CHAPTER 11
RISK AND RETURN
CHAPTER 11 QUIZ
CHAPTER ORGANIZATION
Previously we learned some important lessons from capital market history. Most importantly, there is a reward, on
average, for bearing risk. We called this reward a risk premium. The second lesson is that this risk premium is larger for
riskier investments. Thus far, we have concentrated mainly on the return behavior of a few large portfolios. We need to
expand our consideration to include individual assets. Specifically, we have two tasks to accomplish. First, we have to
define risk and then discuss how to measure it. We then must quantify the relationship between an asset's risk and its
required return.
When we examine the risks associated with individual assets, we find there are two types of risk: systematic and
unsystematic. This distinction is crucial because, as we will see, systematic risk affects almost all assets in the
economy, at least to some degree, while unsystematic risk affects at most a small number of assets. We then develop
the principle of diversification, which shows that highly diversified portfolios will tend to have almost no unsystematic
risk.
The principle of diversification has an important implication: To a diversified investor, only systematic risk matters. It
follows that in deciding whether or not to buy a particular individual asset, a diversified investor will only be concerned
with that asset's systematic risk. This is a key observation, and it allows us to say a great deal about the risks and
returns on individual assets. In particular, it is the basis for a famous relationship between risk and return called the
security market line, or SML. To develop the SML, we introduce the equally famous “beta” coefficient, one of the
centerpieces of modern finance. Beta and the SML are key concepts because they supply us with at least part of the
answer to the question of how to go about determining the required return on an investment.
11.1

Expected Returns and Variances
Expected Return – The return on a risky asset expected in the future. Given all possible outcomes for a
particular investment, the average rate of return is called the expected return. The actual return can differ from
the expected return.
The expected return is a weighted average of the possible rates of return:
E(R) = (Pr1
x
R1) + (Pr2
x
R2) + (Pr3
x
R3) + … + (PrT
x
RT) ,
where T is the number of possible states of the economy, Pr1, Pr2, Pr3, and PrT are the probabilities of the
respective states of the economy, and R1, R2, R3, and RT are the possible rates of return. Further, we define the
expected risk premium as the expected return less the risk-free rate:
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Risk premium = Expected return – Risk-free rate = E(R) – Rf
Based on the following information, calculate the expected return.
E(R) = [.20 x (-.07)] + [.55 x .13] + [.25 x .30] = (-.014) + (.0715) + (.075) = .1325 = 13.25%

Calculating the Variance
n
Var(R)   2   p i (R i  E(R)) 2
i 1
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.
Some 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). We 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 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.
Based on the following information, calculate the variance and standard deviation.
From our previous calculations, E(R) = .1325 or 13.25%.
Var(R) = .20(-.07 - .1325)2 + .55(.13 - .1325)2 + .25(.30 - .1325)2
Var(R) = (.00820125) + (.00000344) + (.00701406) = .01521875
SD(R) = (.01521875).5 = .123364 = 12.34%
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11.2
Portfolios – A portfolio is a group of assets held by an investor.

Portfolio Weights – The respective percentages of a portfolio’s total value invested in each of the assets in the
portfolio. Portfolio weights must sum to 1.0.

Portfolio Expected Returns – The expected return for a portfolio is the weighted average of the expected returns
of the assets which are included in the portfolio:
E(Rp) = [x1 x E(R1)] + [x2 x E(R2)] + [x3 x E(R3)] + … + [xn x E(Rn)] ,
where n is the number of assets in the portfolio, x1, x2, x3, and xn are the portfolio weights, and E(R1), E(R2),
E(R3), and E(Rn) are the expected returns on assets 1 through n.
You own a portfolio that is 20 percent invested in Stock X, 45 percent in Stock Y, and 35 percent in Stock Z. The
expected returns on these three stocks are 10 percent, 14 percent, and 16 percent, respectively. What is the expected
return on the portfolio?
E(Rp) = (.20 x .10) + (.45 x .14) + (.35 x .16) = .02 + .063 + .056 = .139 = 13.90%

