Practice Problems
FIN 5210: Investments
Fall 2014
Problem Set 3: Risk Analysis and Project Evaluation
1.
Given the best available expert opinion, it appears 95% certain that returns from an
investment project will be somewhere in the range from 0% to 30%. Assuming that
the returns follow a symmetric normal probability distribution, translate this estimate
into a mean and standard deviation.
2.
Given the best available expert opinion, it appears 95% certain that returns from an
investment project will be somewhere in the range from -5% to +40%. Assuming
that the returns follow a symmetric normal probability distribution, translate this
estimate into a mean and standard deviation.
3.
An investment can be described as having expected return of 10% with standard
deviation of 2%. Assuming the probability distribution for the risky investment is
symmetric normal, what is the probability that the risky investment will earn a rate
of return of 12% or higher?
4.
An investment has an expected rate of return of 12% with standard deviation of 3%.
The rate of return on U.S. Treasury bills is 6%. What is the approximate probability
that the actual rate of return on the investment will be higher than the T-bill rate,
assuming that the probability distribution is symmetric normal?
5.
An investment can be described as having expected return of 10% with standard
deviation of 2%. Money can be invested without risk in Treasury Bills at 8%.
Assuming the probability distribution for the risky investment is symmetric normal,
what is the probability that the risky investment will perform better than the Treasury
Bills?
6.
Calculate the expected return for a portfolio made up of equal proportions of
investments with individual expected returns of 10%, 12%, and 14%, respectively.
7.
Calculate the expected return for a portfolio with $200 invested in stock A, $300 in
stock B, and $500 in stock C. Expected returns for the individual stocks are 10% for
stock A, 12% for stock B, and 14% for stock C.
8.
Suppose a portfolio is made up of equal proportions of investments whose returns
distributions have standard deviation of 10%, 12%, and 14%, respectively. Which of
the following could possibly be the standard deviation of the returns for the
portfolio?
a. 14%
c. 9%
b. 15%
d. 20%
Prof. Kensinger
page 1
FIN 5210: Investments
Practice Problems
9.
Fall 2014
Suppose a portfolio includes $200 invested in stock A, $300 in stock B, and $500 in
stock C. Standard deviations for the individual stocks are 10% for stock A, 12% for
stock B, and 14% for stock C. Which of the following could possibly be the standard
deviation of the returns for the portfolio?
a. 14%
c. 8%
b. 13%
d. 15%
10. Find the expected return and standard deviation for a portfolio made from three
stocks. IBN has expected return of 15% with standard deviation of 5%. ABS has
expected return of 18% with standard deviation of 9%. IBB has expected return of
20% with standard deviation of 10%. The portfolio consists of $200 worth of IBN,
$300 worth of ABS, and $500 worth of IBB. Correlation coefficients for the returns
on the stocks are given in the table below:
IBN
ABS
IBB
IBN
1.0
.40
.30
ABS
IBB
1.0
.50
1.0
11. An investment would cost $10,000, and could be abandoned when the outcome is
discovered, with recovery of $9,000. The present value of net future cash flows for
the three possible outcomes are given below:
Outcome
Unsuccessful
Moderately successful
Highly successful
Present Value
0
$10,000
$25,000
Probability
.1
.6
.3
Would standard deviation of return be an appropriate way to measure risk in this
case? Why?
12. Portfolio A has expected return of 12% with standard deviation 5%, portfolio B has
expected return of 12% with standard deviation 7%, portfolio C has expected return
of 14% with standard deviation 5%. Which of these portfolios is not dominated by
any of the others?
a. Porfolio A
c. Porfolio C
b. Portfolio B
d. None of the above
13. Portfolio A has expected return of 15% with standard deviation 9%, portfolio B has
expected return of 15% with standard deviation 7%, portfolio C has expected return
of 14% with standard deviation 8%. Which of these portfolios is not dominated by
any of the others?
a. Porfolio A
c. Porfolio C
b. Portfolio B
d. None of the above
Prof. Kensinger
page 2
Practice Problems
FIN 5210: Investments
Fall 2014
14. Someone has told this client that the way for a risk-averse person to invest is to
select a portfolio made up of low-risk stocks such as utilities, and municipal bonds.
