ch07

Chapter 7Risk and Return

Learning Objectives

1. Explain the relation between risk and return.

2. Describe the two components of a total holding period return, and calculate this

return for an asset.

3. Explain what an expected return is, and calculate the expected return for an asset.

4. Explain what the standard deviation of returns is, explain why it is especially useful

in finance, and be able to calculate it.

5. Explain the concept of diversification.

6. Discuss which type of risk matters to investors and why.

7. Describe what the Capital Asset Pricing Model (CAPM) tells us and how to use it to

evaluate whether the expected return of an asset is sufficient to compensate an

investor for the risks associated with that asset.

I. Chapter Outline

7.1 Risk and Return

The greater the risk, the larger the return investors require as compensation for

bearing that risk.

Higher risk means you are less certain about the ex post level of compensation.

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7.2 Quantitative Measures Return

A. Holding Period Returns

The total holding period return consists of two components: (1) capital

appreciation and (2) income.

The capital appreciation component of a return, RCA:

The income component of a return RI:

The total holding period return is simply

B. Expected Returns

Expected value represents the sum of the products of the possible outcomes

and the probabilities that those outcomes will be realized.

The expected return, E(RAsset), is an average of the possible returns from an

investment, where each of these returns is weighted by the probability that it

will occur:

If each of the possible outcomes is equally likely (that is, p1 = p2 = p3 = … = pn

= p = 1/n), this formula reduces to: .

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7.3 The Variance and Standard Deviation as Measures of Risk

A. Calculating the Variance and Standard Deviation

The variance (2) squares the difference between each possible occurrence

and the mean (squaring the differences makes all the numbers positive) and

multiplies each difference by its associated probability before summing them

up:

If all of the possible outcomes are equally likely, then the formula becomes:

Take the square root of the variance to get the standard deviation ().

B. Interpreting the Variance and Standard Deviation

The normal distribution is a symmetric frequency distribution that is

completely described by its mean (average) and standard deviation.

The normal distribution is symmetric in that the left and right sides are mirror

images of each other. The mean falls directly in the center of the distribution,

and the probability that an outcome is a particular distance from the mean is

the same whether the outcome is on the left or the right side of the

distribution.

The standard deviation tells us the probability that an outcome will fall a

particular distance from the mean or within a particular range:

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Number of Standard

Deviations from the Mean

Fraction of Total

Observations

1.000 68.26%

1.645 90%

1.960 95%

2.575 99%

C. Historical Market Performance

The key point is that, on average, annual returns have been higher for riskier

securities. For instance, Exhibit 7.4 shows that small stocks, which have the

largest standard deviation of total returns, also have the largest average return. On

the other end of the spectrum, Treasury bills have the smallest standard deviation

and the smallest average annual return.

7.4 Risk and Diversification

By investing in two or more assets whose values do not always move in the same

direction at the same time, an investor can reduce the risk of his or her

investments, or portfolio. This is the idea behind the concept of diversification.

A. Single-Asset Portfolios

Returns for individual stocks from one day to the next have been found to be

largely independent of each other and approximately normally distributed.

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A first pass at comparing risk and return for individual stocks is the

coefficient of variation, CV,

A lower value for the CV is what we are looking for.

B. Portfolios with More Than One Asset

The coefficient of variation has a critical shortcoming that is not quite evident

when we are only considering a single asset.

The expected return of a portfolio is made up of two assets:

The expected return of a portfolio is made up of multiple assets:

The expected return of each asset must be found before applying either of the

two above formulas. The fraction of the portfolio invested in each asset must

also be known.

The prices of two stocks in a portfolio will rarely, if ever, change by the same

amount and in the same direction at the same time.

When the stock prices move in opposite directions, the change in the price of

one stock offsets at least some of the change in the price of the other stock.

As a result, the level of risk for a portfolio of the two stocks is less than the

average of the risks associated with the individual shares.

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R1,2 is the covariance between stocks 1 and 2. The covariance is a measure of

how the returns on two assets covary, or move together:

The covariance calculation is very similar to the variance calculation. The

difference is that, instead of squaring the difference between the value from

each outcome and the expected value for an individual asset, we calculate the

product of this difference for two different assets.

In order to ease the interpretation of the covariance, we divide the covariance

by the product of the standard deviations of the returns for the two assets. This

gives us the correlation coefficient between the returns on the two assets,

The value of the correlation between the returns on two assets will always

have a value between –1 and +1.

A negative correlation means that the returns tend to have opposite

signs.

A positive correlation means that when the return on one asset is

positive, the return on the other asset also tends to be positive.

A correlation of 0 means that the returns on the assets are not

correlated.

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If we have imperfect correlation between assets, or a correlation coefficient

less than +1, then we have a benefit from diversification by holding more than

one asset with different risk characteristics.

As we add more and more stocks to a portfolio, calculating the variance

becomes increasingly complex because we have to account for the covariance

between each pair of assets.

C. The Limits of Diversification

If the returns on the individual stocks added to our portfolio do not all change

in the same way, then increasing the number of stocks in the portfolio will

reduce the standard deviation of the portfolio returns even further.

However, the decrease in the standard deviation for the portfolio gets smaller

and smaller as more assets are added.

As the number of assets becomes very large, the portfolio standard deviation

does not approach zero. It only decreases up to a point.

That is because investors can diversify away risk that is unique to the

individual assets, but they cannot diversify away risk that is common to all

assets.

The risk that can be diversified away is called diversifiable, unsystematic, or

unique risk, and the risk that cannot be diversified away is called

nondiversifiable or systematic risk.

Most of the risk-reduction benefits from diversification can be achieved in a

portfolio with 15 to 20 assets.

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7.5 Systematic Risk

With complete diversification, all of the unique risk is eliminated from the

portfolio, but the investor still faces systematic risk.

A. Why Systematic Risk Is All That Matters

Diversified investors face only systematic risk, whereas investors whose

portfolios are not well diversified face systematic risk plus unsystematic risk.

Because diversified investors face less risk, they will be willing to pay higher

prices for individual assets than other investors.

