Chapter 13 – Return, Risk, and the Security Market Line
Chapter 13
RETURN, RISK, AND THE SECURITY MARKET LINE
CHAPTER WEB SITES
Section Web Address
13.2 www.thestreet.com
13.5 www.investopedia.com/university
13.6 www.investools.com
moneycentral.msn.com
finance.yahoo.com
money.cnn.com
CHAPTER ORGANIZATION
13.1 Expected Returns and Variances
Expected Return
Calculating the Variance
13.2 Portfolios
Portfolio Weights
Portfolio Expected Returns
Portfolio Variance
13.3 Announcements, Surprises, and Expected Returns
Expected and Unexpected Returns
Announcements and News
13.4 Risk: Systematic and Unsystematic
Systematic and Unsystematic Risk
Systematic and Unsystematic Components of Return
13.5 Diversification and Portfolio Risk
The Effect of Diversification: Another Lesson from Market History
The Principle of Diversification
Diversification and Unsystematic Risk
Diversification and Systematic Risk
13.6 Systematic Risk and Beta
The Systematic Risk Principle
Measuring Systematic Risk
Portfolio Betas
13.7 The Security Market Line
Beta and the Risk Premium
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Chapter 13 – Return, Risk, and the Security Market Line
The Security Market Line
13.8 The SML and the Cost of Capital: A Preview
The Basic Idea
The Cost of Capital
13.9 Summary and Conclusions
ANNOTATED CHAPTER OUTLINE
Lecture Tip: You may find it useful to emphasize the economic
foundations of the material in this chapter. Specifically, we
assume:
-Investor rationality: Investors are assumed to prefer more money
to less and less risk to more, all else equal. The result of this
assumption is that the ex ante risk-return trade-off will be upward
sloping.
-As risk-averse return-seekers, investors will take actions
consistent with the rationality assumptions. They will require
higher returns to invest in riskier assets and are willing to accept
lower returns on less risky assets.
-Similarly, they will seek to reduce risk while attaining the desired
level of return, or increase return without exceeding the maximum
acceptable level of risk.
1. Expected Returns and Variances
A. Expected Return
Let n denote the total number of states of the economy; Ri the
return in state i; and pi the probability of state i. Then the expected
return, E(R), is given by:
E(R)=∑
i=1
n
piRi
Example:
State of economy Probability Return (%) Product
+1% change in
GDP
.25 -5 -1.25
+2% change in
GDP
.50 15 7.5
+3% change in
GDP
.25 35 8.75
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Chapter 13 – Return, Risk, and the Security Market Line
Sums 1.00 E(R) = 15%
Projected risk premium = expected return minus the risk-free rate
= E(R) – Rf
B. Calculating the Variance
Var (R)=σ2=∑
i=1
n
pi(Ri− E(R))2
Variance measures the dispersion of points around the mean of a
distribution. In this context, we are attempting to characterize the
variability of possible future security returns around the expected
return. In other words, we are trying to quantify risk and return.
Variance measures the total risk of the possible returns.
State of Economy Probability Return (%) Squared
Deviation
Product
+1% change in GDP .25 -5 400 100
+2% change in GDP .50 15 0 0
+3% change in GDP .25 35 400 100
Total 1.00 E(R) = 15 2 = 200
Standard deviation = square root of variance = 14.14%
Lecture Tip: Some students experience confusion in understanding
the mathematics of the variance calculation. They may have the
feeling that they should divide the variance of an expected return
by (n-1). Point out that the probabilities account for this division.
We divide by n-1 in the historical variance because we are looking
at a sample. If we looked at the entire population (which is what
we are doing with expected values), then we would divide by n (or
multiply by 1/n) to get our historical variance. This is the same as
saying that the “probability” of occurrence is the same for all
observations and is equal to 1/n.
2. Portfolios
A. Portfolio Weights
A portfolio is a collection of assets, such as stocks and bonds, held
by an investor.
Portfolios can be described by the percentage investment in each
asset. These percentages are called portfolio weights.