The argument presented is that a broad index fund is “too risky” while Treasury bills
have “too little” return. What is wrong with this advice?
15. Smith and Jones have very different attitudes toward risk. Smith is very risk averse,
while Jones is willing to bear substantial risk in order to participate in an opportunity
for above-average reward. According to Nobel-laureate economist James Tobin,
however, they could both be completely satisfied by building their investment
strategies around an investment in a market index mutual fund. Explain how it is
possible to please two completely different sets of tastes and preferences using the
same basic ingredients.
16. A new client seeking advice on investment strategy has completed an interview
designed to gain insight into personal preferences concerning the trade-off between
potential investment performance and risk. Based on the interview you have
concluded that the investor is willing to tolerate some risk exposure, and would be
happy with a portfolio offering a 95% confidence interval ranging from –7.5% to
+22.5%. Currently the best estimates for the stock market index indicate a 95%
confidence interval ranging from –20% to +40%. The return on a fixed-interest fund
specializing in short-term Treasuries is 5% with no variability. This client wants to
make an initial investment of $100,000 allocated between an index fund and the
fixed-interest fund. Based on accepted financial theory, how much of this money
should go into the index fund and how much should go into Treasuries?
17. Another new client has completed an interview from which you have concluded that
the investor would be happy with a portfolio offering a 95% confidence interval
ranging from 2% to 10%. (In the invesor’s words, “I’d be willing to get half the Tbill rate, but no less.”) This time the best estimates for the stock market indicate a
95% confidence interval ranging from –4% to +28%, while the return on a fixedinterest fund specializing in short-term Treasuries is 4% with no variability. This
client also wants to make an initial investment of $100,000 allocated between an
index fund and the short-term Treasuries fund. Based on accepted financial theory,
how much of this money should go into the index fund and how much should go into
Treasuries?
18. In order to get the benefits of diversification, one needs to have a portfolio that
includes equal proportions of how many securities?
a. at least 100
c. at least 1000
b. at least 250
d. 12 to 15
19. Calculate the beta for a portfolio made from equal proportions of securities with
individual betas of .75, 1.0, and 1.25.
Prof. Kensinger
page 3
FIN 5210: Investments
Practice Problems
Fall 2014
20. Calculate the beta for a portfolio with $200 invested in stock A, $300 in stock B, and
$500 in stock C. Betas for the individual stocks are .75 for stock A, 1.0 for stock B,
and 1.25 for stock C.
21. When the T-Bill rate is 9% and the expected return for the market portfolio is 12%,
what is the opportunity cost of capital for an investment with beta of 1.5?
22. Assume the T-Bill rate is 10% and the expected return on the market portfolio is
18%. An investment is twice as risky as the market portfolio, in terms of its
systematic risk. What is the opportunity cost of capital for this investment?
23. An initial public offering (IPO) under analysis by Ajax Pension Fund is in a
relatively new technology, and 75% of its risk is considered to be unsystematic. The
95% confidence interval for its rate of return ranges from 4% to 40%, and the
probability distribution is symmetric normal. The expected return for the market
portfolio is being quoted at 15% with standard deviation of 5%. Estimate the beta
for the IPO, and then recommend whether to accept or reject.
24. Another IPO under analysis by Ajax Pension Fund is in an established technology,
and only 19% of its risk is considered to be unsystematic. The 95% confidence
interval for its rate of return ranges from -5% to 35%, and the probability distribution
is symmetric normal. The expected return for the market portfolio is being quoted at
15% with standard deviation of 5%. Estimate the beta for this IPO, and then
recommend whether to accept or reject.
25. Large State Pension Fund recently installed an evaluation system centered on the
capital asset pricing model. Data for actual outcomes of several managers and
accompanying market conditions during the evaluation period are given below. See
if you can identify any possible problem areas in the fund managers’ performance.