Therefore, expected returns on individual assets will be lower than the total

risk (systematic plus unsystematic risk) of those assets suggests they should

be.

The bottom line is that only systematic risk is rewarded in asset markets, and

this is why we are only concerned about systematic risk when we think about

the relation between risk and return in finance.

B. Measuring Systematic Risk

If systematic risk is all that matters when we think about expected returns,

then we cannot use the standard deviation as a measure of risk since the

standard deviation is a measure of total risk.

Since systematic risk is, by definition, risk that cannot be diversified away, the

systematic risk (or market risk) of an individual asset is really just a measure

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of the relation between the returns on the individual asset and the returns on

the market.

We quantify the relation between the returns on a stock and the general

market by finding the slope of the line of best fit between the returns of the

stock and the general market.

We call the slope of the line of best fit beta.

If the beta of an asset is:

Equal to one, then the asset has the same systematic risk as the market.

Greater than one, then the asset has more systematic risk than the market.

Less than one, then the asset has less systematic risk than the market.

Equal to zero, then the asset has no systematic risk.

7.6 Compensation for Bearing Systematic Risk

The difference between required returns on government securities and

required returns for risky investments represents the compensation investors

require for taking risk: E(Ri) = Rrf + Compensation for taking riski.

If we recognize that the compensation for taking risk varies with asset risk,

and that systematic risk is what matters, we find:

E(Ri) = Rrf + (Units of systematic riski Compensation per unit of systemic risk)

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If beta, β, is the appropriate measure for the number of units of systematic

risk, we find:

Compensation for taking risk = β Compensation per unit of systemic risk

The required rate of return on the market, over and above that of the risk-free

return, represents compensation required by investors for bearing a market

(systematic) risk:

Compensation per unit of systemic risk = E(Rm) – Rrf

Which brings us to the equation for expected return:

E(Ri) = Rrf + βi(E(Rm) – Rrf)

7.7 The Capital Asset Pricing Model

The Capital Asset Pricing Model (CAPM) is a model that describes the

relation between risk and expected return: E(Ri) = Rrf + βi(E(Rm) – Rrf).

A. The Security Market Line

Security Market Line (SML) is the line described by:

E(Ri) = Rrf + βi(E(Rm) – Rrf)

The SML illustrates what the CAPM predicts the expected total return should

be for various values of beta. The actual expected total return depends on the

price of the asset: . If an asset’s price implies that the

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expected return is greater than that predicted by the CAPM, that asset will plot

above the SML.

B. The Capital Asset Pricing Model and Portfolio Returns

The expected return for a portfolio:

E(Rn Asset portfolio) = Rrf + βn Asset portfolio(E[Rm] – Rrf)

The above can be found by applying the expected return and the beta of a

portfolio:

The expected return of a portfolio:

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II. Suggested and Alternative Approaches to the Material

This is a key chapter in that it addresses the measurable aspect of the risk and return relationship

for assets. The chapter begins with simple statistical concepts of expected return and

variance/standard deviation. The chapter quickly points out that these simple statistical tools do

not properly capture the correct measure of risk for an asset. Covariance and correlation

coefficients are then introduced for their impact on the variance of a multiasset portfolio where

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the benefits of diversification can be easily seen. The difference between systematic and

nonsystematic risk is then discussed as well as the need for a new measure of the systematic risk

component, separate from the nonsystematic risk component. Beta is then introduced as well as

the intuitional arguments for the CAPM and the SML and the SML equation. Applications for all

of the concepts are spread throughout the chapter.

The basis for how an instructor should approach this chapter depends on the student’s

statistical background. For advanced students, a review of the concepts will suffice before

proceeding. For less-prepared students, the instructor may need to take a day of lecture to

reinforce concepts that should have been learned in other courses. Regardless of the level of

student preparedness, this chapter holds important intuitional arguments that will escape the

students if they are unable to focus at a level beyond the simple statistical concepts. Those

arguments will then be required for material such as the cost of capital, capital budgeting, and

dividends. Therefore, it is imperative that the instructor adequately cover the material in this

chapter.

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III. Summary of Learning Objectives

1. Explain the relation between risk and return.

Investors require greater returns for taking greater risk. They prefer the investment with

the highest possible return for a given level of risk or the investment with the lowest risk

for a given level of return.

2. Describe the two components of a total holding period return, and calculate this

return for an asset.

The total holding period return on an investment consists of a capital appreciation

component and an income component. This return is calculated using Equation 7.1. It is

important to recognize that investors do not care whether they receive a dollar of return

through capital appreciation or as a cash dividend. Investors value both sources of return

equally.

3. Explain what an expected return is, and calculate the expected return for an asset.

The expected return is a weighted average of the possible returns from an investment,

where each of these returns is weighted by the probability that it will occur. It is

calculated using Equation 7.2.

4. Explain what the standard deviation of returns is and why it is especially useful in

finance, and be able to calculate it.

The standard deviation of returns is a measure of the total risk associated with the returns

from an asset. It is useful in evaluating returns in finance because the returns on many

assets tend to be normally distributed. The standard deviation of returns provides a

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convenient measure of the dispersion of returns. In other words, it tells us about the

probability that a return will fall within a particular distance from the expected value or

within a particular range. To calculate the standard deviation, the variance is first

calculated using Equation 7.3. The standard deviation of returns is then calculated by

taking the square root of the variance.

5. Explain the concept of diversification.

Diversification is a strategy of investing in two or more assets whose values do not

always move in the same direction at the same time in order to reduce risk. Investing in a

portfolio containing assets whose prices do not always move together reduces risk

because some of the changes in the prices of individual assets offset each other. This can

cause the overall volatility in the value of the portfolio to be lower than if it were

invested in a single asset.

6. Discuss which type of risk matters to investors and why.

Investors only care about systematic risk. This is because they can eliminate unique risk

by holding a diversified portfolio. Diversified investors will bid up prices for assets to the

point at which they are just being compensated for the systematic risks they must bear.

7. Describe what the Capital Asset Pricing Model (CAPM) tells us and how to use it to

evaluate whether the expected return of an asset is sufficient to compensate an

investor for the risks associated with that asset.

