



                  CHAPTER 5 – The Mathematics of Diversification           8 - 125
         The New Efficient Frontier
                           Borrowing Portfolios

                                                                   The line can be
                                                                   extended to risk
                                                                   levels beyond
ER
     Lending Portfolios Borrowing Portfolios                       „T‟ by
                                                                   borrowing at RF
                                                                   and investing it
                    T
                                                                   in T. This is a
              A                                                    levered
                                                                   investment that
                                                                   increases both
RF                                                                 risk and
                                                                   expected return
                                                                   of the portfolio.

                                 Risk



                  CHAPTER 5 – The Mathematics of Diversification           8 - 126
     The New Efficient Frontier
              The New (Super) Efficient Frontier

                                                               This is now
                                                               called the with
                                                               Clearly RFnew
                       Capital Market Line                     (or super)
                                                               T (the market
                                                                  The optimal
                                                               efficient offers
                                                               portfolio)frontier
ER                                                               risky portfolio
                       B2                                      of risky
                                                               a series of
                                                                   (the market
                                                               portfolios.
                                                               portfolio ‘M’)
                                                                  portfolio
          T                 B
                                                               combinations
                                                               Investors can
                                                               that dominate
     A2                                                        achieve any
                                                               those produced
                                                               one of these
                                                               by RF and A.
     A                                                         portfolio
RF
                                                               combinations
                                                               Further, they
                                                               by borrowing or
                                                               dominate all but
                                                   σρ          investing in RF
                                                               one portfolio on
                                                               in combination
                                                               the efficient
                                                               frontier! market
                                                               with the
                                                               portfolio.
              CHAPTER 5 – The Mathematics of Diversification            8 - 127
          The New Efficient Frontier
     The Implications – Separation Theorem – Market Portfolio

• All investors will only hold individually-determined
  combinations of:
   – The risk free asset (RF) and
   – The model portfolio (market portfolio)
• The separation theorem
   – The investment decision (how to construct the portfolio of risky
     assets) is separate from the financing decision (how much
     should be invested or borrowed in the risk-free asset)
   – The tangent portfolio T is optimal for every investor regardless of
     his/her degree of risk aversion.
• The Equilibrium Condition
   – The market portfolio must be the tangent portfolio T if everyone
     holds the same portfolio
   – Therefore the market portfolio (M) is the tangent portfolio (T)


                  CHAPTER 5 – The Mathematics of Diversification    8 - 128
     The New Efficient Frontier
                  The Capital Market Line


                                                            The CML is that
                              CML                           set of superior
                                                               The optimal
                                                            portfolio
ER                                                            risky portfolio
                                                            combinations
                                                                (the market
                                                            that are ‘M’)
                                                               portfolio
       M                                                    achievable in
                                                            the presence of
                                                            the equilibrium
                                                            condition.
RF


                                                σρ




           CHAPTER 5 – The Mathematics of Diversification           8 - 129
The Capital Asset Pricing Model

  The Hypothesized Relationship
    between Risk and Return
    The Capital Asset Pricing Model
                                 What is it?




– An hypothesis by Professor William Sharpe
   • Hypothesizes that investors require higher rates of return for greater levels of
     relevant risk.
   • There are no prices on the model, instead it hypothesizes the relationship
     between risk and return for individual securities.
   • It is often used, however, the price securities and investments.




                 CHAPTER 5 – The Mathematics of Diversification                8 - 131
                    The Capital Asset Pricing Model
                                               How is it Used?



           – Uses include:
                 • Determining the cost of equity capital.
                 • The relevant risk in the dividend discount model to estimate a stock‟s intrinsic
                   (inherent economic worth) value. (As illustrated below)


Estimate Investment’s        Determine Investment’s        Estimate the             Compare to the actual
Risk (Beta Coefficient)      Required Return               Investment’s Intrinsic   stock price in the market
                                                           Value


           COVi,M                                                   D1
    i 
            σM
                             ki  RF  ( ERM  RF )  i      P0                       Is the stock
                                                                  kc  g
             2
                                                                                       fairly priced?




                               CHAPTER 5 – The Mathematics of Diversification                    8 - 132
   The Capital Asset Pricing Model
                            Assumptions


– CAPM is based on the following assumptions:
      1. All investors have identical expectations about expected
         returns, standard deviations, and correlation coefficients for all
         securities.
      2. All investors have the same one-period investment time
         horizon.
      3. All investors can borrow or lend money at the risk-free rate of
         return (RF).
      4. There are no transaction costs.
      5. There are no personal income taxes so that investors are
         indifferent between capital gains an dividends.
      6. There are many investors, and no single investor can affect
         the price of a stock through his or her buying and selling
         decisions. Therefore, investors are price-takers.
      7. Capital markets are in equilibrium.

             CHAPTER 5 – The Mathematics of Diversification           8 - 133
    Market Portfolio and Capital Market Line

•    The assumptions have the following
     implications:
     1. The “optimal” risky portfolio is the one that is
        tangent to the efficient frontier on a line that is drawn
        from RF. This portfolio will be the same for all
        investors.
     2. This optimal risky portfolio will be the market
        portfolio (M) which contains all risky securities.


                          (Figure on the next slide illustrates the Market Portfolio ‘M’)

                  CHAPTER 5 – The Mathematics of Diversification                   8 - 134
          The Capital Market Line


     ER
                                 CML

                                                              The CML is that
                                            ERM  RF            of achievable
                                                             setThe market
                                k P  RF             P
ERM                    M                                        The CML the
                                                                   portfolio
                                                               portfolio ishas
                                            M                optimal risky
                                                                  standard
                                                                combinations
                                                                 deviation of
                                                                 portfolio, it
                                                             that are possible
                                                              portfolio returns
                                                              when investing
                                                             contains all risky
                                                                 in as the
                                                                     only two
                                                               securities and
RF                                                               assets (the
                                                                independent
                                                              lies tangent (T)
                                                              on variable.
                                                                   the portfolio
                                                              market efficient
                                                                    frontier.
                                                             and the risk-free
                                                   σρ            asset (RF).

                        σM



            CHAPTER 5 – The Mathematics of Diversification          8 - 135
   The Capital Asset Pricing Model
   The Market Portfolio and the Capital Market Line (CML)



– The slope of the CML is the incremental expected
  return divided by the incremental risk.


                                               ER M - RF
                Slope of the CML 
                                                    M


– This is called the market price for risk. Or
– The equilibrium price of risk in the capital market.
             CHAPTER 5 – The Mathematics of Diversification   8 - 136
    The Capital Asset Pricing Model
    The Market Portfolio and the Capital Market Line (CML)


– Solving for the expected return on a portfolio in the presence of a
  RF asset and given the market price for risk :


                                     ERM - RF 
                    E ( RP )  RF             P
                                     σM       

– Where:
    • ERM = expected return on the market portfolio M
    • σM = the standard deviation of returns on the market portfolio
    • σP = the standard deviation of returns on the efficient portfolio being
      considered


                CHAPTER 5 – The Mathematics of Diversification          8 - 137
         The Capital Market Line
    Using the CML – Expected versus Required Returns




– In an efficient capital market investors will require a
  return on a portfolio that compensates them for the
  risk-free return as well as the market price for risk.
– This means that portfolios should offer returns along
  the CML.