Manager
1
1
2
2
3
3
Fund
A
B
C
D
E
F
βj
0.4
1.1
0.8
1.7
0.5
1.3
Rj
12%
18%
13%
15%
13%
25%
RTbill
9%
10%
10%
9%
9%
10%
R mkt
14%
15%
15%
13%
14%
15%
26. If you were certain that the market was going to rise, what sort of beta would you
prefer for your portfolio?
27. If you were certain that the market was going to drop, what sort of beta would you
prefer for your portfolio?
28. If you thought the market might be turning up, but still were frightened of a
downturn, what sort of beta would you prefer for your portfolio?
Prof. Kensinger
page 4
Practice Problems
FIN 5210: Investments
Fall 2014
29. The last questions are mini-cases for discussion in class.
The Case of the Inventive Financial Analyst
Blimps and Bags, Inc., is in the news. It seems that Tyrone Gasbag, the newly
appointed chief financial officer, has created an innovative way to “put risk analysis back
into risk analysis,” as he says. He has programmed a computer to create different
combinations of cost and revenue streams using a random number generator, in order to
produce what he calls a “synthetic history” of proposed new investment projects. The
objective is to get an idea of the range of possible outcomes for a project. After
generating several hundred possibilities, the computer calculates the mean return and
standard deviation of return for the project from the simulated data.
Tyrone's critics contend that his approach misses something important. They say
this is so even if one is willing to assume that the computer's estimates of expected return
and standard deviation of return agree with those made by the market. Are Tyrone's
critics correct?
The Case of the Cautious Managers
The management of Bio-Tech, Inc. rejected a project that had a beta of .5 and an
expected return of 20%. At the time, the T-Bill rate was 10%, and the expected return on
the market portfolio was 15%. Management said that the project had a very large amount
of risk, since the 95% confidence interval for its rate of return ranged from a low of -30%
to a high of +70%. If the project were adopted, it would represent one fourth of the
company's assets, and managers were fearful that an unpleasant outcome might
dramatically harm the quarterly earnings statement.
Did management act in the best interests of stockholders?
Prof. Kensinger
page 5
Practice Problems
FIN 5210: Investments
Solutions: Set 3
1.
Expected Return = 15%, standard deviation = 30% / 4 = 7.5%
2.
Expected Return = 17.5%, standard deviation = 45% / 4 = 11.25%
3.
The 12% target is 1 standard deviation above the mean. The probability that the
outcome would be 1 or more standard deviations above the mean is about 1/6 =
16.67%. Therefore the answer is 16.67%.
4.
The T-bill rate is 2 standard deviations below the mean. The probability that the
outcome would be 2 or more standard deviations below the mean is about 2.5%.
Therefore the answer is 100%-2.5% = 97.5%.
5.
The T-bill rate is 1 standard deviation below the mean. The probability that the
outcome would be 1 or more standard deviations below the mean is about 1/6 =
16.67%. Therefore the answer is 100%-16.67% = 83.33%.
6.
Expected return = 1/3 (10% + 12% + 14%) = 12%
7.
Expected return = (.2 x 10%) + (.3 x 12%) + (.5 x 14%) = 12.6%
8.
C. Since diversification reduces risk, the standard deviation for a portfolio is less
than the weighted average of the standard deviations for the individual stocks in the
portfolio. The weighted average for this portfolio is 12% (the numbers in this
calculation are the same as problem 1). Choice C is the only one that could possibly
be correct, because all the other choices exceed the weighted average.
9.
C. Since diversification reduces risk, the standard deviation for a portfolio is less
than the weighted average of the standard deviations for the individual stocks in the
portfolio. The weighted average for this portfolio is 12.6% (the numbers in this
calculation are the same as problem 2). Choice C is the only one that could possibly
be correct, because all the other choices exceed the weighted average.