The CAPM tells us that the relation between systematic risk and return is linear and that

the risk-free rate of return is the appropriate return for an asset with no systematic risk.

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From the CAPM we know what rate of return investors will require for an investment

with a particular amount of systematic risk (beta). This means that we can use the

expected return predicted by the CAPM as a benchmark for evaluating whether expected

returns for individual assets are sufficient. If the expected return for an asset is less than

that predicted by the CAPM, then the asset is an unattractive investment because its

return is lower than the CAPM indicates it should be. By the same token, if the expected

return for an asset is greater than that predicted by the CAPM, then the asset is an

attractive investment because its return is higher than it should be.

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IV. Summary of Key Equations

Equation Description Formula

7.1 Total holding period return

7.2 Expected return on an asset

7.3 Variance of return on an asset

7.4 Coefficient of variation

7.5 Expected return for a portfolio

7.6Variance for a two-asset

portfolio

7.7 Covariance between two assets

7.8 Correlation between two assets

7.9Expected return and systematic

riskE(Ri) = Rrf + βi(E(Rm) – Rrf)

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7.10 Portfolio beta

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V. Before You Go On Questions and Answers

(Note to instructor: There are no questions for Section 7.1.)

Section 7.2

1. What are the two components of a total holding period return?

Capital appreciation and income. This can be seen in Equation 7.1.

2. How is the expected return on an investment calculated?

The expected return is calculated as a weighted-average of the possible returns on an

investment (outcomes) where the weights are the probabilities that each of the possible

returns will be realized.

Section 7.3

1. What is the relation between the variance and the standard deviation?

The standard deviation is the square root of the variance. Alternatively, the variance

equals the standard deviation squared.

2. What relation do we generally observe between risk and return when we examine

historical returns?

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Investments with higher risks tend to have higher returns. This is illustrated in Exhibit

7.3.

3. How would we expect the standard deviation of the return on an individual stock to

compare with the standard deviation of the return on a stock index?

We would expect the standard deviation of the return on an individual stock to be greater

than the standard deviation of the return on a stock index. For an illustration, see Exhibit

7.5.

Section 7.4

1. What does the coefficient of variation tell us?

The coefficient of variation is a measure of risk per unit of return. It tells us the amount of

risk, defined as the standard deviation of returns, associated with each 1 percent of

expected return for an asset. A larger coefficient of variation indicates greater risk for

each 1 percent of return.

2. What are the two components of total risk?

The two components of total risk are unique risk and systematic risk. Unique risk is risk

that is unique to a particular asset. This is the risk that can be eliminated through

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diversification. Systematic risk is risk that is common to all assets and cannot be

diversified away.

3. Why does the total risk of a portfolio not approach zero as the number of assets in a

portfolio becomes very large?

Because the systematic risk associated with the individual assets cannot be diversified

away. With a very large number of assets, the total risk of a portfolio will approach the

weighted average (where the weights are the fractions of total portfolio value represented

by each asset) of the systematic risks associated with each asset.

Section 7.5

1. Why are returns on the stock market used as a benchmark in measuring systematic risk?

Because the stock market portfolio is the most diversified portfolio for which good return

data are available, it is the portfolio that both comes closest to eliminating all unique risk

and has good return data. This makes the returns on the stock market a practical choice as

a benchmark for measuring the systematic risk of individual assets.

2. How is beta estimated?

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Beta is estimated using regression analysis in which returns on an individual asset are

regressed against returns on the market. Beta is the slope of the regression line. This is

illustrated in Exhibit 7.11.

3. How would you interpret a beta of 1.5 for an asset? A beta of 0.75?

A beta of 1.5 indicates that the asset has 1.5 times as much systematic risk as the market.

A beta of 0.75 indicates that the asset has 75 percent as much systematic risk as the

market.

(Note to instructor: There are no questions for Section 7.6.)

Section 7.7

1. How is the expected return on an asset related to its systematic risk?

CAPM tells us that there is a linear relation between expected return and systematic risk.

With zero systematic risk, the expected return equals the risk-free rate. For systematic risk

greater than zero, the expected return on an asset increases as its systematic risk increases

and this increase is linear. This relation is illustrated in Equation 7.9.

2. What name is given to the relation between risk and expected return implied by the

CAPM?

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This plot is called the Security Market Line, or SML, and is illustrated in Exhibit 7.12.

3. If an asset’s expected return does not plot on the line in question 2 above, what does that

imply about its price?

If the expected return on an asset does not plot on the SML, then this indicates that the

expected return on the asset is either too low or two high in view of its systematic risk. If

the asset plots below the SML, the expected return is too low and the price is too high. If

the asset plots above the SML, the expected return is too high and its price is too low.

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VI. Self-Study Problems

7.1 Kaaran made a friendly wager with a colleague that involves the result from flipping a

coin. If heads comes up, Kaaran must pay her colleague $15. Otherwise, her colleague will

pay Kaaran $15. What is Kaaran’s expected cash flow, and what is the variance of that

cash flow if the coin has an equal probability of coming up heads or tails? Suppose

Kaaran’s colleague is willing to handicap the bet by paying her $20 if the coin toss results

in tails. If everything else remains the same, what are Kaaran’s expected cash flow and the

variance of that cash flow?

Solution:

Part 1: E(cash flow) = (0.5 x –$15) + (0.5 x $15) = 0

σ2cash flow = [0.5 x (–$15 - $0)2] + [0.5 x ($15 – $0)2] = $225

Part 2: E(cash flow) = (0.5 x –$15) + (0.5 x $20) = $2.50

σ2cash flow = [0.5 x (–$15 – $2.50)2] + [0.5 x ($20 – $2.50)2] = $306.25

7.2 You know that the price of CFI, Inc., stock will be $12 exactly one year from today. Today

the price of the stock is $11. Describe what must happen to the price of CFI, Inc., today in

order for an investor to generate a 20 percent return over the next year. Assume that CFI

does not pay dividends.

