Corporate Finance 9th Edition Ross Solutions Manual
Corporate Finance 9th Edition Ross Solutions Manual
Corporate Finance 9th Edition Ross Solutions Manual
Solutions Manual
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Chapter12
Problem 1
For a large-company stock mutual fund, would you expect the betas to be positive or negative for
Step-by-step solution
step 1 of 1
For a large-company stock fund, we would expect the beta for the market risk premium to be near one since large company returns account for a large part of the total market
return on a market-value basis. We would expect the betas for the SMB and HML risk factors to be low, and possibly negative, since large company stock returns are not highly
related to small company stock returns and large company stocks tend to be more oriented toward value stocks.
Problem 1ACQ
Systematic versus Unsystematic Risk Describe the difference between systematic risk and
unsystematic risk.
Step-by-step solution
step 1 of 1
Systematic risk is risk that cannot be diversified away through formation of a portfolio. Generally, systematic risk factors are those factors that affect a large number
of firms in the market, however, those factors will not necessarily affect all firms equally. Unsystematic risk is the type of risk that can be diversified away through
portfolio formation. Unsystematic risk factors are specific to the firm or industry. Surprises in these factors will affect t he returns of the firm in which you are interested,
but they will have no effect on the returns of firms in a different industry and perhaps little effect on other firms in the same industry.
Problem 1SQP
Factor Models A researcher has determined that a two-factor model is appropriate to determine
the return on a stock. The factors are the percentage change in GNP and an interest rate. GNP
is expected to grow by 3.5 percent, and the interest rate is expected to be 2.9 percent. A stock
has a beta of 1.2 on the percentage change in GNP and a beta of ‒.8 on the interest rate. If the
expected rate of return on the stock is 11 percent, what is the revised expected return on the
stock if GNP actually grows by 3.2 percent and interest rates are 3.4 percent?
Step-by-step solution
step 1 of 1
Since we have the expected return of the stock, the revised expected return can be determined using the innovation, or surprise, in the risk factors. So, the revised expected
return is:
R = 10.24%
Problem 2
The Fama-French factors and risk-free rates are available at Ken French’s Web site:
www.dartmouth.edu/~kfrench. Download the monthly factors and save the most recent 60 months for
each factor. The historical prices for each of the mutual funds can be found on various Web sites,
including finance.yahoo.com. Find the prices of each mutual fund for the same time as the
Fama–French factors and calculate the returns for each month. Be sure to include dividends. For
each mutual fund, estimate the multi- factor regression equation using the Fama-French factors.
How well do the regression estimates explain the variation in the return of each mutual fund?
Step-by-step solution
step 1 of 1
The following shows the results of the regression estimates for the period between January 2001 and December 2005. The actual answer to the case will change based on current
market returns.
Fidelity Magellan:
Regression Statistics
Multiple R 0.96923
R Square 0.93941
Adjusted R Square 0.93617
Standard Error 0.01265
Observations 60
ANOVA
df SS MS F Significance F
Regression 3 0.139004 0.046335 289.4198 4.74E-34
Residual 56 0.008965 0.00016
Total 59 0.14797
Regression Statistics
Multiple R 0.97008
R Square 0.94106
Adjusted R Square 0.93790
Standard Error 0.01177
Observations 60
ANOVA
df SS MS F Significance F
Regression 3 0.123813 0.041271 298.0359 2.19E-34
Residual 56 0.007755 0.000138
Total 59 0.131568
Coefficients Standard Error t Stat P-value
Intercept 0.00157 0.001548 1.01372 0.315076
Mkt-RF 1.047432 0.041898 24.99949 4.63E-32
SMB 0.322687 0.076658 4.209435 9.36E-05
HML 0.082383 0.080861 1.018823 0.312668
Regression Statistics
Multiple R 0.93283
R Square 0.87017
Adjusted R Square 0.86321
Standard Error 0.01874
Observations 60
ANOVA
df SS MS F Significance F
Regression 3 0.131768 0.043923 125.108 8.46E-25
Residual 56 0.01966 0.000351
Total 59 0.151429
Problem 2ACQ
APT Consider the following statement: For the APT to be useful, the number of systematic risk
factors must be small. Do you agree or disagree with this statement? Why?
