Risk Return and Portfolio Theory by liaoqinmei

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```									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 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
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

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
– 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
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
– 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
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

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
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,
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,
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,
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
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
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
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
Selected

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

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
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
U.S. Stocks                                                13.15        11.76     17.54

Source: Data from Canadian Institute of Actuaries

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
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

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

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.
save the data to your hard drive.

CHAPTER 5 – The Mathematics of Diversification   8 - 221
Alcan Example

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
Alcan Example

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
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
Analysis Pac
• When you go back to the tools menu, you should
now find the Data Analysis bar, under that find
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
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‟
• 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
– 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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