Solution:

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The expected return for CFI based on today’s stock price is ($12 – $11)/$11 = 9.09%.

Therefore, you require a higher return. Since the stock price one year from today is fixed,

then the only way that you will generate a 20 percent return is if the price of the stock drops

today. Consequently, the price of the stock today must drop to $10. It is found by solving the

following: 0.2 = ($12 – x)/ x, or x = $10.

7.3 Two men are making a bet according to the outcome of a coin toss. You know that the

expected outcome of the bet is that one man will lose $20. Suppose you know that if that

same man wins the coin toss, he will receive $80. How much will he pay out if he loses the

coin toss?

Solution:

Since you know that the probability of any coin toss outcome is equal to 0.5, you can

solve the problem by setting up the following equation:

–$20 = (0.5 x $80) + (0.5 x x)

and solving for x:

0.5 x x = -$20 – (0.5 x $80)

x = [-$20 – (0.5 x $80)]/0.5 = $120

which means that he pays $120 if he loses the bet.

7.4 The expected value of a normal distribution of prices for a stock is $50. If you are 90

percent sure that the price of the stock will be between $40 and $60, then what is the

variance of the prices for the stock?

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Solution:

Since you know that 1.645 standard deviations around the expected return captures 90

percent of the distribution, you can set up either of the following equations:

$40 = $50 – 1.645σ or $60 = $50 + 1.645σ

and solve for σ. Doing this with either equation yields

σ = $6.079 and σ2 = 36.954

7.5 The JCHart Co. common shares have an expected return of 25 percent and a coefficient of

variation of 2.0. What is the variance of JCHart Co. common share returns?

Solution:

Since the coefficient of variation = CVi = σRi /E(Ri), substituting in the coefficient of

variation and E(Ri) allows us to solve for σ2return as follows:

2.0 = σRi/0.25

σRi = 0.5

σ2Ri = (0.5)2 = 0.25

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VII.Critical Thinking Questions

7.1 Given that you know the risk as well as the expected return for two stocks, discuss what

process you might utilize to determine which of the two stocks is a better buy. You may

assume that the two stocks will be the only assets held in your portfolio.

You should be looking to maximize your expected return on an investment given the

level of risk that such an investment requires the investor to bear. Therefore, you should

compare the expected return and risk associated with each of the two stocks. If the stocks

have the same expected return, then choose the stock with the lower risk. If the stocks

have the same risk, then choose the stock with the greatest expected return. If the

expected return and risk of the two assets have no common level, perhaps you should

compare the ratio of the risk/expected return to see which stock contains the least risk per

unit of expected return.

7.2 What is the difference between the expected rate of return and the required rate of return?

What does it mean if they are different for a particular asset at a particular point in time?

The required rate of return is the rate of return that investors require to compensate them

for the risk associated with an investment. The expected return will not necessarily equal

the required rate of return. The expected return can be lower, in which case the return will

not be sufficient to compensate the investor for the risk associated with the investment if

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the expected return is realized. It can also be higher, in which case the expected return

will be greater than that necessary to compensate the investor for the riskiness of the

asset.

7.3 Suppose that the standard deviation of the returns on the shares of stock at two different

companies is exactly the same. Does this mean that the required rate of return will be the

same for these two stocks? How might the required rate of return on the stock of a third

company be greater than the required rates of return on the stocks of the first two

companies even if the standard deviation of the returns of the third company’s stock is

lower?

No. Because some risk can be diversified away, it is possible that two stocks with the

same standard deviation of returns can have different required rates of return. One of these

stocks can have a higher systematic risk than the other stock and, therefore, a higher

required rate of return. The third stock can have a higher required rate of return if its

systematic risk is greater than the systematic risk of the stock in the other two companies.

7.4 The correlation between stocks A and B is 0.50, while the correlation between stocks A

and C is –0.5. You already own stock A and are thinking of buying either stock B or stock

C. If you want your portfolio to have the lowest possible risk, would you buy stock B or

C? Would you expect the stock you choose to affect the return that you earn on your

portfolio?

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You would buy stock C because it would result in your portfolio having a lower beta. If

you buy stock C, the required return for your portfolio would be lower than the required

return would be if you bought stock B. If the expected returns on stocks C and B equal

their required returns, then you would expect your portfolio to earn less with stock C.

7.5 The model where we know the return on a security for each possible outcome is overly

simplistic in many ways. However, even though we cannot possibly predict all possible

outcomes, this fact has little bearing on the risk-free return. Explain why.

The risk-free security delivers the same return in all states of the world. Even though we

do not know all of the possible states of the world in future periods, we do know that the

U.S. government will be able to repay its borrowing in every state of the world. Therefore,

the shortcoming of the model does not affect the risk-free security’s return.

7.6 Which investment category has shown the greatest degree of risk in the United States since

1926? Explain why that makes sense in a world where the price of a small stock is likely to

be more adversely affected by a particular negative event than the price of a corporate

bond. Use the same type of explanation to help explain other investment choices since

1926.

Small stocks have generally been riskier than large stocks, long-term corporate bonds,

long-term government bonds, intermediate government bonds, and short-term government

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bonds. The explanation for this can be best understood if we realize that small stocks will

be affected to a greater extent than the list of investments above by either good or bad

states of the world. These good and bad effects translate into a distribution with greater

spread or greater risk than the other investments.

7.7 You are concerned about one of the investments in your fully diversified portfolio. You

just have an uneasy feeling about the CFO, Iam Shifty, of that particular firm. You do

believe, however, that the firm makes a good product and that it is appropriately priced by

the market. Should you be concerned about the effect on your portfolio if Shifty embezzles

a portion of the firm’s cash?

The risk of Shifty embezzling is a nonsystematic risk that will most likely be offset by a

more fortunate event affecting another holding in your portfolio. Therefore, Shifty’s

actions should not affect the risk that you bear by investing in your diversified portfolio

(systematic risk).

7.8 The CAPM is used to price the risk in any asset. Our examples have focused on stocks, but

we could also price the expected rate of return for bonds. Explain how debt securities are

also subject to systematic risk.

