Chapter 7 Risk and Return
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 o f 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?
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 o f 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 7.4.
Section 7.4 1.
What does the coefficient of variation tell us and how is it related to the Sharpe Ratio?
The coefficient of variation is a measure of risk per unit of return. return. It tells us the amount of risk, defined as the standard deviation of returns, associated with each 1 percent of o f expected return for an asset. asset. A larger coefficient of variation indicates greater risk risk for each 1 percent of return.
The Sharpe ratio differs from the coefficient of variation in two ways. First, instead of using the total return in the calculation, with the Sharpe ratio we use the difference between the total return and the risk free rate. Second, the Sharpe ratio is the inverse of the coefficient of variation ratio that would be calculated using this adjusted return. Computing the inverse of the coefficient of variation yields a me asure return per unit of risk, which some investors find more intuitive than a measure of risk for each 1 percent of return.
2.
What are the two components of total risk?
The two components of total risk are unique un ique risk and systematic risk. Unique risk is risk that is unique to a particular asset. This is the risk that can be eliminated through
diversification. Systematic risk is risk that is common to all assets and canno t 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 nu mber 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 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 d iversified portfolio for which good return data are available, it is the portfolio po rtfolio 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 individua l assets.
2.
How is beta estimated?
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.
1.
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?
This plot is called the Security Market Line, or SML, and is illustrated in Exhibit 7.11.
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, SM L, then this indicates that the expected return on the asset is either too low or too 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, SM L, the expected return is too high and an d its price is too low.
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 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 K aaran’s expected cash flow and the variance of that cash flow?
Solution:
Part 1: E(cash flow) = (0.5 × – × – $15) $15) + (0.5 × $15) = 0 σ2cash flow = [0.5 × ( – – $15 $15 - $0)2] + [0.5 × ($15 – ($15 – $0) $0)2] = $225
Part 2: E(cash flow) = (0.5 × – × – $15) $15) + (0.5 × $20) = $2.50 σ2cash flow = [0.5 × ( – – $15 – $15 – $2.50) $2.50)2] + [0.5 × ($20 – ($20 – $2.50) $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 p rice of CFI, Inc., today in order for an investor to generate a 20 percent return over the next nex t year. Assume that CFI does not pay dividends.
Solution:
The expected return for CFI based on today’s stock price price is ($12 – ($12 – $11)/$11 $11)/$11 = 9.09 percent, which is lower than 20 percent. 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 toda y must drop to $10. It is found by solving the following: 0.2 = ($12 – ($12 – x x)/ x, or x x = $10.
7.3
The expected value of a normal distribution of prices for a stock is $50. If 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?
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 – $50 – 1.645σ 1.645σ
or
$60 = $50 + 1.645σ
and solve for σ. Doing this with either equation yields σ = $6.079 and σ2 = 36.954
7.4
You must choose between investing in stock A and stock B. You have already used CAPM to calculate the rate of return you should expect to receive for each stock given their systematic risk and decided that the expected return for both exceeds that predicted by CAPM by the same amount. In other words, both are equally attractive investments for a diversified investor. However, since you are still in school and do not have a lot of money, your investment portfolio is not diversified. You have decided to invest in the stock that has the highest expected return per unit of total risk. If the expected return and standard deviation of returns for stock A are 10 percent and 25 percent, respectively, and the expected return and standard deviation of returns for stock B are 15 percent and 40 percent, respectively, which should you choose? Assume that the risk free rate is 5 percent.
Solution: A comparison of the Sharpe ratios for the two stocks will tell you which has the
highest expected return per unit of total risk.
SA =
E(R A ) - R rf σ R
0.10 0.05 0.25
0.20
A
SB =
E(RB )-R rf σ R
0.15 0.15 0.05 0.05 0.40
0.25
B
Stock B has the highest expected ex pected return per unit of risk.
7.5
CSB, Inc., has a beta of 1.35. If the expected market return is 14.5 percent and the risk-free risk-free rate is 5.5 percent, what is the appropriate required return of CSB (using the CAPM)?