             CHAPTER 5 – The Mathematics of Diversification   8 - 138
              The Capital Asset Pricing Model
                     Expected and Required Rates of Return


                                                                            A is an overvalued
                                                                            C
                                                                            B a portfolio that
                                                                            offers andExpected
                                                                            undervalued
                                                                            portfolio. expected
                    Required                                                return equal than
                                                                                    is Expected
                                                                            portfolio.lessto the
                   Return on C
        ER                                       CML                        the required return.
                                                                            required return.
                                                                            return is greater
                                                                            than the required
 Expected
                            A                                               Selling pressure will
return on A                                                                 return.
                                                                            cause the price to
                                                                            Demand foryield to
                                                                            fall and the
                                         C                                  Portfolio A will
                                                                            rise until expected
 Required                                                                   increase driving up
                                                                            equals the required
return on A
               B                                                            return.
                                                                            the price, and
                                                                            therefore the
                    Expected
                   Return on C                                              expected return will
        RF
                                                                            fall until expected
                                                                            equals required
                                                                            (market equilibrium
                                                                            condition is
                                                                            achieved.)
                                          σρ



                           CHAPTER 5 – The Mathematics of Diversification                8 - 139
     The Capital Asset Pricing Model
       Risk-Adjusted Performance and the Sharpe Ratios

– William Sharpe identified a ratio that can be used to assess the risk-
  adjusted performance of managed funds (such as mutual funds and
  pension plans).
– It is called the Sharpe ratio:


                                              ER P - RF
                             Sharpe ratio 
                                                P


– Sharpe ratio is a measure of portfolio performance that describes how
  well an asset‟s returns compensate investors for the risk taken.
– It‟s value is the premium earned over the RF divided by portfolio
  risk…so it is measuring valued added per unit of risk.
– Sharpe ratios are calculated ex post (after-the-fact) and are used to
  rank portfolios or assess the effectiveness of the portfolio manager in
  adding value to the portfolio over and above a benchmark.

                CHAPTER 5 – The Mathematics of Diversification         8 - 140
   The Capital Asset Pricing Model
            Sharpe Ratios and Income Trusts



– Table (on the following slide) illustrates return,
  standard deviation, Sharpe and beta coefficient for
  four very different portfolios from 2002 to 2004.
– Income Trusts did exceedingly well during this time,
  however, the recent announcement of Finance
  Minister Flaherty and the subsequent drop in Income
  Trust values has done much to eliminate this
  historical performance.



            CHAPTER 5 – The Mathematics of Diversification   8 - 141
          Income Trust Estimated Values


Income Trusts Estimated Values


                                               Return        σP         Sharpe    β


Median income trusts                           25.83%     18.66%         1.37     0.22
Equally weighted trust portfolio               29.97%      8.02%         3.44     0.28
S&P/TSX Composite Index                         8.97%     13.31%         0.49     1.00
Scotia Capital government bond index            9.55%      6.57%         1.08    20.02

 Source: Adapted from L. Kryzanowski, S. Lazrak, and I. Ratika, " The True
 Cost of Income Trusts," Canadian Investment Review 19, no. 5 (Spring
 2006), Table 3, p. 15.




                       CHAPTER 5 – The Mathematics of Diversification                    8 - 142
 CAPM and Market Risk

The Capital Asset Pricing Model
 Diversifiable and Non-Diversifiable Risk

• CML applies to efficient portfolios
• Volatility (risk) of individual security returns are caused
  by two different factors:
   – Non-diversifiable risk (system wide changes in the economy and
     markets that affect all securities in varying degrees)
   – Diversifiable risk (company-specific factors that affect the returns
     of only one security)
• Figure 9 – 7 illustrates what happens to portfolio risk as
  the portfolio is first invested in only one investment, and
  then slowly invested, naively, in more and more
  securities.


                  CHAPTER 5 – The Mathematics of Diversification    8 - 144
      The CAPM and Market Risk
                    Portfolio Risk and Diversification



Total Risk (σ)

                                                                             Market or
                                                                           systematic
                 Unique (Non-systematic) Risk
                                                                            risk is risk
                                                                           that cannot
                                                                          be eliminated
                                                                             from the
                                                                           portfolio by
                                                                          investing the
                                         Market (Systematic) Risk
                                                                          portfolio into
                                                                             more and
                                                                             different
                                                                            securities.
                                                   Number of Securities




                   CHAPTER 5 – The Mathematics of Diversification             8 - 145
                       Relevant Risk
                Drawing a Conclusion from Figure

• Figure demonstrates that an individual securities‟
  volatility of return comes from two factors:
   – Systematic factors
   – Company-specific factors
• When combined into portfolios, company-specific risk is
  diversified away.
• Since all investors are „diversified‟ then in an efficient
  market, no-one would be willing to pay a „premium‟ for
  company-specific risk.
• Relevant risk to diversified investors then is systematic
  risk.
• Systematic risk is measured using the Beta Coefficient.
                CHAPTER 5 – The Mathematics of Diversification   8 - 146
Measuring Systematic Risk
  The Beta Coefficient

The Capital Asset Pricing Model
            (CAPM)
            The Beta Coefficient
                What is the Beta Coefficient?



• A measure of systematic (non-diversifiable) risk
• As a „coefficient‟ the beta is a pure number and
  has no units of measure.




             CHAPTER 5 – The Mathematics of Diversification   8 - 148
                      The Beta Coefficient
       How Can We Estimate the Value of the Beta Coefficient?



•   There are two basic approaches to estimating
    the beta coefficient:

    1. Using a formula (and subjective forecasts)
    2. Use of regression (using past holding period returns)

    (Figure 9 – 8 on the following slide illustrates the characteristic line used to estimate
                                                                          the beta coefficient)




                       CHAPTER 5 – The Mathematics of Diversification                    8 - 149
          The CAPM and Market Risk
                The Characteristic Line for Security A


           Security A Returns (%)



                      6


                      4                                                                      The plotted
                                                                                             The slope of




                                                                      Market Returns (%)
                                                                                            points are the
                                                                                           the regression
                                                                                              coincident
                                                                                             line is beta.
                      2
                                                                                           rates of return
                                                                                              The line the
                                                                                            earned on of
                      0                                                                        best fit is
                                                                                             investment
-6   -4    -2          0            2          4           6      8                            known in
                                                                                           and the market
                      -2                                                                    portfolio over
                                                                                           finance as the
                                                                                            past periods.
                                                                                            characteristic
                                                                                                 line.
                      -4


                      -6



                 CHAPTER 5 – The Mathematics of Diversification                                  8 - 150
The Formula for the Beta Coefficient


 Beta is equal to the covariance of the returns of the
 stock with the returns of the market, divided by the
 variance of the returns of the market:



                          COVi,M  i , M  i
                     i        
                           σM
                            2
                                  M




           CHAPTER 5 – The Mathematics of Diversification   8 - 151
                       The Beta Coefficient
                   How is the Beta Coefficient Interpreted?

•   The beta of the market portfolio is ALWAYS = 1.0

•   The beta of a security compares the volatility of its returns to the volatility of
    the market returns:

     βs = 1.0                   -    the security has the same volatility as the market as a whole

     βs > 1.0                   -    aggressive investment with volatility of returns greater than
                                     the market

     βs < 1.0                   -    defensive investment with volatility of returns less than the
                                     market

     βs < 0.0                   -    an investment with returns that are negatively correlated with
                                     the returns of the market



                      Table 9 – 2 illustrates beta coefficients for a variety of Canadian Investments



                       CHAPTER 5 – The Mathematics of Diversification                          8 - 152
                       Canadian BETAS
                                       Selected


Canadian BETAS

Company                                      Industry Classification            Beta


Abitibi Consolidated Inc.                    Materials - Paper & Forest         1.37
Algoma Steel Inc.                                  Materials - Steel            1.92
Bank of Montreal                                 Financials - Banks             0.50
Bank of Nova Scotia                              Financials - Banks             0.54
Barrick Gold Corp.                     Materials - Precious Metals & Minerals   0.74
BCE Inc.                               Communications - Telecommunications      0.39
Bema Gold Corp.                        Materials - Precious Metals & Minerals   0.26
CIBC                                             Financials - Banks             0.66
Cogeco Cable Inc.                         Consumer Discretionary - Cable        0.67
Gammon Lake Resources Inc.             Materials - Precious Metals & Minerals   2.52
Imperial Oil Ltd.                        Energy - Oil & Gas: Integrated Oils    0.80

 Source: Research Insight, Compustat North American database, June 2006.



