# Financial Management_15_

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

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

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

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

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

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

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

6
Step 1: Expected Return

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

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

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

8
Step   3: Square deviations

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

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

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

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

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

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

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

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

14
Examples of Different Correlation
Coefficients-Perfect Positive Correlation

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Perfect Negative Correlation

16
Zero Correlation

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

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

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

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

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

Returnon portfolio :
rp  w ArA  w BrB

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

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

Var (Portfolio)=

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

21
Standard Deviation of a Portfolio

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

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(11.6)

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

(11.9)

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

27
Set of Portfolios Composed of Holdings
in Supertech and Slowpoke

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 Thereare a few important points
concerning this graph:

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

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

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

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

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

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

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

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

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

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

Supertech
 = -1.0

 = 1.0
Slowpoke
 = -0.1639

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

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

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

expected return.

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

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

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

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

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

return
100%
stocks

rf
100%
bonds

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

rf
100%
bonds


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

return

rf

P

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

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

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

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

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

   The standard deviation is:

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

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

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

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

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

53
Part 7: Market Equilibrium

Expected return
M

rf

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

rf

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

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

Cov( Ri , RM )
bi 
 ( RM )
2

56
Estimating b with Regression

Security Returns

Slope = bi
Return on
market %

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

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

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

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

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

Ri  RF  βi  ( R M  RF )

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

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

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

Ri  RF  βi  ( R M  RF )

RM

RF

1.0             b

63
Relationship Between Risk & Return

15%
Expected
return

3%

1.5     b

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

65
 The   expected returns on the two securities
are:

66
 Three
CAPM should be mentioned:

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

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

(11.18)

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

(11.19)

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

69

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