A firm’s ability to repay its debt obligations will be affected in a very similar, and yet

lessened, way than the firm’s stock will be affected. That is, systematic and nonsystematic

factors will also affect returns for debt securities. However, if held in a diversified

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portfolio, then only the systematic risk component will be borne and then compensated for

bearing. Therefore, we can use the CAPM to price debt securities.

7.9 In recent years, investors have correctly agreed that the market portfolio consists of more

than just a group of U.S. stocks and bonds. If you are an investor who only invests in U.S.

stocks, describe the effects on the risk in your portfolio.

If the market portfolio is composed of all assets, then the U.S.-only portfolio will probably

have a small amount of nonsystematic risk that is not providing return compensation for

that risk. Therefore, the portfolio is bearing too much risk given its expected returns.

7.10 You may have heard the statement that you should not include your home as an asset in

your investment portfolio. Assume that your house will comprise up to 75 percent of your

assets in the early part of your investment life. Evaluate omitting it from your portfolio

when calculating the risk of your overall investment portfolio.

From a systematic risk measurement perspective, omitting the beta of your real estate

investment, which does not have a beta equal to zero, could have a serious impact on your

portfolio’s perceived systematic risk. In a volatile real estate market, you could be

understating the risk in your portfolio, and in a flat real estate market, you could be

overestimating the risk in your portfolio.

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VIII. Questions and Problems

BASIC

7.1 Returns: Describe the difference between a total holding period return and an expected

return.

Solution:

The holding period return is the total return over some investment or “holding” period. It

consists of a capital appreciation component and an income component. The holding

period return reflects past performance. The expected return is a return that is based on

the probability-weighted average of the possible returns from an investment. It describes

a possible return (or even a return that may not be possible) for a yet to occur investment

period.

7.2 Expected returns: John is watching an old game show on rerun television called Let’s

Make a Deal in which you have to choose a prize behind one of two curtains. One of the

curtains will yield a gag prize worth $150, and the other will give a car worth $7,200. The

game show has placed a subliminal message on the curtain containing the gag prize,

which makes the probability of choosing the gag prize equal to 75 percent. What is the

expected value of the selection, and what is the standard deviation of that selection?

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Solution:

E(prize) =0 .75($150) + (0.25) ($7,200) = $1,912.50

σ2prize = 0.75($150 – $1,912.50)2 + (0.25) ($7,200 – $1,912.50)2

= $9,319,218.75 =>

σprize = ($9,319,218.75)1/2 = $3,052.74

7.3 Expected returns: You have chosen biology as your college major because you would

like to be a medical doctor. However, you find that the probability of being accepted into

medical school is about 10 percent. If you are accepted into medical school, then your

starting salary when you graduate will be $300,000 per year. However, if you are not

accepted, then you would choose to work in a zoo, where you will earn $40,000 per year.

Without considering the additional educational years or the time value of money, what is

your expected starting salary as well as the standard deviation of that starting salary?

Solution:

E(salary) = 0.9($40,000) + (0.1) ($300,000) = $66,000

σ2salary = 0.9($40,000 – $66,000)2 + (0.1) ($300,000 – $66,000)2 = $6,084,000,000

σsalary = ($6,084,000,000)1/2 = $78,000

7.4 Historical market: Describe the general relation between risk and return that we observe

in the historical bond and stock market data.

Solution:

32

The general axiom that the greater the risk, the greater the return describes the historical

returns of the bond and stock market. If we look at Exhibit 7.4 in the text, we see that

small stocks have averaged the greatest returns but that they also have the greatest

standard deviation for the returns. When compared to large stocks, the average return and

standard deviation of the small stocks are greater. Large stock average returns and

standard deviation numbers are larger than those of long-term government bonds, which

are larger than those of intermediate-term government bonds, which in turn are larger

than those of U.S. Treasury bills. The comparison shows that the riskier the investment

category, the greater the average return as well as standard deviation of returns.

7.5 Single-asset portfolios: Stocks A, B, and C have expected returns of 15 percent, 15

percent, and 12 percent, respectively, while their standard deviations are 45 percent, 30

percent, and 30 percent, respectively. If you were considering the purchase of each of

these stocks as the only holding in your portfolio, then which stock should you choose?

Solution:

Since the holding will be made in a completely undiversified portfolio, then we can

calculate the risk per unit of return for each stock, the coefficient of variation, and choose

the stock with the lowest value.

CV(RA) = 0.45/0.15 = 3.0

CV(RB) = 0.30/0.15 = 2.0

CV(RC) = 0.30/0.12 = 2.5 ===> Choose B

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Alternatively, we could have noted that the expected return for A and B was the same,

with A having a greater degree of risk. B and C have the same degree of risk, but B has a

greater expected return. This would lead you to the conclusion, just as our coefficient of

variation calculations did, that Stock B is superior.

7.6 Diversification: Describe how investing in more than one asset can reduce risk through

diversification.

Solution:

An investor can reduce the risk of his or her investments by investing in two or more

assets whose values do not always move in the same direction at the same time. This is

because the movements in the values of the different investments will partially cancel

each other out.

7.7 Systematic risk: Define systematic risk.

Solution:

Risk that cannot be diversified away is called systematic risk. It is the only type of risk

that exists in a diversified portfolio, and it is the only type of risk that is rewarded in asset

markets.

7.8 Measuring systematic risk: Susan is expecting the returns on the market portfolio to be

negative in the near term. Since she is managing a stock mutual fund, she must remain

34

invested in a portfolio of stocks. However, she is allowed to adjust the beta of her

portfolio. What kind of beta would you recommend for Susan’s portfolio?

Solution:

If we confine our analysis to portfolios with positive beta values, and since beta describes

how much and what direction our portfolio is expected to vary with the market portfolio,

then Susan should construct a very low beta portfolio. In that case, Susan’s portfolio is

not expected to have losses quite as large as that of the market portfolio. A large beta

portfolio would have larger losses than that of the market portfolio. If Susan could

construct a negative beta portfolio, then she would like to construct as negative a

portfolio beta as possible.