                     CHAPTER 5 – The Mathematics of Diversification                    8 - 153
Risk-Based Models and the Cost of
         Common Equity

  Estimating the Cost of Equity Using
               the CAPM
 Risk-Based Models and the Cost of Common
                  Equity
      Using the CAPM to Estimate the Cost of Common Equity

• CAPM can be used to estimate the required return
  by common shareholders.
• It can be used in situations where DCF methods will
  perform poorly (growth firms)
• CAPM estimate is a „market determined‟ estimate
  because:
   – The RF (risk-free) rate is the benchmark return and is measured
     directly, today as the yield on 91-day T-bills
   – The market premium for risk (MRP) is taken from current market
     estimates of the overall return in the market place less RF (ERM
     –RF)

                 CHAPTER 5 – The Mathematics of Diversification   8 - 155
    Risk-Based Models and the Cost of Common
                     Equity
        Using the CAPM to Estimate the Cost of Common Equity


•   As a single-factor model, we estimate the common shareholder‟s
    required return based on an estimate of the systematic risk of the
    firm (measured by the firm‟s beta coefficient)



                                       K e  RF  MRP   e

•   Where:
     Ke = investor‟s required rate of return
     βe = the stock‟s beta coefficient
     Rf = the risk-free rate of return
     MRP = the market risk premium (ERM - Rf )


                      CHAPTER 5 – The Mathematics of Diversification   8 - 156
 Risk-Based Models and the Cost of Common
                  Equity
                 Estimating the Market Risk Premium


                            K e  RF  MRP   e


• Rf is „observable‟ (yield on 91-day T-bills)
• Getting an estimate of the market risk premium is one of the
  more difficult challenges in using this model.
   – We really need a „forward‟ looking of MRP or a „forward‟ looking
     estimate of the ERM
• One approach is to use an estimate of the current, expected
  MRP by examining a long-run average that prevailed in the
  past.
• Table illustrates the % returns on S&P/TSX Composite
  annually for the first five years of this century.

                   CHAPTER 5 – The Mathematics of Diversification       8 - 157
Risk-Based Models and the Cost of Common
                 Equity
   Using the CAPM to Estimate the Cost of Common Equity


  Returns on the S&P/TSX Composite Index                            Investors are
                                                                    It would be
                                                                  unlikely to expect
                                  Returns                         better to use
                                                                   negative returns
                                                                      average
                                                                      on the stock
               2000                                                 market. If they
                                  7.5072%                             realized
                                                                  did, no one would
               2001               -12.572%                         returns over
                                                                      hold shares!
               2002               -12.438%                            an entire
               2003                                               Who would have
                                                                  business/mar
                                  26.725%
                                                                   guessed before
               2004               14.480%
                                                                     ket cycle.
                                                                  hand, there would
               2005                                                    be two
                                  24.127%
                                                                  consecutive years
                                                                    of aggregate
                                                                   market losses?

                                                                  Such is the reality
                                                                  of investing since
                                                                   none of us are
                                                                      clairvoyant.
                 CHAPTER 5 – The Mathematics of Diversification                   8 - 158
Risk-Based Models and the Cost of Common
                 Equity
         Using the CAPM to Estimate the Cost of Common Equity

                    Long-run average rates of return are more reliable.

Average Investment Returns and Standard Deviations (1938 to 2005)

                                                             Annual     Annual    Standard
                                                            Arithmetic Geometric Deviation of
                                                           Average (%) Mean (%)    Annual
                                                                                 Returns (%)

   Government of Canada Treasury Bills                         5.20         5.11      4.32
   Government of Canada Bonds                                  6.62         6.24      9.32
   Canadian Stocks                                            11.79        10.60     16.22
   U.S. Stocks                                                13.15        11.76     17.54

   Source: Data from Canadian Institute of Actuaries



          The consensus Canadian
       Average risk premium ofis that the              Canadian MRP over the long-term
         stocks over bonds was 5.17%
          bond yield (an observable                    yield) is between 4.0 and 5.5%.

                              CHAPTER 5 – The Mathematics of Diversification                 8 - 159
   Risk-Based Models and the Cost of Common
                    Equity
           Using the CAPM to Estimate the Cost of Common Equity

Long-Run Financial Projections


                            Financial Forecasts                     Average Annual Percent Return

     Bank of Canada Overnight Rate                                                 4.50
     Cash: 3-Month T-bills                                                         4.40
     Income: Scotia Universe Bond Index                                            5.60
     Canadian Equities: S&P/TSX Composite Index                                    7.30
     U.S. Equities: S&P 500 Index                                                  7.80
     International Non-U.S. Equities: MSCI EAFE Index                              7.50
     Source: TD Economics




          The Scotia Universeis very Index is a long-term bond index that
          An estimate of ERM Bond important.
          contains Canada‟s and corporate bonds with default risk.
          TD Economicson a risk-adjusted basis,above estimates for„forward‟
          Nevertheless, recently generated the the TD forecast of MRP is
          looking rates. an arithmetic risk premium of 4.3%
          consistent with
                                  CHAPTER 5 – The Mathematics of Diversification                    8 - 160
 Risk-Based Models and the Cost of Common
                  Equity
                         Estimating Betas




• After obtaining estimates of the two important
  market rates (Rf and MRP), an estimate for the
  company beta is required.

• Figure on the following slide illustrates that
  estimated betas for major sub-indexes of the
  S&P/TSX have varied widely over time:


             CHAPTER 5 – The Mathematics of Diversification   8 - 161
Risk-Based Models and the Cost of Common
                 Equity
Estimated Betas for Sub Indexes of the S&P/TSX Composite Index




               CHAPTER 5 – The Mathematics of Diversification   8 - 162
 Risk-Based Models and the Cost of Common
                  Equity
 Estimated Betas for Sub Indexes of the S&P/TSX Composite Index

• Actual data for the preceding Figure is presented in
  Table on the following slide:
• You should note:
   – IT sub index shows rapidly increasing betas
   – Other sub index betas show constant or decreasing trends.
• Reasons:
   – The weighted average of all betas = 1.0 (by definition they are
     the market)
   – If one sub index is changing…that change alone affects all
     others in the opposite direction.
• What Happened in the 1995 – 2005 decade?
   – The internet bubble of the late 1990s resulted in rapid growth in
     the IT sector till it burst in the early 2000s.

                  CHAPTER 5 – The Mathematics of Diversification   8 - 163
    Risk-Based Models and the Cost of Common
                     Equity
                                                    Estimating Betas
                                                                                                IT
                                                                                              Bubble
Table 20-15 S&P/TSX Sub Index Beta Estimates

         Energy       Materials     Industrials ConsDisc ConsStap      Health   Fin    IT     Telco    Utilities
1995       0.93          1.41           1.19         0.82      0.68     0.36    0.92   1.25    0.53      0.67
1996       0.93          1.28           1.10         0.83      0.66     0.39    1.02   1.36    0.61      0.65
1997       0.98          1.33           0.97         0.82      0.62     0.60    0.93   1.56    0.62      0.53
1998       0.85          1.12           0.94         0.80      0.60     1.02    1.11   1.40    0.92      0.55
1999       0.91          1.04           0.78         0.73      0.43     1.00    1.00   1.55    1.11      0.30
2000       0.67          0.74           0.73         0.69      0.23     1.10    0.79   1.78    0.92      0.14
2001       0.50          0.60           0.82         0.68      0.10     0.98    0.67   2.12    0.94     -0.03
2002       0.43          0.57           0.86         0.73      0.11     0.99    0.67   2.27    0.92     -0.06
2003       0.27          0.42           0.91         0.74      -0.04    0.85    0.39   2.75    0.82     -0.26
2004       0.17          0.42           1.04         0.81      -0.02    0.84    0.41   2.89    0.55     -0.14
2005       0.48          0.78           1.12         0.84      0.14     0.74    0.58   2.71    0.71     -0.01
Source: Data from Financial Post Corporate Analyzer Database




                                    CHAPTER 5 – The Mathematics of Diversification                        8 - 164
 Risk-Based Models and the Cost of Common
                  Equity
                         Nortel Stock Price



• Nortel‟s stock price reflects the IT bubble and
  crash.




           (See Figure on the following slide for Nortel Stock Price history)




              CHAPTER 5 – The Mathematics of Diversification           8 - 165
Risk-Based Models and the Cost of Common
                 Equity
                     Nortel Stock Price




          CHAPTER 5 – The Mathematics of Diversification   8 - 166
 Risk-Based Models and the Cost of Common
                  Equity
  IT Bubble effect on Betas of Other Companies Outside the Sector


• The bubble in IT stocks has driven down the betas in
  other sectors.
• This is demonstrated in Rothman‟s beta over the 1966 –
  2004 period.
• Remember, Rothman‟s is a stable company and it‟s beta
  should be expected to remain constant.