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.
Consider the following information:
a. Your portfolio is invested 30 percent each in A and C and 40 percent in B. What is the expected return of the
portfolio?
This portfolio does not have an equal weight in each asset. We first need to find the return of the portfolio in each state
of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products
to get the portfolio return in each state of the economy. Doing so, we get:
Boom: E(Rp) = .30(.30) + .40(.45) + .30(.33) = .3690 or 36.90%
Good:
E(Rp) = .30(.12) + .40(.10) + .30(.15) = .1210 or 12.10%
Poor: E(Rp) = .30(.01) + .40(–.15) + .30(–.05) = –.0720 or –7.20%
Bust: E(Rp) = .30(–.20) + .40(–.30) + .30(–.09) = –.2070 or –20.70%
And the expected return of the portfolio is:
E(Rp) = .15(.3690) + .45(.1210) + .35(–.0720) + .05(–.2070) = .0743 or 7.43%
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b. What is the variance of this portfolio? The standard deviation?
To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared
deviations from the expected return. We then multiply each possible squared deviation by its probability, and then sum.
The result is the variance. So, the variance and standard deviation of the portfolio is:
p2 = .15(.3690 – .0743)2 + .45(.1210 – .0743)2 + .35(–.0720 – .0743)2 + .05(–.2070 – .0743)2 = .02546
p = (.02546).5 = .1596 or 15.96%
11.3

Announcements, Surprises, and Expected Returns
Expected and Unexpected Returns
Total return = R = expected return + unexpected return = E(R) + U,
where R stands for the actual total return for the year, E(R) represents the expected part of the return, and U
stands for the unexpected part of the return.
Thus, total return differs from expected return because of surprises, or “news.” This is one of the reasons that
realized returns differ from expected returns.

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 form efficient.
It is easy to see the effect of unexpected news on stock prices and returns. Consider the following cases: (1) On August
9, 2000 it was announced that Eli-Lilly’s Prozac patent would not be extended, overturning a lower court decision. This
was unexpected and the price dropped from $108.531 to $76.875 (a 29% drop) in one day. (2) On September 22, 2000,
Intel issued an earnings warning and its stock price dropped from $61.468 to $47.937 (a 22% drop) in one day. On
September 30, 2004, Merck announced a voluntary recall of its arthritis and acute pain medication VIOXX following a
reported increased incidence of cardiovascular events. The stock dropped from $45.07 to $33 (a 37% drop). There are
plenty of other examples in which unexpected news causes a change in price and expected returns.
11.4
Risk: Systematic and Unsystematic

Systematic and Unsystematic Risk
Risk consists of surprises. There are two kinds of surprises:
o Systematic risk is a surprise that affects a large number of assets, although at varying degrees. It is
sometimes called market risk or nondiversifiable risk.
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o Unsystematic risk is a surprise that affects a small number of assets (or one). It is sometimes called
unique, asset-specific risk, or diversifiable risk.
Example: Changes in GDP, interest rates, and inflation are examples of systematic risk. Strikes, accidents, and
takeovers are examples of unsystematic risk.

Systematic and Unsystematic Components of Return
R = E(R) + U
The unexpected return, U, has systematic and unsystematic portions. By definition, the unsystematic risk of one
asset is unrelated to the unsystematic risk of another.
R = E(R) + Systematic portion + Unsystematic portion
11.5

Diversification and Portfolio Risk
The Effect of Diversification: Another Lesson from Market History – Diversification is the process of
combining assets to reduce risk. Studies of common stocks listed on the NYSE demonstrate that the standard
deviation of a portfolio decreases as the number of securities in the portfolio increases.
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
The Principle of Diversification – Spreading an investment across a number of assets will eliminate some, but
not all, of the risk. 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 a portfolio of assets is
nondiversifiable risk.

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.