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Chapter 13 – Return, Risk, and the Security Market Line
Example: If two securities in a portfolio have a combined value of
$10,000 with $6000 invested in IBM and $4000 invested in GM,
then the weight on IBM = 6/10 = .6 and the weight on GM = 4/10
= .4.
B. Portfolio Expected Returns
The expected return on a portfolio is the sum of the product of the
expected returns on the individual securities and their portfolio
weights. Let wj be the portfolio weight for asset j and m be the
total number of assets in the portfolio; then
E(R)=∑
j=1
m
wjE(Rj)
This formula also works if you drop the expectations and just
compute the portfolio return in each state of the economy. This is
necessary for the calculation of the portfolio variance in the next
section.
C. Portfolio Variance
Unlike expected return, the variance of a portfolio is NOT the
weighted sum of the individual security variances. Combining
securities into portfolios can reduce the total variability of returns.
Example: Consider a portfolio with equal amounts invested in
three stocks:
State of
Economy
Probability Return on
A (%)
Return on
B (%)
Return on
C (%)
Return on
Portfolio (%)
+1% change
in GDP
.25 -5 0 20 5
+2% change
in GDP
.50 15 10 10 11.7
+3% change
in GDP
.25 35 20 0 18.3
Expected
Return
15 10 10 11.7
Variances and standard deviations:
Var(A) = .25(-5-15)2 + .5(15-15)2 + .25(35-15)2 = 200
Std. Dev.(A) = 14.14
Var(B) = .25(0 – 10)2 + .5(10 – 10)2 + .25(20 – 10)2 = 50
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Chapter 13 – Return, Risk, and the Security Market Line
Std. Dev.(B) = 7.07
Var(C) = .25(20 – 10)2 + .5(10-10)2 + .25(0-10)2 = 50
Std. Dev.(C) = 7.07
Var(portfolio) = .25(5-11.7)2 + .5(11.7-11.7)2 + .25(18.3-11.7)2 =
22.1125
Std. Dev.(portfolio) = 4.70
Notice that the portfolio variance is less than any of the individual
variances.
Lecture Tip: In most business programs, a course in elementary
statistics is a prerequisite for the introductory finance course. And,
while students are sometimes fuzzy on the details, they usually
remember the general concept of the correlation coefficient (and
hopefully the covariance). They almost always remember that the
correlation coefficient is bounded by –1 and 1. You may find it
useful to reintroduce them to the concept of correlation here to
deepen their understanding of portfolio variance.
Specifically, for a two-asset portfolio, the portfolio variance is
equal to:
w1
2σ1
2+ w2
2σ2
2+ 2w1w2σ1σ2ρ1,2
or w1
2σ1
2+ w2
2σ2
2+ 2w1w2σ1,2
where 1,2 is the correlation coefficient and 1,2 is the covariance.
When you expand the equation to more assets, you will have a
variance term for each asset and a covariance term for each pair
of assets. As you increase the number of assets, it is easy to see
that the correlation (covariance) between assets is much more
important in determining the portfolio variance than the individual
variances.
Reconsider the previous example.
The following covariances can be computed:
cov(A,B) = 100
cov(A,C) = -100
cov(B,C) = -50
Using the covariances and extending the formula above to three
assets, you can compute a portfolio variance and standard
deviation:
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Chapter 13 – Return, Risk, and the Security Market Line
var = (1/3)2(200) + (1/3)2(50) + (1/3)2(50) + 2(1/3)(1/3)(100) +
2(1/3)(1/3)(-100) + 2(1/3)(1/3)(-50) = 22.22
standard deviation = 4.71%
This is just as we computed earlier, with a slight difference due to
rounding portfolio returns.
Lecture Tip: Here are a few tips to pass along to students suffering
from “statistics overload”:
-The distribution is just the picture of all possible outcomes
-The mean return is the central point of the distribution
-The standard deviation is the average deviation from the mean
-Assuming investor rationality (two-parameter utility functions),
the mean is a proxy for expected return and the standard deviation
is a proxy for total risk.
3. Announcements, Surprises, and Expected Returns
A. Expected and Unexpected Returns
Total return = expected return + unexpected return
Total return differs from expected return because of surprises, or
“news.” This is one of the reasons that realized returns differ from
expected returns.