Solution:
E(R CSB CSB) = R rf rf + CSB[E(R M) - R rf rf ] = 0.055 +1.35 × (0.145 - 0.055)] = 0.1765 or, 17.65%
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 inv estor 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 cho ose 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 ex pected return can be lower, in which case the return will not be sufficient to compensate the investor for the th e risk associated with the investment if the expected return is realized. It can also be higher, in which case the expected return will be greater than that necessary to compensate co mpensate 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 d eviation 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 o f 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 – is – 0.5. 0.5. You already own stock A and are thinking of buying bu ying either stock B or stock C. If you want your portfolio to have the lowest possible risk, would you buy bu y stock B or C? Would you expect the stock you choose to affect the return that you earn on your portfolio?
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 tha n 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. Eve n 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 risk-free security’s return.
7.6
Which investment category has shown the greatest degree deg ree 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 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 non systematic risk that will most likely be offset by a more fortunate event affecting another holding in your portfolio. Therefore, Shifty’s Shifty’s actions should not affect the risk that you bear b ear 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 ex pected rate of return for bonds. Explain Ex plain 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 nonsystematic factors will also affect returns for debt securities. However, if held in a diversified 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 nons ystematic 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. 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.
Questions and Problems BASIC 7.1
Returns: Describe the difference between a total holding period return and an expected
return. LO 2, LO3
Solution:
The holding period return is the total return over some so me investment or ―holding‖ period. It consists of a capital appreciation component and an income component. The holding period return reflects past performance. 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 ma y 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. Behind one
of the curtains is a gag prize priz e worth $ 150, and behind the other is a round-the-world trip 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? LO 3
Solution:
E(prize) = (0 .75 × $150) + (0.25 (0.25 × $7,200) = $1,912.50 σ2 prize = (0.75 × ($150 – ($150 – $1,912.50) $1,912.50)2) + (0.25 × ($7,200 – ($7,200 – $1,912.50) $1,912.50)2 ) = $9,319,218.75
σ 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 $300,0 00 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 standa rd deviation of that starting salary? LO 3
Solution:
E(salary) = 0.9($40,000) + (0.1) ($300,000) = $66,000 σ2salary = 0.9($40,000 – 0.9($40,000 – $66,000) $66,000)2 + (0.1) ($300,000 – ($300,000 – $66,000) $66,000)2 = $$5,476,208,400σ $$5,476,208,400σsalary = ($5,476,208,400)1/2 = $74,002.74
7.4
t hat we observe Historical market: Describe the general relation between risk and return that in the historical bond and stock market data. LO 3, LO4
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 av erage returns and standard deviation numbers are larger than those of long-term government bonds, which are larger than those of intermediate-term government bonds, wh ich in turn are larger than those of U.S. Treasury bills. The comparison comp arison shows that the riskier the investment category, the greater the average return as well w ell 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 and the risk free rate is 0 percent, p ercent, which stock should you choose? LO 4
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(R A) = 0.45/0.15 = 3.0 CV(R B) = 0.30/0.15 = 2.0 CV(R C) = 0.30/0.12 = 2.5 ===> Choose B 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. LO 5
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. LO 6
Solution:
Risk that cannot be diversified away is called c alled systematic risk. It is the only type of risk that exists in a diversified portfolio, and it is the only o nly 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 S ince she is managing a stock mutual fund, she must remain 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? portfolio? LO 6
Solution:
If we confine our analysis to portfolios with positive beta valu es, and since beta describes how much and what direction dire ction our portfolio is expected to vary with the market portfolio, then Susan should construct a very low beta portfolio. portfolio. In that case, Susan’s portfolio is not expected to have losses quite as large as that of the market portfolio. po rtfolio. A large beta portfolio would have larger losses than that of the market portfolio. If 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. LO 6
Solution:
Since the beta of an y 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 e ach 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 beta of zero means that our T-bill has no systematic risk. That is logical logical given that we know that a T-bill has no risk at all since it is a riskless asset.
7.10
n ot much Measuring systematic risk: If the expected rate of return for the market is not greater than the risk-free rate of return, what is the general level of compensation for bearing systematic risk? LO 6
Solution:
Such a situation suggests that return compensation for investing in a n asset is determined more by the risk-free risk-free return than by the market’s compensatio co mpensation n 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. LO 7
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(R i) = R rf – R rf rf + i(E(R m) – R rf ) .