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.
Total risk = Nondiversifiable risk + Diversifiable risk = Systematic risk + Unsystematic risk
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11.6

Systematic Risk and Beta
The Systematic Risk Principle - The systematic risk principle: The reward for bearing risk depends only on the
systematic risk of the investment. The implication: The expected return on an asset depends only on that asset’s
systematic risk. A corollary: No matter how much total risk an asset has, its expected return depends only on its
systematic risk.
The text states that “The underlying rationale for this principle is straightforward: Since unsystematic risk can be
eliminated at virtually no cost (by diversifying), there is no reward for bearing it.” This is a crucial point that is
consistent with observed behavior. The rapid growth in mutual funds suggests that investors aggressively seek to
diversify their holdings. Further, barriers to diversification are minimal: many funds will open accounts with initial
deposits as small as $500. Many company-sponsored retirement plans will open accounts with much less.

Measuring Systematic Risk – The systematic risk of an asset is measured by its beta coefficient (β), which
measures the amount of systematic risk for a particular asset relative to the amount of systematic risk for the
average asset (by definition, βM, the beta of the “market portfolio” equals 1.0). Since systematic risk is the
relevant risk for a well-diversified investor, the expected return is dependent on β. This is true regardless of the
stock’s standard deviation.
Beta coefficients for selected companies
Beta, Beta, Who's Got the Beta?
Based on what we've studied so far, you can see that beta is a pretty important topic. You might wonder then, are all
published betas created equal? Read on for a partial answer to this question.
We did some checking on betas and found some interesting results. The Value Line Investment Survey is one of the
best-known sources for information on publicly traded companies. However, with the explosion of online investing,
there has been a corresponding increase in the amount of investment information available online. We decided to
compare the betas presented by Value Line to those reported by Yahoo! Finance (finance.yahoo.com) and CNN Money
(money.cnn.com). What we found leads to an important note of caution.
Consider Amazon.com, the big online retailer. Its beta reported on the Internet was 3.75, which is much larger than
Value Line's beta of 1.25. Amazon.com wasn't the only stock that showed a divergence in betas from different sources.
In fact, for most of the technology companies we looked at, Value Line reported betas that were significantly lower
than their online cousins. For example, the online beta for Dell was 1.36, but Value Line reported 0.95. The online beta
for computer antivirus company McAfee was 2.88 versus a Value Line beta of 1.60. Value Line's betas are not always
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lower. For example, the online beta for Yahoo! was 0.68, compared to Value Line's 1.55.
We also found some unusual, and even hard to believe, estimates for beta. Starwood Hotels had a very low online beta
of 0.00, while Value Line reported 1.35. The online estimate for Hormel Foods, the famous maker of Spam (the lunch
meat, not junk e-mail), was 0.06, compared to Value Line's 0.75. Perhaps the most outrageous reported betas were the
online betas for the International Fight League and Nano Jet Corp., with betas of 77.4 and –64.06 (notice the minus
sign!), respectively. Value Line did not report a beta for these companies. How do you suppose we should interpret a
beta of –64.06?
There are a few lessons to be learned from all of this. First, not all betas are created equal. Some are computed using
weekly returns and some using daily returns. Some are computed using 60 months of stock returns; some consider
more or less. Some betas are computed by comparing the stock to the S&P 500 index, while others use alternative
indices. Finally, some reporting firms (including Value Line) make adjustments to raw betas to reflect information
other than just the fluctuation in stock prices.
The second lesson is perhaps more subtle. We are interested in knowing what the betas of the stocks will be in the
future, but betas have to be estimated using historical data. Anytime we use the past to predict the future, there is the
danger of a poor estimate. As we will see later in the chapter (and the next one), it is very unlikely that International
Fight League has a beta anything like 77.4 or that Nano Jet Corp. has a beta of –64.06. Instead, the estimates are almost
certainly poor ones. The moral of the story is that, as with any financial tool, beta is not a black box that should be
taken without question.
The point that “the market does not reward risks that are borne unnecessarily,” should be strongly emphasized. Many
investment companies offer investors a choice between income-oriented mutual funds, containing both bonds and
stocks in established companies with higher dividend payouts, and growth-oriented funds that are typically composed
of stocks of smaller companies that retain most of their earnings for reinvestment in the firm. Investors that desire
growth-oriented funds typically assume a much greater degree of systematic risk and expect higher returns. However,
both types of funds eliminate the unsystematic portion of risk through diversification.