B. Announcements and News
Announcement – The release of information not previously
available. Announcements have two parts: the expected part and
the surprise part.
The expected part is “discounted” information used by the market
to estimate the expected return, while the surprise is news that
influences the unexpected return.
Discounted information is information that is already included in
the expected return (and the price). The tie-in to efficient markets
is obvious. The assumption here is that markets are semistrong
efficient.
Lecture Tip: It is easy to see the effect of unexpected news on
stock prices and returns. Consider the following two cases: (1) On
November 17, 2004 it was announced that K-Mart would acquire
Sears in an $11 billion deal. Sears’ stock price jumped from a
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Chapter 13 – Return, Risk, and the Security Market Line
closing price of $45.20 on November 16 to a closing price of
$52.99 (a 17.2% increase) and K-Mart’s stock price jumped from
$101.22 on November 16 to a closing price of $109.00 on
November 17 (a 7.69% increase). Both stocks traded even higher
during the day. Why the jump in price? Unexpected news, of
course. (2) On November 18, 2004, Williams-Sonoma cut its sales
and earnings estimates for the fourth quarter of 2004 and its share
price dropped by 6%. There are plenty of other examples where
unexpected news causes a change in price and expected returns.
4. Risk: Systematic and Unsystematic
A. Systematic and Unsystematic Risk
Risk consists of surprises. There are two kinds of surprises:
Systematic risk is a surprise that affects a large number of assets,
although at varying degrees. It is sometimes called market risk.
Unsystematic risk is a surprise that affects a small number of assets
(or one). It is sometimes called unique or asset-specific risk.
Example: Changes in GDP, interest rates, and inflation are
examples of systematic risk. Strikes, accidents, and takeovers are
examples of unsystematic risk.
Lecture Tip: You can expand the discussion of the difference
between systematic and unsystematic risk by using the example of
a strike by employees. Students will generally agree that this is
unique or unsystematic risk for one company. However, what if the
UAW stages the strike against the entire auto industry. Will this
action impact other industries or the entire economy? If the
answer to this question is yes, then this becomes a systematic risk
factor. The important point is that it is not the event that
determines whether it is systematic or unsystematic risk; it is the
impact of the event.
B. Systematic and Unsystematic Components of Return
Total return = expected return + unexpected return
Total return = expected return + systematic portion + unsystematic
portion
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Chapter 13 – Return, Risk, and the Security Market Line
5. Diversification and Portfolio Risk
A. The Effect of Diversification: Another Lesson from Market History
Portfolio variability can be quite different from the variability of
individual securities.
A typical single stock on the NYSE has a standard deviation of
annual returns around 49%, while the typical large portfolio of
NYSE stocks has a standard deviation of around 20%.
Video Note: “Portfolio Management” looks at the value of diversification.
B. The Principle of Diversification
Principle of Diversification – States that combining imperfectly
correlated assets can produce a portfolio with less variability than
the typical individual asset.
The portion of variability present in a single security that is not
present in a portfolio of securities is called diversifiable risk. The
level of variance that is present in portfolios of assets is
nondiversifiable risk.
International Note: Common sense suggests that, to the extent
that national economies are less than perfectly positively
correlated, there may be diversification benefits to be had by
investing in foreign securities. As a result, the potential for risk
reduction is greater when you include international stocks in your
portfolio.
C. Diversification and Unsystematic Risk
When securities are combined into portfolios, their unique or
unsystematic risks tend to cancel out, leaving only the variability
that affects all securities to some degree. Thus, diversifiable risk is
synonymous with unsystematic risk. Large portfolios have little or
no unsystematic risk.
D. Diversification and Systematic Risk
Systematic risk cannot be eliminated by diversification since it
represents the variability due to influences that affect all securities
to some degree. Therefore, systematic risk and nondiversifiable
risk are the same.
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Chapter 13 – Return, Risk, and the Security Market Line
Total risk = nondiversifiable risk + diversifiable risk = systematic
risk + unsystematic risk
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