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 exp ected return for a stock with a beta equal to 1.5? What is the market risk premium for the set of circumstances described? LO 7
Solution:
Following the CAPM prediction: (R cs – R rf (0.1 – 0.04) 0.04) = 0.13 cs) = R rf rf + β (E(R M) – R rf ) = 0.04 + 1.5 × (0.1 –
The market risk premium is (E(R M) – R – R rf rf ) = 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? mac hine? Assume 250 working days in a year and ignore taxes. LO 3
Solution:
E(Revenue) = 1,000 × 0.05 x $.10 × 250 days = $1,250 Therefore, the most Jose should pay for the machine is $1,250.
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? LO 4
Solution:
90% is 1.645 standard deviations from the mean.
Thus, the range can be determined as:
75 (1.645 .645 (7)) (7)) So, the lower end is: 75 – 75 – (1.645 (1.645 × (7)) = 63.49 And the upper end is: 75 + (1.645 × (7)) = 86.52
So, 90 percent of the grades should fall between 63.49 and 86.52.
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 a =25 percent if the economy econo my 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? LO 4
Solution:
E(R i)
= (0.4)(0.3) + (0.5) (0.1) + (0.1) ( – – .25) .25) = 0.145
σ2return = (0.4)(0.3 – (0.4)(0.3 – 0.145) 0.145)2 + (0.5) (0.1 – (0.1 – 0.145) 0.145)2 + (0.1) ( – – 0.25 – 0.25 – 0.145) 0.145)2 = 0.02623
σreturn
7.16
= (0.02623)1/2 = 0.16194
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 econo my can affect the return on the investment. Using the table tabl e of returns and probabilities below, find the expected return and the standard s tandard deviation of the return on Barbara’s Ba rbara’s investment.
Probability
Return
Boom
0.1
25.00%
Good
0.4
15.00%
Level
0.3
10.00%
Slump
0.2
-5.00%
LO 4
Solution:
E(R i) = 0.1 × (0.25) + (0.4) × (0.15) + (0.3) × (0.1) + (0.2) – (– 0.05) 0.05) = 0.105 σ2return = 0.1 × (0.25 – (0.25 – 0.105) 0.105)2 + (0.4) × (0.15 – (0.15 – 0.105) 0.105)2 + (0.3) × (0.1 – (0.1 – 0.105) 0.105)2 + (0.2) × ( – – 0.5 – 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 th e following table of states, probabilities, and returns to determine the exp ected return on Ben’s gold investment. Probability
Return
Boom
0.1
40.00%
Good
0.2
30.00%
OK
0.3
15.00%
Level
0.2
2.00%
Slump
0.2
-12.00%
LO 4
Solution
E(R i) = 0.1 × (0.4) + (0.2) (0.2) × (0.3) + (0.3) × (0.15) + (0.2) × (0.02) (0.02) + (0.2) × – (– 0.12) 0.12) = 0.125
σ2return = 0.1 × (0.4 – (0.4 – 0.125) 0.125)2 + (0.2) × (0.3 – (0.3 – 0.125) 0.125)2 + (0.3) × (0.15 – (0.15 – 0.125) 0.125)2 + (0.2) × (0.02 – (0.02 – 0.125)2 + (0.2) × ( – – 0.12 – 0.12 – 0.125) 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 the coefficient of variation for each of the investments in these problems. LO 3, LO 4
Solution:
Coefficient of variation = σReturn / E(R i)
Problem 15: 0.16194/0.145 = 1.11684 (using the exact values rather than the printed)
Problem 16: 0.08789/0.105 = 0.83707 (using the exact values rather than the printed)
Problem 17: 0.16759/0.125 = 1.34069 (using the exact values rather than the printed)
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 covariance of 0.12, 0, and – and – 0.12 0.12 — in in order to understand how the variance of such a portfolio would react. Do the calculation for all three cases (0.12, 0 and – 0.12), 0.12), assuming an equal proportion of each stock in the portfolio. LO 5
Solution: Var Var ( R2 asset por t ) x1 1 2
2
x22 22 2 x1 x2 12
Case 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
Case 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
Case 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 below, 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)
Good
0.35
0.30
0.50
OK
0.50
0.10
0.10
Poor
0.15
-0.25
-0.30
LO 5
Solution:
Cov(R v(R A ,R B ) AB
0.35 (0.3 0.1175) (0.5 0.18) 0.5 (0.1 0.1175) (0.1 0.18)
0.15 ( 0.2 0.25 0.1175) (0.3 0.3 0.18) 0.0476
7.21
Compensation for bearing systematic risk: You have constructed a diversified
portfolio of stocks such that there is no nonsystematic risk. risk. Explain why the expected return of that portfolio should be greater than the expected return of a risk-free security. LO 6
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. LO 6
Solution:
Cov(Return1 ,Return2 ) R 12 n
(Return1,i E(Re (Return1 ) (Re (Return 2,i E(Return 2 ) p (Re i 1
i
We know that all state probabilities must be greater than zero, and thus the source of a zero covariance cannot be b e 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 bu ying every asset in the market, we can ca n completely eliminate all types of risk, thereby creating a synthetic Treasury bill. LO 6
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 nons ystematic 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.