Portfolio Betas - While portfolio variance is not a weighted average of the individual asset betas, portfolio betas
are a weighted average of the individual asset betas.
βp = (x1 x β1) + (x2 x β2) + (x3 x β3) + … + (xn x βn) ,
where n is the number of assets in the portfolio, x1, x2, x3, and xn are the portfolio weights, and β1, β2, β3, and βn
are the betas of the assets in the portfolio.
Example:
Stock
Amount Invested
IBM
GM
Wal-Mart
$6,000
$4,000
$2,000
Portfolio
$12,000
Portfolio Weight
Beta Product
$6,000 / $12,000 = 50%
1.47
$4,000 / $12,000 = 33.33% 1.19
$2,000 / $12,000 = 16.67% 0.91
100%
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.735
.397
.152
1.284
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11.7

The Security Market Line - FinSim – The graphic representation of the relationship between systematic risk and
expected return.
Beta and the Risk Premium - The relationship between a portfolio’s expected return, E(Rp), and its systematic
risk, βp, is linear. The slope of the line is the reward-to-risk ratio, which indicates what an investor’s
compensation would be for taking on risk. The reward-to-risk ratio for a given asset equals its risk premium
divided by its beta.
A riskless asset has a beta of 0.When a risky asset with  > 0 is combined with a riskless asset, the resulting
expected return is the weighted sum of the expected returns and the portfolio beta is the weighted sum of the
betas. By varying the amount invested in each asset, we can get an idea of the relationship between portfolio
expected returns and betas. As can be seen, all of the risk-return combinations lie on a straight line. Remember
that the equation for a line is:
y = mx + b
where y = expected return, x = beta, m = slope* = (E(RM) – Rf), and b = y-intercept = risk-free rate
*Note: the slope is NOT beta.
Expected Returns and Systematic Risk
The Reward-to-Risk Ratio is the expected return per unit of systematic risk. In other words, it is the ratio of risk
premium to systematic risk. The basic argument is that since systematic risk is all that matters in determining
expected return, the reward-to-risk ratio must be the same for all assets. If it were not, people would buy the
asset with the higher reward-to-risk ratio (driving the price up and the return down).
The fundamental result is that, in a competitive market where only systematic risk affects E(R), the reward-torisk ratio must be the same for all assets in the market. Consequently, the expected returns and betas of all assets
must plot on the same straight line.
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
The Security Market Line - The line that gives the expected return/systematic risk combinations of assets in a
well-functioning, active financial market.
Consider a portfolio of all the assets in the market. This portfolio, by definition, has “average” systematic risk
with a beta of 1. Since all assets must lie on the SML when appropriately priced, the market portfolio must also
lie on the SML.
Let the expected return on the market portfolio = E(RM).
The slope of the SML = reward-to-risk ratio = [E(RM) – Rf] / M = [E(RM) – Rf] / 1 = E(RM) – Rf
The term E(RM) - Rf is often referred to as the market risk premium, since it is the difference between the
expected return on a market portfolio and the risk-free rate.
If we let E(Ri) and βi stand for the expected return and beta, respectively, on any asset in the market, then we
know that asset must plot on the SML. As a result, we know that its reward-to-risk ratio is the same as the
overall market's:
E(R i )  R f
βi
 E(R M )  R f
Rearranging terms, the equation for the SML can be written as:
E(Ri) = Rf + [E(RM) – Rf] x βi ,
where E(Ri) and βi are the expected return and beta, respectively, for any asset (i).
This is the Capital Asset Pricing Model (CAPM), and indicates the expected return for any asset. The CAPM
demonstrates that the expected return for a given asset is a function of the following:
1. the pure time value of money, Rf
2. the reward for bearing systematic risk, [E(RM) – Rf]
3. the amount of systematic risk, βi