                        (See Figure on the following slide for Rothman‟s beta history)

                 CHAPTER 5 – The Mathematics of Diversification                 8 - 167
Risk-Based Models and the Cost of Common
                 Equity
               Rothman‟s Beta Estimates




          CHAPTER 5 – The Mathematics of Diversification   8 - 168
 Risk-Based Models and the Cost of Common
                  Equity
       Adjusting Beta Estimates and Establishing a Range


• When betas are measured over the period of a
  sector bubble or crash, it is necessary to adjust
  the beta estimates of firms in other sectors.
• Take the industry grouping as a major input,
  plus the individual company beta estimate.
  – Using current MRP and Rf Develop estimates of Ke
    using the range of Company betas prior to the bubble
    or crash



               CHAPTER 5 – The Mathematics of Diversification   8 - 169
           The Beta of a Portfolio


The beta of a portfolio is simply the weighted average of the
betas of the individual asset betas that make up the portfolio.


   [9-8]         P  wA  A  wB  B  ...  wn  n


Weights of individual assets are found by dividing the value of
the investment by the value of the total portfolio.




            CHAPTER 5 – The Mathematics of Diversification        8 - 170
The Security Market Line

The Capital Asset Pricing Model
            (CAPM)
        The CAPM and Market Risk
                    The Security Market Line (SML)

– The SML is the hypothesized relationship between return (the
  dependent variable) and systematic risk (the beta coefficient).
– It is a straight line relationship defined by the following formula:



      [9-9]                ki  RF  ( ERM  RF )  i

– Where:
    ki = the required return on security ‘i’
    ERM – RF = market premium for risk
    Βi = the beta coefficient for security ‘i’

              (See Figure 9 - 9 on the following slide for the graphical representation)

                   CHAPTER 5 – The Mathematics of Diversification                 8 - 172
          The CAPM and Market Risk
              The Security Market Line (SML)




     ER                ki  RF  ( ERM  RF )  i
                         M                                      The SML
                                                               The SML is
ERM                                                              uses the
                                                                  used to
                                                                   beta
                                                                  predict
                                                              coefficient as
                                                                 required
                                                              the measure
                                                               returns for
                                                               of relevant
                                                                individual
RF
                                                                   risk.
                                                                securities


                       βM = 1                   β


             CHAPTER 5 – The Mathematics of Diversification         8 - 173
            The CAPM and Market Risk
                  The SML and Security Valuation



                                                                  Required B is an
                                                                  Similarly,
                                                                  A is an returns
       ER         ki  RF  ( ERM  RF )  i                      overvalued
                                                                  are forecast using
                                                                  undervalued
                                                                  security.
                                                                  security because
                                                                  this equation.
                                              SML                 its expected
                                                                  Investor‟s will sell
                                                                  You can see that
                                                                  return is greater
                                                                  to lock in gains,
                                                                  the required
                                                                  than the required
Expected    A                                                     but the selling
                                                                  return on any
Return A                                                          return.
                                                                  security is a
                                                                  pressure will
Required
Return A         B
                                                                  Investors will
                                                                  function ofmarket
                                                                  cause the its
  RF
                                                                  „flock‟to fall,risk (β)
                                                                  systematicand bid
                                                                  price to A
                                                                  causing the
                                                                  and market
                                                                  up the price
                                                                  causing expected
                                                                  factors (RF and to
                                                                  expected return
                                                                  return premium
                                                                  marketto it equals
                                                                  rise until fall till it
            βA   βB                                    β          equals
                                                                  the required
                                                                  for risk)the
                                                                  required return.
                                                                  return.
                 CHAPTER 5 – The Mathematics of Diversification                  8 - 174
         The CAPM in Summary
                      The SML and CML



– The CAPM is well entrenched and widely used by
  investors, managers and financial institutions.
– It is a single factor model because it based on the
  hypothesis that required rate of return can be
  predicted using one factor – systematic risk
– The SML is used to price individual investments and
  uses the beta coefficient as the measure of risk.
– The CML is used with diversified portfolios and uses
  the standard deviation as the measure of risk.


            CHAPTER 5 – The Mathematics of Diversification   8 - 175
Alternative Pricing Models

The Capital Asset Pricing Model
            (CAPM)
                  Challenges to CAPM

• Empirical tests suggest:
   – CAPM does not hold well in practice:
       • Ex post SML is an upward sloping line
       • Ex ante y (vertical) – intercept is higher that RF
       • Slope is less than what is predicted by theory
   – Beta possesses no explanatory power for predicting stock
     returns (Fama and French, 1992)
• CAPM remains in widespread use despite the foregoing.
   – Advantages include – relative simplicity and intuitive logic.
• Because of the problems with CAPM, other models have
  been developed including:
   – Fama-French (FF) Model
   – Abitrage Pricing Theory (APT)

                   CHAPTER 5 – The Mathematics of Diversification    8 - 177
   Alternative Asset Pricing Models
                  The Fama – French Model



– A pricing model that uses three factors to relate
  expected returns to risk including:
      1. A market factor related to firm size.
      2. The market value of a firm‟s common equity (MVE)
      3. Ratio of a firm‟s book equity value to its market value of
         equity. (BE/MVE)
– This model has become popular, and many think it
  does a better job than the CAPM in explaining ex
  ante stock returns.



             CHAPTER 5 – The Mathematics of Diversification           8 - 178
    Alternative Asset Pricing Models
                  The Arbitrage Pricing Theory

– A pricing model that uses multiple factors to relate expected
  returns to risk by assuming that asset returns are linearly related
  to a set of indexes, which proxy risk factors that influence
  security returns.


      [9-10]    ERi  a0  bi1 F1  bi1 F1  ...  bin Fn


– It is based on the no-arbitrage principle which is the rule that two
  otherwise identical assets cannot sell at different prices.
– Underlying factors represent broad economic forces which are
  inherently unpredictable.



               CHAPTER 5 – The Mathematics of Diversification     8 - 179
    Alternative Asset Pricing Models
              The Arbitrage Pricing Theory – the Model

– Underlying factors represent broad economic forces which are
  inherently unpredictable.



                    ERi  a0  bi1 F1  bi1 F1  ...  bin Fn



– Where:
    •   ERi = the expected return on security i
    •   a0 = the expected return on a security with zero systematic risk
    •   bi = the sensitivity of security i to a given risk factor
    •   Fi = the risk premium for a given risk factor

– The model demonstrates that a security‟s risk is based on its sensitivity
  to broad economic forces.


                   CHAPTER 5 – The Mathematics of Diversification          8 - 180
    Alternative Asset Pricing Models
            The Arbitrage Pricing Theory – Challenges


– Underlying factors represent broad economic forces
  which are inherently unpredictable.
– Ross and Roll identify five systematic factors:
       1.   Changes in expected inflation
       2.   Unanticipated changes in inflation
       3.   Unanticipated changes in industrial production
       4.   Unanticipated changes in the default-risk premium
       5.   Unanticipated changes in the term structure of interest rates


•   Clearly, something that isn‟t forecast, can‟t be used
    to price securities today…they can only be used to
    explain prices after the fact.

                CHAPTER 5 – The Mathematics of Diversification        8 - 181
          Summary and Conclusions

In this chapter you have learned:

  – How the efficient frontier can be expanded by introducing risk-
    free borrowing and lending leading to a super efficient frontier
    called the Capital Market Line (CML)
  – The Security Market Line can be derived from the CML and
    provides a way to estimate a market-based, required return for
    any security or portfolio based on market risk as measured by
    the beta.
  – That alternative asset pricing models exist including the Fama-
    French Model and the Arbitrage Pricing Theory.