Step-by-step solution
step 1 of 1
Any return can be explained with a large enough number of systematic risk factors. However, for a factor model to be useful as a practical matter, the number of factors
Factor Models Suppose a three-factor model is appropriate to describe the returns of a stock.
b.Suppose unexpected bad news about the firm was announced that causes the stock price to drop
by 2.6 percent. If the expected return on the stock is 10.8 percent, what is the total return
on this stock?
Step-by-step solution
step 1 of 2
m = 2.81%
step 2 of 2
b.The unsystematic return is the return that occurs because of a firm specific factor such as the bad news about the company. So, the unsystematic return of the stock
is –2.6 percent. The total return is the expected return, plus the two components of unexpected return: the systematic risk portion of return and the unsy stematic portion.
R = + m + ε
R = 10.80% + 2.81% – 2.6%
R = 11.01%
Problem 3
What do you observe about the beta coefficients for the different mutual funds? Comment on any
similarities or differences.
Step-by-step solution
step 1 of 1
Problem 3ACQ
APT David McClemore, the CFO of Ultra Bread, has decided to use an APT model to estimate the required
return on the company’s stock. The risk factors he plans to use are the risk premium on the stock
market, the inflation rate, and the price of wheat. Because wheat is one of the biggest costs
Ultra Bread faces, he feels this is a significant risk factor for Ultra Bread. How would you evaluate
his choice of risk factors? Are there other risk factors you might suggest?
Step-by-step solution
step 1 of 1
The market risk premium and inflation rates are probably good choices. The price of wheat, while a risk factor for Ultra Products, is not a market risk factor and will
not likely be priced as a risk factor common to all stocks. In this case, wheat would be a firm specific risk factor, not a market risk factor. A better model would employ
macroeconomic risk factors such as interest rates, GDP, energy prices, and industrial production, among others.
Problem 3SQP
Factor Models Suppose a factor model is appropriate to describe the returns on a stock. The current
expected return on the stock is 10.5 percent. Information about those factors is presented in
b.The firm announced that its market share had unexpectedly increased from 23 percent to 27 percent.
Investors know from past experience that the stock return will increase by .45 percent for every
1 percent increase in its market share. What is the unsystematic risk of the stock?
Step-by-step solution
step 1 of 3
m = 1.06%
step 2 of 3
b.The unsystematic is the return that occurs because of a firm specific factor such as the increase in market share. If ε is the unsystematic risk portion of the return,
then:
ε = 0.45(27% – 23%)
ε = 1.80%
step 3 of 3
c.The total return is the expected return, plus the two components of unexpected return: the systematic risk portion of return and the unsystematic portion. So, the total
R = + m + ε
R = 13.36%
Problem 4
If the market is efficient, what value would you expect for alpha? Do your estimates support market
efficiency?
Step-by-step solution
step 1 of 1
If the market is efficient, all assets should have an alpha of zero. In this case, none of the three funds has a statisticall y significant positive alpha, so the evidence
Problem 4ACQ
Systematic and Usystematic Risk You own stock in the Lewis-Striden Drug Company. Suppose you had
a.The government would announce that real GNP had grown 1.2 percent during the previous quarter.
c.Interest rates would rise 2.5 percentage points. The returns of Lewis-Striden are negatively
d.The president of the firm would announce his retirement. The retirement would be effective six
months from the announcement day. The president is well liked: In general, he is considered an
e. Research data would conclusively prove the efficacy of an experimental drug. Completion of
the efficacy testing means the drug will be on the market soon.
a.The government announced that real GNP grew 2.3 percent during the previous quarter.
b.The government announced that inflation over the previous quarter was 3.7 percent.
e.Research results in the efficacy testing were not as strong as expected. The drug must be tested
for another six months, and the efficacy results must be resubmit-ted to the FDA.