7.9 Measuring systematic risk: Describe and justify what the value of the beta of a U.S.

Treasury bill should be.

Solution:

Since the beta of any asset is the slope of the line of best fit for the plot of an asset against

that of the market return, then we can use that logic to help us understand the beta of a T-

bill. If we purchased a T-bill five years ago and held the same T-bill through each of the

last 60 months, then the return for each of those 60 months would be exactly the same.

Therefore, the vertical axis coordinates of each of the monthly returns would have the

same value and the slope (beta) of the line of best fit would be zero. The meaning of a

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beta of zero means that our T-bill has no systematic risk. That is logical given that we

know that a T-bill has no risk at all since it is a riskless asset.

7.10 Measuring systematic risk: If the expected rate of return for the market is not much

greater than the risk-free rate of return, what is the general level of compensation for

bearing systematic risk?

Solution:

Such a situation suggests that return compensation for investing in an asset is determined

more by the risk-free return than by the market’s compensation for bearing systematic

risk. This means that the price for bearing systematic risk is very low. This may be caused

by a very low perceived level of risk in the market or by an abundance of funds in the

market seeking to be invested in risky assets.

7.11 CAPM: Describe the Capital Asset Pricing Model (CAPM) and what it tells us.

Solution:

The CAPM is a model that describes the relation between systematic risk and the

expected return. The model tells us that the expected return on an asset with no

systematic risk equals the risk-free rate. As systematic risk increases, the expected return

increases linearly with beta. The CAPM is written as E(Ri) = Rrf + i(E(Rm) – Rrf) .

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7.12 The Security market line: If the expected return on the market is 10 percent and the

risk-free rate is 4 percent, what is the expected return for a stock with a beta equal to 1.5?

What is the market risk premium for the set of circumstances described?

Solution:

Following the CAPM prediction:

(Rcs) = Rrf + β (E(RM) – Rrf) = 0.04 + 1.5(0.1 – 0.04) = 0.13

The market risk premium is (E(RM) – Rrf) = 0.06

INTERMEDIATE

7.13 Expected returns: Jose is thinking about purchasing a soft drink machine and placing it

in a business office. He knows that there is a 5 percent probability that someone who

walks by the machine will make a purchase from the machine, and he knows that the

profit on each soft drink sold is $0.10. If Jose expects a thousand people per day to pass

by the machine and requires a complete return of his investment in one year, then what is

the maximum price that he should be willing to pay for the soft drink machine? Assume

250 working days in a year and ignore taxes.

Solution:

E(Revenue) = 1,000 x 0.05 x $.10 x 250 days = $1,250

Therefore, the most Jose should pay for the machine is $1,250.

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7.14 Interpreting the variance and standard deviation: The distribution of grades in an

introductory finance class is normally distributed, with an expected grade of 75. If the

standard deviation of grades is 7, in what range would you expect 90 percent of the

grades to fall?

Solution:

95% is 1.96 standard deviations from the mean

75 – 1.96(7) = 61.28

7.15 Calculating the variance and standard deviation: Kate recently invested in real estate

with the intention of selling the property one year from today. She has modeled the

returns on that investment based on three economic scenarios. She believes that if the

economy stays healthy, then her investment will generate a 30 percent return. However, if

the economy softens, as predicted, the return will be 10 percent, while the return will be –

25 percent if the economy slips into a recession. If the probabilities of the healthy, soft,

and recessionary states are 0.4, 0.5, and 0.1, respectively, then what are the expected

return and the standard deviation for Kate’s investment?

Solution:

E(Ri) = (0.4)(0.3) + (0.5) (0.1) + (0.1) (–.25) = 0.145

σ2return = (0.4)(0.3 – 0.145)2 + (0.5) (0.1 – 0.145)2 + (0.1) (–0.25 – 0.145)2

= 0.02623

σreturn = (0.02623)1/2 = 0.16194

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7.16 Calculating the variance and standard deviation: Barbara is considering investing in a

stock, and is aware that the return on that investment is particularly sensitive to how the

economy is performing. Her analysis suggests that four states of the economy can affect

the return on the investment. Using the table of returns and probabilities below, find the

expected return and the standard deviation of the return on Barbara’s investment.

Probability Return

Boom 0.1 25.00%

Good 0.4 15.00%

Level 0.3 10.00%

Slump 0.2 -5.00%

Solution:

E(Ri) = 0.1(0.25) + (0.4) (0.15) + (0.3) (0.1) + (0.2) (–o.05) = 0.105

σ2return = 0.1(0.25 – 0.105)2 + (0.4) (0.15 – 0.105)2 + (0.3) (0.1 – 0.105)2 + (0.2) (–0.5 – 0.105)2

= 0.00773

σreturn = (0.00773)1/2 = 0.08789

7.17 Calculating the variance and standard deviation: Ben would like to invest in gold and

is aware that the returns on such an investment can be quite volatile. Use the following

table of states, probabilities, and returns to determine the expected return on Ben’s gold

investment.

Probability Return

Boom 0.1 45.00%

39

Good 0.2 30.00%

OK 0.3 15.00%

Level 0.2 2.00%

Slump 0.2 -12.00%

Solution

E(Ri) = 0.1(0.4) + (0.2) (0.3) + (0.3) (0.15) + (0.2) (0.02) + (0.2) (–0.12) = 0.125

σ2return = 0.1(0.4 – 0.125)2 + (0.2) (0.3 – 0.125)2 + (0.3) (0.15 – 0.125)2 + (0.2) (0.02 – 0.125)2 +

(0.2) (–0.12 – 0.125)2

= 0.02809

σreturn = (0.02809)1/2 = 0.16759

7.18 Single-asset portfolios: Using the information from Problems 7.15, 7.16, and 7.17,

calculate each coefficient of variation.