                 CHAPTER 5 – The Mathematics of Diversification   8 - 182
Estimating the Ex Ante (Forecast) Beta

             APPENDIX 1
 Calculating a Beta Coefficient Using Ex Ante
                   Returns

• Ex Ante means forecast…
• You would use ex ante return data if historical rates of
  return are somehow not indicative of the kinds of returns
  the company will produce in the future.
• A good example of this is Air Canada or American
  Airlines, before and after September 11, 2001. After the
  World Trade Centre terrorist attacks, a fundamental shift
  in demand for air travel occurred. The historical returns
  on airlines are not useful in estimating future returns.




               CHAPTER 5 – The Mathematics of Diversification   8 - 184
                Appendix 1 Agenda

• The beta coefficient
• The formula approach to beta measurement
  using ex ante returns
  –   Ex ante returns
  –   Finding the expected return
  –   Determining variance and standard deviation
  –   Finding covariance
  –   Calculating and interpreting the beta coefficient


                CHAPTER 5 – The Mathematics of Diversification   8 - 185
               The Beta Coefficient

• Under the theory of the Capital Asset Pricing Model total
  risk is partitioned into two parts:
   – Systematic risk
   – Unsystematic risk – diversifiable risk

                       Total Risk of the Investment



               Systematic Risk                Unsystematic Risk

• Systematic risk is non-diversifiable risk.
• Systematic risk is the only relevant risk to the diversified
  investor
• The beta coefficient measures systematic risk

                CHAPTER 5 – The Mathematics of Diversification    8 - 186
               The Beta Coefficient
                               The Formula




         Covariance of Returns between stock ' i' returns and the market
Beta 
                       Variance of the Market Returns




                           COVi,M  i , M  i
   [9-7]              i        
                            σM
                             2
                                   M



                CHAPTER 5 – The Mathematics of Diversification             8 - 187
              The Term – “Relevant Risk”

•   What does the term “relevant risk” mean in the context of the CAPM?
     – It is generally assumed that all investors are wealth maximizing risk
       averse people
     – It is also assumed that the markets where these people trade are highly
       efficient
     – In a highly efficient market, the prices of all the securities adjust instantly
       to cause the expected return of the investment to equal the required
       return
     – When E(r) = R(r) then the market price of the stock equals its inherent
       worth (intrinsic value)
     – In this perfect world, the R(r) then will justly and appropriately
       compensate the investor only for the risk that they perceive as
       relevant…
     – Hence investors are only rewarded for systematic risk.
       NOTE: The amount of systematic risk varies by investment. High systematic risk
       occurs when R-square is high, and the beta coefficient is greater than 1.0

                       CHAPTER 5 – The Mathematics of Diversification                   8 - 188
The Proportion of Total Risk that is Systematic

 • Every investment in the financial markets vary with
   respect to the percentage of total risk that is systematic.

 • Some stocks have virtually no systematic risk.
    – Such stocks are not influenced by the health of the economy in
      general…their financial results are predominantly influenced by
      company-specific factors.
    – An example is cigarette companies…people consume cigarettes
      because they are addicted…so it doesn‟t matter whether the
      economy is healthy or not…they just continue to smoke.
 • Some stocks have a high proportion of their total risk that
   is systematic
    – Returns on these stocks are strongly influenced by the health of
      the economy.
    – Durable goods manufacturers tend to have a high degree of
      systematic risk.

                CHAPTER 5 – The Mathematics of Diversification           8 - 189
The Formula Approach to Measuring the Beta


                      Cov(k i k M )
               Beta 
                       Var(k M )
   You need to calculate the covariance of the returns between the
   stock and the market…as well as the variance of the market
   returns. To do this you must follow these steps:
     • Calculate the expected returns for the stock and the market
     • Using the expected returns for each, measure the variance
        and standard deviation of both return distributions
     • Now calculate the covariance
     • Use the results to calculate the beta


                CHAPTER 5 – The Mathematics of Diversification       8 - 190
              Ex ante Return Data
                                A Sample

A set of estimates of possible returns and their respective
                probabilities looks as follows:


  Possible                                                      Since the beta
Future State                     Possible  Possible             relates the stock
                                                                By observation
                                                                returns to the
   of the                      Returns on Returns on            market returns,
                                                                you can see the
 Economy     Probability        the Stock the Market            the greater range
                                                                range is much
                                                                of stock returns
Boom               25.0%             28.0%             20.0%                 the
                                                                greater for the
                                                                changing in
                                                                stockdirection as
                                                                same than the
Normal             50.0%             17.0%             11.0%    market and they
                                                                the market
Recession          25.0%            -14.0%             -4.0%    indicates the beta
                                                                move in the
                                                                will be direction.
                                                                same greater
                                                                than 1 and will be
                                                                positive.
                                                                (Positively
                                                                correlated to the
                                                                market returns.)
               CHAPTER 5 – The Mathematics of Diversification             8 - 191
The Total of the Probabilities must Equal 100%

This means that we have considered all of the possible outcomes in
                 this discrete probability distribution


            Possible
          Future State                       Possible  Possible
             of the                        Returns on Returns on
           Economy     Probability          the Stock the Market
          Boom                 25.0%             28.0%             20.0%
          Normal               50.0%             17.0%             11.0%
          Recession            25.0%            -14.0%             -4.0%
                            100.0%


                  CHAPTER 5 – The Mathematics of Diversification           8 - 192
   Measuring Expected Return on the Stock
         From Ex Ante Return Data
The expected return is weighted average returns from the
                    given ex ante data

               (1)             (2)             (3)               (4)
             Possible
           Future State                     Possible
              of the                      Returns on
            Economy     Probability        the Stock (4) = (2)*(3)
           Boom           25.0%      28.0%                         0.07
           Normal         50.0%      17.0%                        0.085
           Recession      25.0%     -14.0%                       -0.035
             Expected return on the Stock =                       12.0%



                CHAPTER 5 – The Mathematics of Diversification            8 - 193
  Measuring Expected Return on the Market
         From Ex Ante Return Data
The expected return is weighted average returns from the
                    given ex ante data

               (1)            (2)             (3)                (4)
             Possible
           Future State                   Possible
              of the                     Returns on
            Economy     Probability      the Market (4) = (2)*(3)
           Boom           25.0%      20.0%                         0.05
           Normal         50.0%      11.0%                        0.055
           Recession      25.0%      -4.0%                        -0.01
            Expected return on the Market =                        9.5%



                CHAPTER 5 – The Mathematics of Diversification            8 - 194
 Measuring Variances, Standard Deviations of
         the Forecast Stock Returns
Using the expected return, calculate the deviations away from the mean, square those
       deviations and then weight the squared deviations by the probability of their
    occurrence. Add up the weighted and squared deviations from the mean and you
                                have found the variance!


       (1)          (2)            (3)           (4)           (5)            (6)          (7)
     Possible                                                                           Weighted
   Future State                  Possible                                                  and
      of the                   Returns on                                   Squared      Squared
    Economy     Probability     the Stock (4) = (2)*(3)    Deviations      Deviations   Deviations
   Boom             25.0%             0.28         0.07    0.16     0.0256                0.0064
   Normal           50.0%             0.17        0.085    0.05     0.0025               0.00125
   Recession        25.0%            -0.14       -0.035   -0.26     0.0676                0.0169
         Expected return (stock) =              12.0% Variance (stock)=                  0.02455
                                              Standard Deviation (stock) =                15.67%


                          CHAPTER 5 – The Mathematics of Diversification                         8 - 195
Measuring Variances, Standard Deviations of
       the Forecast Market Returns
           Now do this for the possible returns on the market



     (1)          (2)            (3)           (4)              (5)         (6)          (7)
   Possible                                                                           Weighted
 Future State                 Possible                                                   and
    of the                   Returns on                                   Squared      Squared
  Economy     Probability    the Market (4) = (2)*(3)       Deviations   Deviations   Deviations
 Boom           25.0%                0.2         0.05      0.105 0.011025             0.002756
 Normal         50.0%               0.11        0.055      0.015 0.000225             0.000113
 Recession      25.0%              -0.04        -0.01     -0.135 0.018225             0.004556
    Expected return (market) =                  9.5% Variance (market) =              0.007425
                                              Standard Deviation (market)=               8.62%




                        CHAPTER 5 – The Mathematics of Diversification                      8 - 196
                        Covariance

From Chapter 8 you know the formula for the covariance
between the returns on the stock and the returns on the
market is:


                               n                      _        _
    [8-12]     COVAB   Probi (k A,i  ki )(k B ,i - k B )
                              i 1




Covariance is an absolute measure of the degree of „co-
movement‟ of returns.