g.A competitor announced that it will begin distribution and sale of a medicine that will compete
Discuss how each of the actual occurrences affects the return on your Lewis-Striden stock. Which
Step-by-step solution
step 1 of 8
a.Real GNP was higher than anticipated. Since returns are positively related to the level of GNP, returns should rise based on this factor.
step 2 of 8
b.Inflation was exactly the amount anticipated. Since there was no surprise in this announcement, it will not affect Lewis-Striden returns.
step 3 of 8
c.Interest rates are lower than anticipated. Since returns are negatively related to interest rates, the lower than expected rate is good news. Returns should rise due
to interest rates.
step 4 of 8
d.The President’s death is bad news. Although the president was expected to retire, his retirement would not be effective for six months. During that period he would
still contribute to the firm. His untimely death means that those contributions will not be made. Since he was generally considered an asset to the firm, his death will
cause returns to fall. However, since his departure was expected soon, the drop might not be very large.
step 5 of 8
e.The poor research results are also bad news. Since Lewis-Striden must continue to test the drug, it will not go into production as early as expected. The delay will
affect expected future earnings, and thus it will dampen returns now.
step 6 of 8
f.The research breakthrough is positive news for Lewis Striden. Since it was unexpected, it will cause returns to rise.
step 7 of 8
g.The competitor’s announcement is also unexpected, but it is not a welcome surprise. This announcement will lower the returns on Lewis-Striden.
step 8 of 8
The systematic factors in the list are real GNP, inflation, and interest rates. The unsystematic risk factors are the president’s ability to contribute to the firm, the
Problem 4SQP
Multifactor Models Suppose stock returns can be explained by the following three- factor model:
Assume there is no firm-specific risk. The information for each stock is presented here:
β1 β2 β3
Stock A1.45.80 .05
Stock B.73 1.25 –.20
Stock C.89 –.141.24
The risk premiums for the factors are 5.5 percent, 4.2 percent, and 4.9 percent, respectively.
If you create a portfolio with 20 percent invested in stock A, 20 percent invested in stock B,
and the remainder in stock C, what is the expression for the return on your portfolio? If the
Step-by-step solution
step 1 of 1
The beta for a particular risk factor in a portfolio is the weighted average of the betas of the assets. This is true whether the betas are from a single factor model or
F1 = 0.97
F2 = 0.33
F3 = 0.71
Ri = 8.21%
Intermediate
Problem 5
Step-by-step solution
step 1 of 1
Once adjusting for risk, we cannot say any of these three funds performed better since all three alphas are not significantly different from zero at any reasonable level
of confidence.
Problem 5ACQ
Market Model versus APT What are the differences between a k-factor model and the market model?
Step-by-step solution
step 1 of 1
The main difference is that the market model assumes that only one factor, usually a stock market aggregate, is enough to exp lain stock returns, while a k-factor model
Problem 5SQP
Multifactor Models Suppose stock returns can be explained by a two-factor model. The firm-specific
risks for all stocks are independent. The following table shows the information for two diversified
portfolios:
β 1 β2 E(R)
Portfolio A .85 1.15 16%
Portfolio B1.45–.25 12
If the risk-free rate is 4 percent, what are the risk premiums for each factor in this model?
Step-by-step solution
step 1 of 1
where F1 and F2 are the respective risk premiums for each factor. Expressing the return equation for each portfolio, we get:
16% = 4% + 0.85 F1 + 1.15F2
We can solve the system of two equations with two unknowns. Multiplying each equation by the respective F2 factor for the other equation, we get:
F1 = 6.49%
And now, using the equation for portfolio A, we can solve for F2, which is:
F2 = 5.64%
Problem 6ACQ
APT In contrast to the CAPM, the APT does not indicate which factors are expected to determine
the risk premium of an asset. How can we determine which factors should be included? For example,
one risk factor suggested is the company size. Why might this be an important risk factor in an
APT model?