Solution:

Coefficient of variation = σReturn / E(Ri)

Problem 15: 0.16194/0.145 = 1.11684 (using the exact values rather than the printed)



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7.19 Portfolios with more than one asset: Emmy is analyzing a two-stock portfolio that

consists of a Utility stock and a Commodity stock. She knows that the return on the

Utility has a standard deviation of 40 percent, and the return on the Commodity has a

standard deviation of 30 percent. However, she does not know the exact covariance in the

returns of the two stocks. Emmy would like to plot the variance of the portfolio for each

of three cases—covariance of 0.12, 0, and –0.12—in order to understand how the

variance of such a portfolio would react. Do the calculation for each of the extreme cases

(0.12 and –0.12), assuming an equal proportion of each stock in Emmy’s portfolio.

Solution:

Part 1, σ12 = 0.12:

(0.5)2 (0.4)2 + (0.5)2 (0.3)2 + 2(0.5)(0.5)(0.12) = 0.1225

Part 2, ρ = 0.0:

(0.5)2 (0.4)2 + (0.5)2 (0.3)2 + 2(0.5)(0.5)(0.0) = 0.0625

Part 3, σ12 = -0.12:

(0.5)2 (0.4)2 + (0.5)2 (0.3)2 + 2(0.5)(0.5)(-0.12) = 0.0025

7.20 Portfolios with more than one asset: Given the returns and probabilities for the three

possible states listed here, calculate the covariance between the returns of Stock A and

Stock B. For convenience, assume that the expected returns of Stock A and Stock B are

11.75 percent and 18 percent, respectively.

Probability Return(A) Return(B)

41

Good 0.35 0.30 0.50

OK 0.50 0.10 0.10

Poor 0.15 -0.25 -0.30

Solution:

7.21 Compensation for bearing systematic risk: You have constructed a diversified

portfolio of stocks such that there is no nonsystematic risk. Explain why the expected

return of that portfolio should be greater than the expected return of a risk-free security.

Solution:

Your portfolio contains no nonsystematic risk but it does in fact contain systematic risk.

Therefore, the market should compensate the holder of this portfolio for the systematic

risk that the investor bears. The risk-free security has no risk and therefore requires no

compensation for risk bearing. The expected return of the portfolio should therefore be

greater than the return of the risk-free security.

7.22 Compensation for bearing systematic risk: Write out the equation for the covariance in

the returns of two assets, Asset 1 and Asset 2. Using that equation, explain the easiest

way for the two asset returns to have a covariance of zero.

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Solution:

We know that all state probabilities must be greater than zero, and thus the source of a

zero covariance cannot be from the state probabilities. The easiest way for the entire

probability weighted sum to equal zero is for one of the assets, say Number 1(2), to have

a value in all states j that is equal to the expected return of Number 1(2). Another way of

saying that is for one of the assets to have a constant return in all states. If that occurs,

then the second term in the equation will always be equal to zero, causing the sum, or

covariance, to be zero.

7.23 Compensation for bearing systematic risk: Evaluate the following statement: By fully

diversifying a portfolio, such as by buying every asset in the market, we can completely

eliminate all types of risk, thereby creating a synthetic Treasury bill.

Solution:

The statement is false. Even if we could afford such a portfolio and thus completely

diversify our portfolio, we would only be eliminating nonsystematic risk. The systematic

risk generated by the portfolio would remain. Otherwise, the expected rate of return on

the market portfolio would be equal to the risk-free rate of return. We know that to be a

false statement.

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7.24 CAPM: Damien knows that the beta of his portfolio is equal to 1, but he does not know

the risk-free rate of return or the market risk premium. He also knows that the expected

return on the market is 8 percent. What is the expected return on Damien’s portfolio?

Solution:


(Rcs) = Rrf + β (E(RM) – Rrf) = Rrf + E(RM) – Rrf = E(RM) = 0.08

ADVANCED

7.25 David is going to purchase two stocks to form the initial holdings in his portfolio. Iron

stock has an expected return of 15 percent, while Copper stock has an expected return of

20 percent. If David plans to invest 30 percent of his funds in Iron and the remainder in

Copper, then what will be the expected return from his portfolio? What if David invests

70 percent of his funds in Iron stock?

Solution:

Part 1: E(Rport) = (0.3)(0.15) + (0.7)(0.2) = 0.185

Part 2: E(Rport) = (0.7)(0.15) + (0.3)(0.2) = 0.165

7.26 Sumeet knows that the covariance in the return on two assets is –0.0025. Without

knowing the expected return of the two assets, explain what that covariance means.

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Solution:

The covariance measure is dependent on the expected return of the two assets in

questions, so without the expected return of the two assets, it is difficult to characterize

the scale of the covariance. However, since the covariance is negative, we can say that

generally the two assets move in opposite directions, with respect to their own means,

from each other in given states of nature.

7.27 In order to fund her retirement, Glenda requires a portfolio with an expected return of 12

percent per year over the next 30 years. She has decided to invest in Stocks 1, 2, and 3,

with 25 percent in Stock 1, 50 percent in Stock 2, and 25 percent in Stock 3. If Stocks 1

and 2 have expected returns of 9 percent and 10 percent per year, respectively, then what

is the minimum expected annual return for Stock 3 that will enable Glenda to achieve her

investment requirement?

Solution:

The formula for the expected return of a three-stock portfolio is:

Therefore, we can solve as in the following:

0.12 = 0.25(0.09) + 0.5(0.1) + 0.25E(R3)

0.19 = E(R3)

7.28 Tonalli is putting together a portfolio of 10 stocks in equal proportions. What is the

relative importance of the variance for each stock versus the covariance for the pairs of

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stocks? For this exercise, ignore the actual values of the variance and covariance terms

and explain their importance conceptually.

Solution:

The variance of the portfolio will be composed of 10 (n = 10) individual stock variance

terms and 45 ((n2 –n)/2) covariance terms (really 90). Therefore, the vast majority of the

portfolio variance calculation will be determined by the covariance terms of the portfolio

in most cases.

7.29 Explain why investors who have diversified their portfolios will determine the price and,

consequently, the expected return on an asset.

Solution:

If we assume that all investors will seek to be compensated (generate returns) for the level

of risk that they are bearing, then we can see that undiversified investors will require a

greater return for a given investment than diversified investors will. Given that, we can

see that diversified investors will be willing to pay a greater price for an asset than

undiversified investors. Therefore, the diversified investor is the marginal investor whose

purchase will determine the equilibrium price, and therefore the equilibrium return for an

asset.