              CHAPTER 5 – The Mathematics of Diversification       8 - 197
                   Correlation Coefficient
Correlation is covariance normalized by the product of the standard
deviations of both securities. It is a „relative measure‟ of co-movement of
returns on a scale from -1 to +1.

The formula for the correlation coefficient between the returns on the stock
and the returns on the market is:


                                                      COV AB
                                       AB 
                                                        A B
                    [8-13]




The correlation coefficient will always have a value in the range of +1 to -1.
      +1 – is perfect positive correlation (there is no diversification potential when combining these two
          securities together in a two-asset portfolio.)
      - 1 - is perfect negative correlation (there should be a relative weighting mix of these two
          securities in a two-asset portfolio that will eliminate all portfolio risk)



                       CHAPTER 5 – The Mathematics of Diversification                               8 - 198
                            Measuring Covariance
                                    from Ex Ante Return Data

    Using the expected return (mean return) and given data measure the
        deviations for both the market and the stock and multiply them
      together with the probability of occurrence…then add the products
                                        up.

    (1)       (2)        (3)         (4)            (5)         (6)         (7)        (8)           "(9)

 Possible             Possible                                         Deviations   Deviations
   Future             Returns                    Possible               from the     from the
State of the           on the       (4) =       Returns on              mean for     mean for
 Economy     Prob.     Stock       (2)*(3)      the Market (6)=(2)*(5) the stock    the market   (8)=(2)(6)(7)
Boom      25.0%         28.0%          0.07        20.0%         0.05       16.0%      10.5%         0.0042
Normal    50.0%         17.0%         0.085        11.0%        0.055        5.0%       1.5%       0.000375
Recession 25.0%        -14.0%        -0.035        -4.0%        -0.01      -26.0%     -13.5%       0.008775
                     E(kstock) =     12.0%     E(kmarket ) =     9.5%           Covariance =        0.01335



                               CHAPTER 5 – The Mathematics of Diversification                       8 - 199
                  The Beta Measured
Using Ex Ante Covariance (stock, market) and Market Variance

Now you can substitute the values for covariance and the
variance of the returns on the market to find the beta of the
                           stock:


                       Cov S, M         .01335
             Beta                             1.8
                        VarM           .007425


• A beta that is greater than 1 means that the investment is aggressive…its
  returns are more volatile than the market as a whole.
• If the market returns were expected to go up by 10%, then the stock
  returns are expected to rise by 18%. If the market returns are expected
  to fall by 10%, then the stock returns are expected to fall by 18%.
               CHAPTER 5 – The Mathematics of Diversification        8 - 200
      Lets Prove the Beta of the Market is 1.0

   Let us assume we are comparing the possible market
         returns against itself…what will the beta be?

    (1)       (2)      (3)       (4)        (5)      (6)          (6)          (7)           (8)

 Possible            Possible            Possible              Deviations   Deviations
   Future            Returns      Cov    Returns    .007425     from the     from the
State of the           Beta (4) =
                      on the
                                         `M,M
                                          on the            1.0
                                                                mean for     mean for    (8)=(2)(6)(7
 Economy     Prob.    Market      Var
                            (2)*(3)                 .007425
                                          Market (6)=(2)*(5)
                                          M                    the stock    the market         )
Boom      25.0%       20.0%       0.05  20.0%         0.05         10.5%      10.5%       0.002756
Normal    50.0%       11.0%      0.055  11.0%        0.055          1.5%       1.5%       0.000113
Recession 25.0%       -4.0%      -0.01  -4.0%        -0.01        -13.5%     -13.5%       0.004556
                     E(kM) =      9.5% E(kM) =        9.5%             Covariance =       0.007425

      Since the variance of the returns on the market is = .007425 …the beta for
                         the market is indeed equal to 1.0 !!!


                         CHAPTER 5 – The Mathematics of Diversification                         8 - 201
      Proving the Beta of Market = 1

If you now place the covariance of the market with itself
             value in the beta formula you get:



                  Cov MM     .007425
           Beta                     1.0
                  Var(R M ) .007425


            The beta coefficient of the market will always be
            1.0 because you are measuring the market returns
            against market returns.

              CHAPTER 5 – The Mathematics of Diversification    8 - 202
Using the Security Market Line

 Expected versus Required Return
How Do We use Expected and Required Rates
               of Return?
 Once you have estimated the expected and required rates of return, you can
 plot them on the SML and see if the stock is under or overpriced.



      % Return
                                                            E(Rs) = 5.0%

        R(ks) = 4.76%
                                                                SML
         E(kM)= 4.2%



  Risk-free Rate = 3%




                                         BM= 1.0        Bs = 1.464



                              Since E(r)>R(r) the stock is underpriced.




                        CHAPTER 5 – The Mathematics of Diversification     8 - 204
How Do We use Expected and Required Rates
               of Return?
•   The stock is fairly priced if the expected return = the required return.
•   This is what we would expect to see „normally‟ or most of the time in an efficient market
    where securities are properly priced.




               % Return

          E(Rs) = R(Rs) 4.76%
                                                                             SML
                   E(RM)= 4.2%



            Risk-free Rate = 3%




                                                       BM= 1.0       BS = 1.464




                                  CHAPTER 5 – The Mathematics of Diversification         8 - 205
               Use of the Forecast Beta
•   We can use the forecast beta, together with an estimate of the risk-
    free rate and the market premium for risk to calculate the investor‟s
    required return on the stock using the CAPM:




         Required Return  RF  βi [E (k M )  RF]


•   This is a „market-determined‟ return based on the current risk-free
    rate (RF) as measured by the 91-day, government of Canada T-bill
    yield, and a current estimate of the market premium for risk (kM – RF)


                    CHAPTER 5 – The Mathematics of Diversification    8 - 206
                        Conclusions


• Analysts can make estimates or forecasts for the returns
  on stock and returns on the market portfolio.
• Those forecasts can be analyzed to estimate the beta
  coefficient for the stock.
• The required return on a stock can then be calculated
  using the CAPM – but you will need the stock‟s beta
  coefficient, the expected return on the market portfolio
  and the risk-free rate.
• The required return is then using in Dividend Discount
  Models to estimate the „intrinsic value‟ (inherent worth)
  of the stock.

               CHAPTER 5 – The Mathematics of Diversification   8 - 207
Calculating the Beta using Trailing
     Holding Period Returns

            APPENDIX 2
The Regression Approach to Measuring the
                 Beta
•   You need to gather historical data about the stock and the market
•   You can use annual data, monthly data, weekly data or daily data.
    However, monthly holding period returns are most commonly used.
     • Daily data is too „noisy‟ (short-term random volatility)
     • Annual data will extend too far back in to time
•   You need at least thirty (30) observations of historical data.
•   Hopefully, the period over which you study the historical returns of the
    stock is representative of the normal condition of the firm and its
    relationship to the market.
•   If the firm has changed fundamentally since these data were produced
    (for example, the firm may have merged with another firm or have
    divested itself of a major subsidiary) there is good reason to believe
    that future returns will not reflect the past…and this approach to beta
    estimation SHOULD NOT be used….rather, use the ex ante approach.
                   CHAPTER 5 – The Mathematics of Diversification         8 - 209
                    Historical Beta Estimation
             The Approach Used to Create the Characteristic Line

     In this example, we have regressed the quarterly returns on the stock against the
       quarterly returns of a surrogate for the market (TSE 300 total return composite
         index) and then using Excel…used the charting feature to plot the historical
                            points and add a regression trend line.
        The ‘cloud’ of plotted points
  represents ‘diversifiable or company
Period HPR(Stock) HPR(TSE 300)                                       Ch a r a c te r istic L in e (Re gr e ssio n )
                 in the
 specific’ risk-4.0% securities returns
2006.4                          1.2%                                                  30.0%
                                a portfolio
 that can be eliminated from-7.0%
2006.3         -16.0%                                                               25.0%
2006.2through diversification. Since
                32.0%          12.0%
                                                                                    20.0%
       company-specific risk can be

                                              Returns on Stock
2006.1          16.0%           8.0%
   eliminated, investors don’t require                                              15.0%
2005.4         -22.0%         -11.0%
    compensation for it according to
2005.3          15.0%          16.0%                                                10.0%
2005.2Markowitz Portfolio Theory.
                28.0%          13.0%                                                 5.0%
2005.1          19.0%           7.0%                                                0.0%
2004.4         -16.0%          -4.0%                             -40.0%     -20.0% -5.0%0.0%          20.0%           40.0%
  The regression line is a line of ‘best
2004.3           8.0%         16.0%
     fit’ that describes the inherent                                              -10.0%
  relationship-3.0%
2004.2           between the-11.0% on
                              returns                                              -15.0%
                              25.0%
2004.1 stock 34.0% returns on the
    the         and the
                                                                                 Returns on TSE 300
     market. The slope is the beta
             coefficient.