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? LO 7
Solution:
Following the CAPM prediction: (R cs – R rf – R rf cs) = R rf rf + β (E(R M) – R rf ) = R rf rf + E(R M) – R rf = E(R M) = 0.08 = 8.0%
7.25
4. 75 percent, the market risk premium CAPM: In February 2011 the risk free rate was 4.75 was 6 percent and the beta bet a for Dell stock was 1.31. What is the expected return that was consistent with the systematic risk associated with the returns on Dell stock? LO 7
Solution: From Equation 7.10:
E(R Dell Dell stock ) = R rf rf + Dell stock × [E(R M) - R rf rf ] E(R Dell Dell stock ) = 0.045 + 1.31 × [0.060] E(R Dell Dell stock ) = 0.045 + 0.079 = 0.124, or 12.4%
7.26
CAPM: The market risk premium is 6 percent and the risk free rate is 5 percent. If the
expected return on a bond is 6.5 percent, what is its beta? LO 7
Solution: From Equation 7.10, we can calculate the beta of the bond as:
E(R bond) = R rf rf + bond × [E(R M) - R rf rf ] 0.065 = 0.050 + bond × 0.060 = 6.0% (0.065 – 0.050)/0.060 0.050)/0.060 = 0.25 bond = (0.065 –
ADVANCED
7.27
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? LO 3
Solution:
Part 1: E(R port) = (0.3) × (0.15) + (0.7) × (0.2) (0.2 ) = 0.185 = 18.5%
Part 2: E(R port) = (0.7) × (0.15) + (0.3) × (0.2) (0.2 ) = 0.165 = 16.5%
7.28
Sumeet knows that the covariance in the return on two assets is – is – 0.0025. 0.0025. Without knowing the expected return of the two assets, explain what that covariance means. LO 3, LO 5
Solution:
The covariance measure is dependent on the expected return of the two assets in questions, so without the expected return of the th e 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 o pposite directions, with respect to their own means, from each other in given states of nature.
7.29
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? LO 3, LO 5
Solution:
The formula for the expected return of a three-stock portfolio is: E ( R3 asset por t ) x1 E ( R1 ) x2 E ( R2 ) x3 E ( R3 )
Therefore, we can solve as in the following: 0.12 = 0.25 × (0.09) + 0.5 × (0.1) + 0.25 × E(R 3) E(R 3) = 0.19 or 19.0%
7.30
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 stocks? For this exercise, ignore the actual values of the variance and covariance cova riance terms and explain their importance conceptually. LO 4, LO 5
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.
7.31
Explain why investors who have diversified dive rsified their portfolios will determine the price and, consequently, the expected return on an asset. LO 3, LO 5
Solution:
If we assume that all investors will seek to be compensated co mpensated (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 p rice 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.
7.32
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, 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. LO 5
Solution:
It will have no effect on the beta of the asset. The beta b eta measures only the systematic risk or variation in the returns of the asset. The prediction p rediction error reflects the nonsystematic risk inherent in the returns of the asset and will consequently not affect the beta of the asset.
7.33
The beta of an asset is equal eq ual to 0. Discuss what the asset must be. LO 7
Solution:
Following the CAPM prediction: (R cs – R rf – R rf cs) = R rf rf + β (E(R M) – R rf ) = R rf rf + 0 (E(R M) – R rf ) = R rf rf
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.
7.34
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? LO 7
Solution:
The beta of the market portfolio p ortfolio 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 ex pected return of a portfolio with a beta equal to 0.5 is 10 percent.