7.30 Brad is about to purchase an additional asset for his well-diversified portfolio. He notices

that when he plots the historical returns of the asset against those of the market portfolio,

46

the line of best fit tends to have a large amount of prediction error for each data point (the

scatter plot is not very tight around the line of best fit). Do you think that this will have a

large or a small impact on the beta of the asset? Explain your opinion.

Solution:

It will have no effect on the beta of the asset. The beta measures only the systematic risk

or variation in the returns of the asset. The prediction error reflects the nonsystematic risk

inherent in the returns of the asset and will consequently not affect the beta of the asset.

7.31 The beta of an asset is equal to 0. Discuss what the asset must be.

Solution:


(Rcs) = Rrf + β (E(RM) – Rrf) = Rrf + 0 (E(RM) – Rrf) = Rrf

Therefore, the expected return on the asset is equal to the risk-free rate of return. The only

way an asset could generate a risk-free rate of return is if the asset had no systematic risk

(otherwise the asset would have to compensate an investor for such risk bearing). This

implies that the asset must be the riskless asset, or, practically speaking, it must be a T-

bill.

7.32 The expected return on the market portfolio is 15 percent, and the return on the risk-free

security is 5 percent. What is the expected return on a portfolio with a beta equal to 0.5?

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Solution:

The beta of the market portfolio is equal to 1. Therefore, we can use the Security Market

Line graph to determine the halfway point between the point (1, 15%) and the point (0,

5%). We can then average the first and second values of the two coordinates to arrive at

((1 + 0)/2, (15% + 5%), 2) or (0.5, 10%), which means that the expected return of a

portfolio with a beta equal to 0.5 is 10 percent.

7.33 Draw the Security Market Line (SML) for the case where the market risk premium is 5

percent and the risk-free rate is 7 percent. Now, suppose an asset has a beta of –1.0 and an

expected return of 4 percent. Plot it on your graph. Is the security properly priced? If not,

explain what we might expect to happen to the price of this security in the market. Next,

suppose another asset has a beta of 3.0 and an expected return of 20 percent. Plot it on the

graph. Is this security properly priced? If not, explain what we might expect to happen to

the price of this security in the market.

Solution:

The Security Market Line (SML) shows the relationship between an asset’s expected

return and its beta. We know the market has a beta of one, and we know the risk-free rate

has a beta of zero. The risk-free rate of return is 7 percent, and the market is expected to

return 5 percent more than this. Therefore, the expected rate of return for the market (a

beta one asset) is 12 percent. To draw this SML, we need only connect the dots:

48

We can see from the following diagram that an asset with expected return of 4 percent

and a beta of –1.0 is underpriced (its expected return is too high). As the market becomes

aware of this underpricing, investors will purchase the asset, bidding up its price until its

expected return falls on the SML. (Recall that as the initial purchase price of an asset

increases, the expected return from purchasing the asset will decrease because you are

paying a higher initial cost for the asset.)

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The investment will fall here in this plot

0%

3%

6%

9%

12%

15%

18%

0 1 2

Beta

Exp

ect

ed

Re

turn

As we can see from the following diagram, an asset with a beta of 3.0 should have an

expected return of 7% + (3)(5%) = 22%. The asset only has an expected return of 20

percent. Therefore, this asset is overpriced. Demand for this asset will be low, driving

down its market price, until the asset’s expected return falls on the SML.

.

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The investment will fall here in this plot

Sample Test Problems

7.1 SLVNT Airlines stock is selling at a current price of $37.50 per share. If the stock does

not pay a dividend and has a 12 percent expected return, then what is the expected price

of the stock one year from today?

Solution:

Using the formula for an asset’s return during a period,

7.2. Stefan’s parents are about to invest their nest egg in a stock that he has estimated to have

an expected return of 9 percent over the next year. If the stock is normally distributed

with a 3 percent standard deviation, in what range will the stock return fall 95 percent of

the time?

Solution:

Since the return distribution for the stock is normal, then a 95 percent confidence level

corresponds to 1.96 standard deviations. Therefore,

0.09 – 1.96(0.03) = 0.0312 or 3.12%

is the return that we would expect to be exceeded 95 percent of the time.

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7.3 Elaine has narrowed her investment alternatives to two stocks (at this time she is not

worried about diversifying): Stock M, which has a 23 percent expected return, and Stock

Y, which has an 8 percent expected return. If Elaine requires a 16 percent return on her

total investment, what proportion of her portfolio will she invest in each stock?

Solution:

If we let x = the proportion of the portfolio invested in M and (1 – x) = the proportion

invested in Y, then we can solve

0.23(x) + 0.08(1 – x) = 0.16 ==> x = 0.53

or 53 percent of the portfolio is to be invested in M, and therefore, 47 percent of the

portfolio is to be invested in Y.

7.4 You have just prepared a graph similar to Exhibit 7.9 comparing historical data for Pear

Computer Corp. and the general market. When you plot the line of best fit for these data,

you find that the slope of that line is 2.5. If you know that the market generated a return

of 12 percent and that the risk-free rate is 5 percent, then what would your best estimate

be for the return of Pear Computer during that same time period?

Solution:

Since the line of best fit has a slope of 2.5, then we know that the beta of Pear Computer

is also 2.5. This tells us that for every 1 percent change in the return on the market, we

can expect the return on Pear to be 2.5 percent. Therefore, our best estimate for the return

on Pear during this time period is 2.5 x 12% = 30%.

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7.5 You know that the CAPM predicts that the return of MoonBucks Tea Corp. is 23.6

percent. If the risk-free rate of return is 8 percent and the expected return on the market is

20 percent, then what is MoonBucks’s beta?

Solution:

Using the CAPM, we find

E(RMoonBucks) = Rrf + ßMoonBucks(E[RM] – Rrf)

0.236 = 0.08 + ßMoonBucks(0.20 – 0.08)

ßMoonBucks= 1.3

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