Step-by-step solution
step 1 of 1
The fact that APT does not give any guidance about the factors that influence stock returns is a commonly-cited criticism. However, in choosing factors, we should choose
factors that have an economically valid reason for potentially affecting stock returns. For example, a smaller company has more risk than a large company. Therefore, the
Market Model The following three stocks are available in the market:
E(R) β
Stock A10.5%1.20
Stock B 13.0 .98
Stock C 15.7 1.37
Market 14.2 1.00
b.What is the return on a portfolio with weights of 30 percent stock A, 45 percent stock B, and
25 percent stock C ?
c.Suppose the return on the market is 15 percent and there are no unsystematic surprises in the
returns. What is the return on each stock? What is the return on the portfolio? .
Step-by-step solution
step 1 of 3
R = + β(RM – ) + ε
Stock A:
R A = + βA(RM – ) + εA
Stock B:
R B = + βB(RM – ) + εB
Stock C:
R C = + βC(RM – ) + εC
step 2 of 3
b.Since we don't have the actual market return or unsystematic risk, we will get a formula with those values as unknowns:
step 3 of 3
c.Using the market model, if the return on the market is 15 percent and the systematic risk is zero, the return for each individual stock is:
R A = 11.46%
R B = 13.78%
R C = 15.70% + 1.37(15% – 14.2%)
R C = 16.80%
To calculate the return on the portfolio, we can use the equation from part b, so:
R P = 13.84%
Alternatively, to find the portfolio return, we can use the return of each asset and its portfolio weight, or:
R P = 13.84%
Problem 7ACQ
CAPM versus APT What is the relationship between the one-factor model and the CAPM?
Step-by-step solution
step 1 of 1
Assuming the market portfolio is properly scaled, it can be shown that the one-factor model is identical to the CAPM.
Problem 7SQP
Portfolio Risk You are forming an equally weighted portfolio of stocks. Many stocks have the same
beta of .84 for factor 1 and the same beta of 1.69 for factor 2. All stocks also have the same
expected return of 11 percent. Assume a two-factor model describes the return on each of these
stocks.
a.Write the equation of the returns on your portfolio if you place only five stocks in it.
b.Write the equation of the returns on your portfolio if you place in it a very large number of
stocks that all have the same expected returns and the same betas.
Step-by-step solution
step 1 of 2
a.Since five stocks have the same expected returns and the same betas, the portfolio also has the same expected return and beta. However, the unsystematic risks might
step 2 of 2
b.Consider the expected return equation of a portfolio of five assets we calculated in part a. Since we now have a very large number of stocks in the portfolio, as:
N → ∞, → 0
Thus:
Challenge
Problem 8ACQ
Factor Models How can the return on a portfolio be expressed in terms of a factor model?
Step-by-step solution
step 1 of 1
It is the weighted average of expected returns plus the weighted average of each security's beta times a factor F plus the weighted average of the unsystematic risks of
Problem 8SQP
APT There are two stock markets, each driven by the same common force, F, with an expected value
of zero and standard deviation of 10 percent. There are many securities in each market; thus,
you can invest in as many stocks as you wish. Due to restrictions, however, you can invest in
only one of the two markets. The expected return on every security in both markets is 10 percent.
The returns for each security, i, in the first market are generated by the relationship:
where e1 is the term that measures the surprises in the returns of stock i in market l. These
surprises are normally distributed; their mean is zero. The returns on security j in the second
where ϵ2j is the term that measures the surprises in the returns of stock j in market 2. These
surprises are normally distributed; their mean is zero. The standard deviation ϵ1i,.of and ϵ2j
a.If the correlation between the surprises in the returns of any two stocks in the first market
is zero, and if the correlation between the surprises in the returns of any two stocks in the
second market is zero, in which market would a risk-averse person prefer to invest? (Note: The
correlation between ϵ1j and ϵ2j. for any i and j is zero, and the correlation between ϵ2j. and
b.If the correlation between ϵ1j and ϵ1j. in the first market is .9 and the correlation between
ϵ2j and ϵ2j in the second market is zero, in which market would a risk-averse person prefer to
invest?
c.If the correlation between ϵ1j and ϵ1j in the first market is zero and the correlation between
ϵ2j and ϵ2j in the second market is.5,in which market would a risk-averse person prefer to invest?
d.In general, what is the relationship between the correlations of the disturbances in the two
markets that would make a risk-averse person equally willing to invest in either of the two markets?