                           CHAPTER 5 – The Mathematics of Diversification                                                     8 - 210
                    Characteristic Line

•   The characteristic line is a regression line that represents the
    relationship between the returns on the stock and the returns on the
    market over a past period of time. (It will be used to forecast the
    future, assuming the future will be similar to the past.)

•   The slope of the Characteristic Line is the Beta Coefficient.

•   The degree to which the characteristic line explains the variability in
    the dependent variable (returns on the stock) is measured by the
    coefficient of determination. (also known as the R2 (r-squared or
    coefficient of determination)).

•   If the coefficient of determination equals 1.00, this would mean that
    all of the points of observation would lie on the line. This would mean
    that the characteristic line would explain 100% of the variability of
    the dependent variable.

•   The alpha is the vertical intercept of the regression (characteristic
    line). Many stock analysts search out stocks with high alphas.

                   CHAPTER 5 – The Mathematics of Diversification           8 - 211
                                Low R2

• An R2 that approaches 0.00 (or 0%) indicates that the
  characteristic (regression) line explains virtually none of the
  variability in the dependent variable.
• This means that virtually of the risk of the security is
  „company-specific‟.
• This also means that the regression model has virtually no
  predictive ability.
• In this case, you should use other approaches to value the
  stock…do not use the estimated beta coefficient.



                   (See the following slide for an illustration of a low r-square)

                 CHAPTER 5 – The Mathematics of Diversification             8 - 212
Characteristic Line for Imperial Tobacco
  An Example of Volatility that is Primarily Company-Specific

                                                       Characteristic
        Returns on
                                                       Line for Imperial
        Imperial
                                                       Tobacco
        Tobacco %
                                                                 • High alpha
                                                                 • R-square is very
                                                                   low ≈ 0.02
                                                                 • Beta is largely
                                                                   irrelevant




                                                               Returns on
                                                               the Market %
                                                               (S&P TSX)


              CHAPTER 5 – The Mathematics of Diversification                          8 - 213
                               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.




                 CHAPTER 5 – The Mathematics of Diversification   8 - 214
Characteristic Line General Motors
        A Positive Beta with Predictive Power

                                                    Characteristic
   Returns on
                                                    Line for GM
   General
   Motors %                                         (high R2)
                                                                • Positive alpha
                                                                • R-square is
                                                                  very high ≈ 0.9
                                                                • Beta is positive
                                                                  and close to 1.0




                                                           Returns on
                                                           the Market %
                                                           (S&P TSX)


          CHAPTER 5 – The Mathematics of Diversification                       8 - 215
An Unusual Characteristic Line
        A Negative Beta with Predictive Power

Returns on a                   Characteristic Line for a stock
Stock %                        that will provide excellent
                               portfolio diversification
                                                             • Positive alpha
                               (high R2)                     • R-square is
                                                               very high
                                                             • Beta is negative
                                                               <0.0 and > -1.0




                                                       Returns on
                                                       the Market %
                                                       (S&P TSX)


           CHAPTER 5 – The Mathematics of Diversification                    8 - 216
                      Diversifiable Risk
                           (Non-systematic Risk)

• Volatility in a security‟s returns caused by company-
  specific factors (both positive and negative) such as:
   – 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 (this
     could be either a positive or negative event, depending on the
     competence of the management team)
   – the patented formula for a new drug discovered by the firm.
• Obviously, diversifiable risk is that unique factor that
  influences only the one firm.
                   CHAPTER 5 – The Mathematics of Diversification          8 - 217
       OK – lets go back and look at raw data
         gathering and data normalization

• A common source for stock of information is Yahoo.com
• You will also need to go to the library a use the TSX Review (a
  monthly periodical) – to obtain:
   – Number of shares outstanding for the firm each month
   – Ending values for the total return composite index (surrogate for the
     market)
• You want data for at least 30 months.
• For each month you will need:
   –   Ending stock price
   –   Number of shares outstanding for the stock
   –   Dividend per share paid during the month for the stock
   –   Ending value of the market indicator series you plan to use (ie. TSE
       300 total return composite index)


                    CHAPTER 5 – The Mathematics of Diversification        8 - 218
Demonstration Through Example

 The following slides will be based on
       Alcan Aluminum (AL.TO)
Five Year Stock Price Chart for AL.TO




       CHAPTER 5 – The Mathematics of Diversification   8 - 220
      Spreadsheet Data From Yahoo

Process:

  – Go to http://ca.finance.yahoo.com
  – Use the symbol lookup function to search for the
    company you are interested in studying.
  – Use the historical quotes button…and get 30 months
    of historical data.
  – Use the download in spreadsheet format feature to
    save the data to your hard drive.


             CHAPTER 5 – The Mathematics of Diversification   8 - 221
       Spreadsheet Data From Yahoo
                             Alcan Example

The raw downloaded data should look like this:




  Date         Open      High       Low       Close      Volume
    01-May-02     57.46       62.39     56.61     59.22    753874
     01-Apr-02      62.9      63.61     56.25       57.9   879210
     01-Mar-02      64.9      66.81     61.68     63.03    974368
     01-Feb-02    61.65       65.67     58.75     64.86    836373
     02-Jan-02    57.15       62.37     54.93     61.85    989030
     03-Dec-01      56.6      60.49      55.2     57.15    833280
     01-Nov-01        49      58.02     47.08     56.69    779509


                CHAPTER 5 – The Mathematics of Diversification   8 - 222
       Spreadsheet Data From Yahoo
                              Alcan Example

The raw downloaded data should look like this:




      Date        Open            High          Low           Close   Volume
     01-May-02     57.46            62.39        56.61          59.22  753874
      01-Apr-02     62.9            63.61        56.25           57.9  879210


                                                                      Volume of
                    Opening price per share, the                    trading done
   The day,         highest price per share during the               in the stock
   month and        month, the lowest price per share               on the TSE in
   year             achieved during the month and the               the month in
                    closing price per share at the end               numbers of
                    of the – The Mathematics of Diversification
                  CHAPTER 5 month                                            lots
                                                                      board 8 - 223
       Spreadsheet Data From Yahoo
                            Alcan Example

From Yahoo, the only information you can use is the closing
  price per share and the date. Just delete the other
  columns.



                  Date               Close
                 01-May-02             59.22
                  01-Apr-02             57.9
                 01-Mar-02             63.03
                 01-Feb-02             64.86
                  02-Jan-02            61.85

               CHAPTER 5 – The Mathematics of Diversification   8 - 224
 Acquiring the Additional Information You Need
                                Alcan Example



In addition to the closing price of the stock on a per share basis,
   you will need to find out how many shares were outstanding at
   the end of the month and whether any dividends were paid
   during the month.

You will also want to find the end-of-the-month value of the
  S&P/TSX Total Return Composite Index (look in the green
  pages of the TSX Review)

You can find all of this in The TSX Review periodical.




