# Formulae for Calculation of Expected Return and Risk by crg76519

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```									Lessons from Capital
Market History

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

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

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

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

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

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

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

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

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

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

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

SAME
ANSWER !   Geometric Average Return :
_                                                                        1

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

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

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

Historical Returns
Historical Averages - Risk and Return

16
Small cap
Equities
Percent Return

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

km
Market         Security Market
for risk
Rf
Real Return

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

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

       (ki  ki )         2

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

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

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

K. Hartviksen
A Normal Probability Distribution

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

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

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

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

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

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

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

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

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

n
R p   xi Ri
i 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Expected Return                                                                                  B

AB = -0.5
12%
AB = -1

8%
AB = 0

AB= +1

A
4%

0%

0%              10%                    20%                30%                      40%

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

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

Portfolio Combinations:

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

Attainable Portfolio Combinations
and Efficient Set of Portfolio Combinations

14.0
Efficient Set
Portfolio Expected Return (%)

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

Expected Return on
the Portfolio

12%

8%

4%

0%

0%   10%               20%                 30%                  40%

Standard Deviation of the Portfolio
The Efficient Frontier

Expected Return on
the Portfolio

12%

8%

4%

0%

0%   10%     20%                    30%                   40%

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

Expected Return on
the Portfolio

12%

8%

4%

Risk-free
rate
0%

0%   10%      20%                30%                   40%

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

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

12%

Capital
8%
Market Line

4%

Risk-free
rate
0%

0%          10%                 20%                  30%                   40%

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

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

Capital
8%
Market Line

4%

Risk-free
rate
0%

0%         10%                 20%                 30%                   40%

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

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

km
Market         Security Market
for risk
Rf
Real Return

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

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

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

km
Market         Security Market
for risk
Rf
Real Return

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

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

TOTAL RISK = SYSTEMATIC RISK + UNSYSTEMATIC RISK

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

n

       (ki  ki )   2

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

n
        (k
i 1
i    k i ) Pi
2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Rf

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

km =12%

Rf = 4%

1.0   Beta Coefficient
Draw the Security Market Line
Return                        SML
%

km =12%

Rf = 4%

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

Return                                                    SML
%
R(k) = 13.6%

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

Rf = 4%

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

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

E(k)

Rf = 4%

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

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

Rf = 4%

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

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

E(k) = 9%                                                         E(k)

Rf = 4%

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

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

ks2

ks1
Rising interest rates will
cause all required rates of
Rf1               will force down stock and
bond prices.

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

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

Bs=1.2        Beta Coefficient

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