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A stock has a beta of 0.9, the expected return on the market is 13 percent, and the risk-free rate is 6 percent. What must
the expected return on this stock be?
E(Ri) = Rf + [E(RM) – Rf] x βi
E(Ri) = .06 + (.13 - .06)(0.9) = .1230 = 12.30%
A stock has an expected return of 17 percent, the risk-free rate is 5.5 percent, and the market risk premium is 8 percent.
What must the beta of this stock be?
.17 = .055 + (.08)(βi)
.17 - .055 = (.08)(βi)
βi = .1150 / .08 = 1.4375
A stock has an expected return of 11.90 percent and a beta of .85, and the expected return on the market is 13 percent.
What must the risk-free rate be?
.1190 = Rf + (.13 - Rf)(.85)
.1190 = Rf + .1105 - .85(Rf)
.1190 - .1105 = .15(Rf)
Rf = .0085 / .15 = .056667 = 5.67%
TABLE 11.9
I.
Summary of risk and return concepts
Total return
The total return on an investment has two components: the expected return and the unexpected return. The
unexpected return comes about because of unanticipated events. The risk from investing stems from the
possibility of an unanticipated event.
II.
Total risk
The total risk of an investment is measured by the variance or, more commonly, the standard deviation of its
return.
III.
Systematic and unsystematic risks
Systematic risks (also called market risks) are unanticipated events that affect almost all assets to some degree
because the effects are economywide. Unsystematic risks are unanticipated events that affect single assets or
small groups of assets. Unsystematic risks are also called unique or asset-specific risks.
IV.
The effect of diversification
Some, but not all, of the risk associated with a risky investment can be eliminated by diversification. The reason
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is that unsystematic risks, which are unique to individual assets, tend to wash out in a large portfolio, but
systematic risks, which affect all of the assets in a portfolio to some extent, do not.
V.
The systematic risk principle and beta
Because unsystematic risk can be freely eliminated by diversification, the systematic risk principle states that
the reward for bearing risk depends only on the level of systematic risk. The level of systematic risk in a
particular asset, relative to the average, is given by the beta of that asset.
VI.
The reward-to-risk ratio and the security market line
The reward-to-risk ratio for Asset i is the ratio of its risk premium, E(Ri) – Rf) to its beta, βi;
In a well-functioning market, this ratio is the same for every asset. As a result, when asset expected returns are
plotted against asset betas, all assets plot on the same straight line, called the security market line (SML).
VII.
The capital asset pricing model
From the SML, the expected return on Asset i can be written:
This is the capital asset pricing model (CAPM). The expected return on a risky asset thus has three components.
The first is the pure time value of money, Rf; the second is the market risk premium, [E(RM) – Rf]; and the third
is the beta for that asset, βi,.
11.8

The SML and the Cost of Capital: A Preview
The Basic Idea – The SML describes investors’ opportunities in the financial markets in terms of systematic
risk (i.e., β) and expected return. As an example, a capital budgeting project must provide an expected return to
shareholders that exceeds the return available for investments of comparable risk in the financial markets.
Remember that in an efficient market, security investments have an NPV = 0, on average. However, this does not imply
that a company’s investments in new projects must have an NPV of zero. Firms that can consistently invest in positive
NPV projects will trade at higher prices, all else equal.

The Cost of Capital – The cost of capital for a capital budgeting project is the minimum required return for the
investment. Since this minimum is the expected return available in the financial markets for investments with a
specified risk level, it is also the shareholders’ opportunity cost. Therefore, the minimum required return on a
new investment is the firm’s cost of capital.
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