chapter 7 risk and return.pdf

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Chapter 7 Risk and Return

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 Lets

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

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

= $9,319,218.75 =>

prize = ($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

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

salary = ($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.

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Solution:

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

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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 Susans 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, Susans 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 markets 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 + (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 Kates investment?

Solution:

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

2return = (0.4)(0.3 0.145)2 + (0.5) (0.1 0.145)2 + (0.1) (0.25 0.145)2

= 0.02623

return = (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 Barbaras 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

2return = 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

return = (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 Bens gold

investment.

Probability Return

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Boom 0.1 45.00%

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

2return = 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

return = (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 = Return / 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 casescovariance of 0.12, 0, and 0.12in 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 Emmys portfolio.

Solution:

1221

2

2

2

2

2

1

2

12 2)( xxxxRVar portasset

Part 1, 12 = 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, = 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, 12 = -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)

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Good 0.35 0.30 0.50

OK 0.50 0.10 0.10

Poor 0.15 -0.25 -0.30

Solution:

0476.0)18.03.)(1175.025.(15.0

)18.01.0)(1175.01.0(5.0)18.05.0)(1175.03.0(35.0),(

ABBA RRCov

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.

Solution:

Comment [BP1]: Add the zero before.25 and .3 in the last line.

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121 2 R

1, 1 2, 2

1

Cov(Return ,Return )

x (Return E(Return ) x (Return E(Return )i

n

i i

i

p

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 Damiens portfolio?

Solution:


(Rcs) = Rrf + (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:

)()()()( 3322113 RExRExRExRE portasset

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,

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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 + (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 assets 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:

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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.)

0%

3%

6%

9%

12%

15%

18%

-1 0 1 2

Exp

ecte

d R

etu

rn

Beta

The investment

will fall here in

this plot

0%

3%

6%

9%

12%

15%

18%

0 1 2

Beta

Exp

ecte

d R

etu

rn

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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 assets expected return falls on the SML.

.

0%

5%

9%

14%

18%

23%

0 1 2 3

Expecte

d R

etu

rn

Beta

The investment

will fall here in

this plot

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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 assets return during a period,

00.42$P50.37$

50.37$P12.0

P

PPRR 1

1

0

01CAT

7.2. Stefans 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%.

Comment [BP2]: Is this number correct? Or did you mean Exhibit 7.10?

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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 MoonBuckss beta?

Solution:

Using the CAPM, we find

E(RMoonBucks) = Rrf + MoonBucks(E[RM] Rrf)

0.236 = 0.08 + MoonBucks(0.20 0.08)

MoonBucks= 1.3

chapter 7 risk and return.pdf

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