Step-by-step solution
step 1 of 4
To determine which investment an investor would prefer, you must compute the variance of portfolios created by many stocks from either market. Because you know that
diversification is good, it is reasonable to assume that once an investor has chosen the market in which she will invest, she will buy many stocks in that market.
Known:
E F = 0 and σ = 0.10
If we assume the stocks in the portfolio are equally-weighted, the weight of each stock is , that is:
X i = for all i
If a portfolio is composed of N stocks each forming 1/N proportion of the portfolio, the return on the portfolio is 1/N times the sum of the returns on the N stocks. To
find the variance of the respective portfolios in the 2 markets, we need to use the definition of variance from Statistics:
In our case:
Note however, to use this, first we must find R P and E(RP). So, using the assumption about equal weights and then substituting in the known equation for Ri:
R P =
R P = (0.10 + βF + εi)
R P = 0.10 + βF +
If:
and
E( a) = a
E(R P) = E
E(R P) = 0.10
Next, substitute both of these results into the original equation for variance:
Var(R P) = E
Var(R P) = E
Var(R P) = E
Var(R P) =
Finally, since we can have as many stocks in each market as we want, in the limit, as N → ∞,
→ 0, so we get:
and, since:
Cov(ε i,εj) = σiσjρ(εi,εj)
Finally we can begin answering the questions a, b, &c for various values of the correlations:
Since Var(R 1P) > Var(R2P), and expected returns are equal, a risk averse investor will prefer to invest in the second market.
step 2 of 4
b.If we assume ρ(ε 1i,ε1j) = 0.9, and ρ(ε2i,ε2j) = 0, the variance of each portfolio is:
Since Var(R 1P) > Var(R2P), and expected returns are equal, a risk averse investor will prefer to invest in the second market.
step 3 of 4
c.If we assume ρ(ε 1i,ε1j) = 0, and ρ(ε2i,ε2j) = .5, the variance of each portfolio is:
Since Var(R 1P) = Var(R2P), and expected returns are equal, a risk averse investor will be indifferent between the two markets.
step 4 of 4
d.Since the expected returns are equal, indifference implies that the variances of the portfolios in the two markets are also equal. So, set the variance equations e qual,
and solve for the correlation of one market in terms of the other:
Therefore, for any set of correlations that have this relationship (as found in part c), a risk adverse investor will be indifferent between the two markets
Problem 9ACQ
Data Mining What is data mining? Why might it overstate the relation between some stock attribute
and returns?
Step-by-step solution
step 1 of 1
Choosing variables because they have been shown to be related to returns is data mining. The relation found between some attribute and returns can be accidental, thus overstated.
For example, the occurrence of sunburns and ice cream consumption are related; however, sunburns do not necessarily cause ice cream consumption, or vice versa. For a factor
to truly be related to asset returns, there should be sound economic reasoning for the relationship, not just a statistical one.
Problem 9SQP
APT Assume that the following market model adequately describes the return- generating behavior
of risky assets:
Here:
RMt = The return on a portfolio containing all risky assets in some proportion at time t.
Short selling (i.e., negative positions) is allowed in the market. You are given the following
information:
AssetβiE(Ri)Var(ϵi)
A .7 8.41% .0100
B 1.212.06 .0144
C 1.513.95 .0225
The variance of the market is .0121, and there are no transaction costs.
types A, B, or C, respectively.
c.Assume the risk-free rate is 3.3 percent and the expected return on the market is 10.6 percent.
d.What equilibrium state will emerge such that no arbitrage opportunities exist? Why?