                  CHAPTER 5 – The Mathematics of Diversification   8 - 225
             Raw Company Data
                          Alcan Example



                                        Closing Price    Cash
                    Issued                for Alcan    Dividends
   Date             Capital                 AL.TO      per Share
  01-May-02        321,400,589                  $59.22      $0.00
   01-Apr-02       321,400,589                  $57.90      $0.15
  01-Mar-02        321,400,589                  $63.03      $0.00
  01-Feb-02        321,400,589                  $64.86      $0.00
   02-Jan-02       160,700,295                $123.70       $0.30
  01-Dec-01        160,700,295                $119.30       $0.00
Number of shares doubled and share price fell by half between
January and February 2002 – this is indicative of a 2 for 1 stock split.
             CHAPTER 5 – The Mathematics of Diversification          8 - 226
   Normalizing the Raw Company Data
                                 Alcan Example




                           Closing
                           Price for    Cash
               Issued       Alcan     Dividends Adjustment          Normalized Normalized
   Date        Capital      AL.TO     per Share   Factor            Stock Price  Dividend
01-May-02    321,400,589       $59.22      $0.00   1.00                   $59.22       $0.00
 01-Apr-02   321,400,589       $57.90      $0.15   1.00                   $57.90       $0.15
01-Mar-02    321,400,589       $63.03      $0.00   1.00                   $63.03       $0.00
01-Feb-02    321,400,589       $64.86      $0.00   1.00                   $64.86       $0.00
 02-Jan-02   160,700,295     $123.70       $0.30   0.50                   $61.85       $0.15
01-Dec-01    145,000,500     $111.40       $0.00   0.45                   $50.26       $0.00


       The adjustment factor is just the value in the issued
       capital cell divided by 321,400,589.



                   CHAPTER 5 – The Mathematics of Diversification                      8 - 227
   Calculating the HPR on the stock from the
                Normalized Data


             Normalized         Normalized                                    ( P  P0 )  D1
                                                                      HPR       1
                                                                                    P0
   Date      Stock Price         Dividend               HPR
                                                                            $59.22 - $57.90  $0.00
01-May-02      $59.22              $0.00               2.28%              
                                                                                    $57.90
 01-Apr-02     $57.90              $0.15              -7.90%               2.28%
01-Mar-02      $63.03              $0.00              -2.82%
01-Feb-02      $64.86              $0.00               4.87%
 02-Jan-02     $61.85              $0.15              23.36%
01-Dec-01      $50.26              $0.00


             Use $59.22 as the ending price, $57.90 as the
             beginning price and during the month of May, no
             dividend was declared.

                     CHAPTER 5 – The Mathematics of Diversification                             8 - 228
   Now Put the data from the S&P/TSX Total
         Return Composite Index in

                                                                      Ending
              Normalized Normalized                                     TSX
   Date       Stock Price  Dividend                     HPR            Value
01-May-02           $59.22       $0.00                   2.28%        16911.33
 01-Apr-02          $57.90       $0.15                  -7.90%        16903.36
01-Mar-02           $63.03       $0.00                  -2.82%        17308.41
01-Feb-02           $64.86       $0.00                   4.87%        16801.82
 02-Jan-02          $61.85       $0.15                  23.36%        16908.11
01-Dec-01           $50.26       $0.00                                16881.75

             You will find the Total Return S&P/TSX Composite
             Index values in TSX Review found in the library.
                     CHAPTER 5 – The Mathematics of Diversification              8 - 229
  Now Calculate the HPR on the Market Index

                                                  ( P  P0 )
                                          HPR       1
                                                       P0
                                                 16,911.33 - 16,903.36
                                                                         Ending
                                               
              Normalized Normalized                   16,903.36            TSX    HPR on
                                                0.05%
   Date       Stock Price  Dividend                            HPR        Value the TSX
01-May-02           $59.22       $0.00                          2.28%    16911.33   0.05%
 01-Apr-02          $57.90       $0.15                         -7.90%    16903.36  -2.34%
01-Mar-02           $63.03       $0.00                         -2.82%    17308.41   3.02%
01-Feb-02           $64.86       $0.00                          4.87%    16801.82  -0.63%
 02-Jan-02          $61.85       $0.15                         23.36%    16908.11   0.16%
01-Dec-01           $50.26       $0.00                                   16881.75

             Again, you simply use the HPR formula using the
             ending values for the total return composite index.
                      CHAPTER 5 – The Mathematics of Diversification                 8 - 230
            Regression In Excel


• If you haven‟t already…go to the tools
  menu…down to add-ins and check off the VBA
  Analysis Pac
• When you go back to the tools menu, you should
  now find the Data Analysis bar, under that find
  regression, define your dependent and
  independent variable ranges, your output range
  and run the regression.


             CHAPTER 5 – The Mathematics of Diversification   8 - 231
                                Regression
                          Defining the Data Ranges



                                                                      Ending
              Normalized Normalized                                     TSX    HPR on
   Date       Stock Price  Dividend                     HPR            Value the TSX
01-May-02           $59.22       $0.00                   2.28%        16911.33   0.05%
 01-Apr-02          $57.90       $0.15                  -7.90%        16903.36  -2.34%
01-Mar-02           $63.03       $0.00                  -2.82%        17308.41   3.02%
01-Feb-02           $64.86       $0.00                   4.87%        16801.82  -0.63%
 02-Jan-02          $61.85       $0.15                  23.36%        16908.11   0.16%
01-Dec-01           $50.26       $0.00                                16881.75

                 dependent variable is the returns on the Stock.
             The independent variable is the returns on the Market.

                     CHAPTER 5 – The Mathematics of Diversification               8 - 232
 Now Use the Regression Function in Excel to
  regress the returns of the stock against the
             returns of the market
  SUMMARY OUTPUT

         Regression Statistics                                               R-square is the
  Multiple R              0.05300947
  R Square                    0.00281                                         coefficient of
  Adjusted R Square       -0.2464875                                         determination =
  Standard Error          5.79609628
  Observations                      6                                         0.0028=.3%
  ANOVA
                             df          SS        MS            F      Significance F
  Regression                        1 0.3786694 0.37866937   0.011271689 0.920560274
  Residual                          4 134.37893 33.5947321
  Total                             5  134.7576

                        CoefficientsStandard Error t Stat     P-value     Lower 95%      Upper 95% Lower 95.0%Upper 95.0%
  Intercept              59.3420816 2.8980481 20.4765686      3.3593E-05 51.29579335     67.38836984 51.2957934 67.38837
  X Variable 1           3.55278937 33.463777 0.10616821     0.920560274 -89.35774428    96.46332302 -89.3577443 96.46332


    Beta                                           The alpha is the
Coefficient is                                    vertical intercept.
   the X-
 Variable 1
                                  CHAPTER 5 – The Mathematics of Diversification                               8 - 233
                 Finalize Your Chart
                              Alcan Example


• You can use the charting feature in Excel to create a
  scatter plot of the points and to put a line of best fit (the
  characteristic line) through the points.
• In Excel, you can edit the chart after it is created by
  placing the cursor over the chart and „right-clicking‟
  your mouse.
• In this edit mode, you can ask it to add a trendline
  (regression line)
• Finally, you will want to interpret the Beta (X-coefficient)
  the alpha (vertical intercept) and the coefficient of
  determination.


                 CHAPTER 5 – The Mathematics of Diversification   8 - 234
                           The Beta
                           Alcan Example




• Obviously the beta (X-coefficient) can simply be
  read from the regression output.
  – In this case it was 3.56 making Alcan‟s returns more
    than 3 times as volatile as the market as a whole.
  – Of course, in this simple example with only 5
    observations, you wouldn‟t want to draw any serious
    conclusions from this estimate.



              CHAPTER 5 – The Mathematics of Diversification   8 - 235
          Summary and Conclusions

In this chapter you have learned:
    – How to measure different types of returns
    – How to calculate the standard deviation and
      interpret its meaning
    – How to measure returns and risk of portfolios and
      the importance of correlation in the diversification
      process.
    – How the efficient frontier is that set of achievable
      portfolios that offer the highest rate of return for a
      given level of risk.

                CHAPTER 5 – The Mathematics of Diversification   8 - 236
Appendix 3

				
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