7.35
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 1.0 and an expected return of 4 percent. percen t. 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. LO 7
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 ma rket (a beta one asset) is 12 percent. To draw this SML, we need only connect the dots: 18% 15%
n r u t e 12% R d 9% e t c e 6% p x E 3% 0% 0
1
2
Beta
We can see from the following diagram that an asset with expected return of 4 percent and a beta of – – 1.0 1.0 is underpriced (its expected return is too high). As the market becomes be comes 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.) 18%
15%
The investment will fall here in this plot
n r u 12% t e R d e 9% t c e p x E 6% 3%
0% -1
0
1
Beta
2
As we can see from the following diagram, dia gram, an asset with a beta of 3.0 should have an expected return of 7% + (3)(5%) = 22%. The asset only has an expected ex pected 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.
23%
18% n r u t e R 14% d e t c e 9% p x E
The investment will fall here in this plot
5%
0% 0
1
2
3
Beta
7.36
If the CAPM describes the relation between systematic systematic risk and investment investment returns, returns, can both an individual asset and the market portfolio of all risky assets have negative expected real rates of return? Why Wh y or why not?
Solution:
A negative expected real rate of o f return implies that the expected total rate of return is less than the risk free rate. 1) It is possible for the beta of an individual stock to be negative. Therefore, from CAPM, E[R i] = R rf rf + i × [E(R M) - R rf rf )], it is also possible for a share of stock to have a negative
expected real rate of return. 2) It is not possible for the market portfolio to have a negative expected real rate of return since the beta for the market equals eq uals 1. LO 7
7.37
You have been provided the following following data on the securities securities of three firms firms and and the market:
Security
E[R i]
Stock A Stock B
0.15 0.15
0.18
Stock C
0.10
0.02
Market portfolio
0.10
0.04
Treasury-bills
0.05
0
σR i i
βi
1.0 0.5
1.5 0.5
Assume the CAPM and SML are true and fill in the missing values in the table. t able. Would you invest in the stock of any an y of the three firms? If so, which one(s) and wh y? LO 7
Solution:
1) The missing missing values in the table table are:
A RA,M ( RA ) RM 1.0 ( RA ) (0.04) RA 0.06 B RB,M ( RB ) RM 0.5 (0.18) (0.04) 2.25 C RC,M ( RC ) RM RC,M 0.02 0.04 RC,M 1.00
R M ,M
R T bill , M
1.00;
0;
M
T bill
1.00
0
2) Using Equation 7.10 you can calculate the expected returns for the different shares: E(R A) = 0.05 + [1.5 × (0.10 - 0.05)] = 0.125, or 12.5%
E(R B) = 0.05 + [2.25 × (0.10 - 0.05)] = 0.1625, or 16.25%
E(R C) = 0.05 + [0.5 × (0.10 - 0.05)] = 0.075, or 7.5%
A comparison of the expected returns that are given in the table above, with the returns that CAPM predicts (which are calculated above), indicates that you should buy stocks A and C and avoid stock B.
.
Sample Test Problems 7.1
Friendly Airlines stock is selling at a current price of $37.5 0 per share. If the stock does d oes 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, R T
7.2.
R CA
P1 P0 P0
0.12
P1 $37.50 $37.50
P1 $42.00
Stefan’s parents are about 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 return on 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,
Thus, the range can be determined as:
0.09 0.09 (1.96 1.96 (0.03 (0.03)) )) So, the lower end is: 0.09 – 0.09 – (1.96 (1.96 × (0.03)) = 0.0312 or 3.12% And the upper end is: 0.09 + (1.96 × (0.03)) = 0.1488 or 14.88%
So, 95 percent of the time the stock return will fall in the range between 3.12 percent. and 14.88 percent.
7.3
Elaine has narrowed her investment alternatives to two t wo 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 – (1 – x x) = the proportion invested in Y, then we can solve 0.23( x 0.08(1 – x 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 gen eral 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%.
7.5
The CAPM predicts that the return of MoonBucks Tea Corp. is 23.6 percent. If the riskfree rate of return is 8 percent and the expected return on the market is 20 percent, then what is MoonBucks’ MoonBucks’ beta?
Solution:
Using the CAPM, we find
E(R MoonBucks – R rf MoonBucks) = R rf rf + ßMoonBucks(E[R M] – R rf ) 0.236 = 0.08 + ßMoonBucks(0.20 – (0.20 – 0.08) 0.08) ßMoonBucks= 1.3