Step-by-step solution
step 1 of 4
a.In order to find standard deviation, σ, you must first find the Variance, since σ = . Recall from Statistics a property of Variance:
If:
and:
Var( a) = 0
Realize that R i,t, RM, and εi,t are random variables, and αi and βi are constants. Then, applying the above properties to this model, we get:
= = .1262 or 12.62%
= = .1784 or 17.84%
= = .2230 or 22.30%
step 2 of 4
b. From the above formula for variance, note that as N → ∞, → 0, so you get:
Var(R i) = Var(RM )
= 0.72(.0121) = 0.005929
= 1.22(.0121) = 0.017424
= 1.52(.0121) = 0.027225
step 3 of 4
= RF + βi( – RF)
which is the CAPM (or APT Model when there is one factor and that factor is the Market). So, the expected return of each asset is:
We can compare these results for expected asset returns as per CAPM or APT with the expected returns given in the table. This shows that assets A&B are accurately priced,
but asset C is overpriced (the model shows the return should be higher). Thus, rational investors will not hold asset C.
step 4 of 4
d. If short selling is allowed, rational investors will sell short asset C, causing the price of asset C to decrease until no arbitrage opportunity exists. In other w ords,
the price of asset C should decrease until the return becomes 14.25 percent.
Problem 10ACQ
Factor Selection What is wrong with measuring the performance of a U.S. growth stock manager
Step-by-step solution
step 1 of 1
Using a benchmark composed of English stocks is wrong because the stocks included are not of the same style as those in a U.S. growth stock fund.
Problem 10SQP
APT Assume that the returns on individual securities are generated by the following two-factor
model:
F1t and F2t are market factors with zero expectation and zero covariance. In addition, assume
that there is a capital market for four securities, and the capital market for these four assets
is perfect in the sense that there are no transaction costs and short sales (i.e., negative
Securityβ 1β2E(R)
1 1.01.5 20%
2 .52.0 20
3 1.0 .5 10
4 1.5.75 10
a.Construct a portfolio containing (long or short) securities 1 and 2, with a return that does
not depend on the market factor, F1t, in any way. (Hint: Such a portfolio will have β1 = 0.)
b.Following the procedure in (a), construct a portfolio containing securities 3 and 4 with a return
that does not depend on the market factor, F1t. Compute the expected return and β2 coefficient
Describe a possible arbitrage opportunity in such detail that an investor could implement it.
d.What effect would the existence of these kinds of arbitrage opportunities have on the capital
markets for these securities in the short run and long run? Graph your analysis.
Step-by-step solution
step 1 of 4
a.Let:
X 1 = 1 – X2
Recall from Chapter 10 that the beta for a portfolio (or in this case the beta for a factor) is the weighted average of the security betas, so
β P1 = X1β11 + X2β21
β P1 = X1β11 + (1 – X1)β21
Now, apply the condition given in the hint that the return of the portfolio does not depend on F1. This means that the portfolio beta for that factor will be 0, so:
β P1 = 0 = X1β11 + (1 – X1)β21
β P1 = 0 = X1(1.0) + (1 – X1)(0.5)
X 1 = – 1
X 2 = 2
R P = X1R1 + X2R2
β P1 = –1(1) + 2(0.5)
β P1 = 0
step 2 of 4
β P2 = 0 = X3β31 + (1 – X3)β41
β P2 = 0 = X3(1) + (1 – X3)(1.5)
and
X 3 = 3
X 4 = –2
β P2 = 3(0.5) – 2(0.75)
β P2 = 0
Note that since both β P1 and βP2 are 0, this is a risk free portfolio!
step 3 of 4
c.The portfolio in part b provides a risk free return of 10%, which is higher than the 5% return provided by the risk free security. To take advantage of this opportunity,
borrow at the risk free rate of 5% and invest the funds in a portfolio built by selling short security four and buying security three with weights (3,–2) as in part b.
step 4 of 4
d.First assume that the risk free security will not change. The price of security four (that everyone is trying to sell short) will decrease, and the price of security
three (that everyone is trying to buy) will increase. Hence the return of security four will increase and the return of security three will decrease.
The alternative is that the prices of securities three and four will remain the same, and the price of the risk-free security drops until its return is 10%.
Finally, a combined movement of all security prices is also possible. The prices of security four and the risk-free security will decrease and the price of security three