10. Expected return = (.2 x 15%) + (.3 x 18%) + (.5 x 20%) = 18.4%
σ2 = (.22 x .052) + (.32 x .092) + (.52 x .102) + 2( .2x .3x .05 x .09 x .4)
+ 2( .2x .5 x .05 x .1 x .3) + 2( .3 x .5 x .09 x .1 x .5) = 0.0052
σ = square root of variance = 7.21%
11. Standard deviation would not be appropriate, because the probability distribution for
the NPV is not symmetric. It is, in fact, very strongly skewed to the right. The table
of outcomes for NPV is as follows:
Outcome
Unsuccessful
Moderately successful
Highly successful
NPV
($1,000)*
0
$15,000
Probability
.1
.6
.3
*In case of an unsuccessful outcome the project would be abandoned, and the loss
would be limited to $1,000.
12. Portfolio C has higher return than any of the other portfolios, and no other portfolio
has lower risk. There portfolio C dominates.
Prof. Kensinger
page 1
Practice Problems
FIN 5210: Investments
Solutions: Set 3
13. Portfolio B has lower risk than any of the other portfolios, and no other portfolio has
higher return. There portfolio B dominates.
14. A specialized portfolio of “low-risk” securities such as the one proposed is
dominated by a position along the capital market line.
15. The answer requires a clear statement of how Tobin’s separation principle works.
You should explain the mechanism for achieving any desired point on the CML, and
why any point on the CML dominates all portfolios of risky assets except the market
portfolio.
16. Index fund: $50,000; Treasuries: $50,000. This problem is an exercise in locating a
desired point on the Capital Market Line (CML). The expected return on the market
portfolio is 10% with standard deviation of 15%. The desired position has expected
return of 7.5% with standard deviation 7.5%, which is exactly midway between the
market portfolio and the risk-free asset along the Capital Market Line. Therefore,
the desired position can be achieved by investing 1/2 of the amount in the index fund
and 1/2 in the fixed fund. The mix can be worked out formally by letting w = the
weight in the index fund. Then 1-w is the weight in the fixed fund. Then there are
two equations to evaluate for a common solution.
17. Index fund: $25,000; Treasuries: $75,000. This is another exercise in locating a
desired point on the Capital Market Line (CML). The desired position has expected
return of 6% with standard deviation 2%. The expected return on the market
portfolio is 12% with standard deviation of 8%, which is exactly four times as much
risk and four times as large a risk premium as the desired position. Therefore, the
desired position can be achieved by investing 1/4 of the amount in the index fund
and 3/4 in Treasuries. The mix can be worked out formally by letting w = the weight
in the index fund. Then 1-w is the weight in Treasuries. Then there are two
equations to evaluate for a common solution:
Return condition: .12w + .04(1–w) = .06
.08 w = .02
w = .25
Risk condition: .08w = .02
w = .25
18. D. See pp. 164-167 of the Jones text for further discussion of this point. Also see
figure 7.3 on page 166.
19. Beta = 1/3 (.75 + 1.0 + 1.25) = 1.0
20. Beta = (.2 x .75) + (.3 x 1.0) + (.5 x 1.25) = 1.075
21. OCC = 9% + 1.5(12% – 9%) = 13.5%
22. OCC = 10% + 2(18% – 10%) = 26.0%
23. Expected return is 22% and β = (9% / 5%) x sqrt (.25) = 0.9. Since the IPO has an
expected return higher than the market potfolio’s, with lower risk, it is attractive.
Accept it.
Prof. Kensinger
page 2
Practice Problems
FIN 5210: Investments
Solutions: Set 3
24. Expected return is 15% and β = (10% / 5%) x sqrt (.81) = 1.8. Since the IPO has an
expected return the same as the market potfolio’s, with nearly twice the risk, it is
unattractive. Reject it.
25. The results for the projects can be analyzed using the table below:
Manager
1
1
2
2
3
3
Fund
A
B
C
D
E
F
Rj
12%
18%
13%
15%
13%
25%
Opportunity Cost
of Capital
11%
15.5%
14%
15.8%
11.5%
16.5%
Analysis: Both of the 2nd manager’s funds under-performed, while the other
managers’ funds all earned more than the opportunity cost of capital. Manager 2 may
have simply been a victim of bad luck. There is, however, the possibility that
something may be wrong with the decision-making process for those funds.
26. Very High.
27. Very negative.
28. Low, perhaps less than 1.
29. Cases are for class discussion.
Prof. Kensinger
page 3