Investment Analysis (FIN 670)
Fall 2009
Homework 4
Instructions: please read carefully
•
•
You should show your work how to get the answer for each calculation question to
get full credit
The due date is Tuesday, Nov 3, 2009. Late homework will not be graded.
Name(s):
Student ID
1. Historically, which security had the lowest standard deviation?
a. U.S. large stocks
b. World bond portfolio
c. U.S. long-term Treasury bonds
d. U.S. Treasury bills
1. d
2. What is the risk premium of a stock that has an expected return of 20%, assuming the rate of
return on Treasury bills is 3%?
a. 20%
b. 23%
c. 17%
d. Cannot be determined.
2.c
3. What is the effective annual rate of return on a bond that has a holding period return of 10%,
assuming it pays coupons semi-annually?
a. 10.25%
b. 10%
c. 21%
d. 8.25%
3.a
4. You purchased 100 shares of ABC stock for $20 per share. One year later you received $1
cash dividend and sold the shares for $22 each. Your holding-period return was
_______________.
a. 5%
b. 10%
c. 15%
d. 20%
4. c
HPR = (1 + 22 – 20)/20 = 15%
5. The geometric average return of 10%, -20%, -10%, and 20% is _______________.
a. 0%
b. 1.26%
c. -1.26%
d. -2%
5. c
[(1+0.1)(1-0.2)(1-0.1)(1+0.2)]^(1/4) -1 = -1.26%
6. The sample standard deviation of returns of 18%, -15%, -10% and 30% is _______________.
a. 15.2%
b. 18.8%
c. 21.7%
d. 25.3%
r = (18 − 15 − 10 + 30) / 4 = 5.75
6. c
σ=
[
]
1
(18 − 5.75)2 + (− 15 − 5.75)2 + (− 10 − 5.75)2 + (30 − 5.75)2 = 21.7
3
7. What is the ending price of a stock if its beginning price was $20, its cash dividend was $2,
and the holding period return on a stock was 10%?
a. $18
b. $20
c. $22
d. $24
7. b
0.1 = (2+P1-20)/20
P1 = 20
8. A complete portfolio holds _______________.
a. all risky assets
b. all risk-free assets
c. risky and risk-free assets
d. bonds and stocks
8. c
9. Which of the following is most correct concerning the standard deviation of a stock's returns?
a. It represents the chance of making negative returns from investing in the stock.
b. It should be zero if the stock has the same return every year.
c. It should be greater than the stock's geometric mean return.
d. All of the above are correct.
9. b
10. The _____________________ return ignores the compounding effect
a. Geometric average
b. Arithmetic average
c. Dollar-weighted
d. Both B and C above
10. b
11. The risk-free asset is proxied by the _______________.
a. Treasury bills
b. AAA corporate bonds
c. inflation-index bonds
d. money market mutual funds
11. a
12. DFI, Inc. has the following probability distribution of holding period returns on its stock.
State of Economy
Probability
HPR
Boom
.25
25%
Normal Growth
.45
15%
Recession
.30
9%
The expected return on DFI's stock is
a. 15.7%.
b. 12.4%.
c. 16.5%.
d. 17.8%.
e. 11.6%.
12. a
HPR = .25 (25%) + .45 (15%) + .30 (9%) = 15.7%
13. DFI, Inc. has the following probability distribution of holding period returns on its stock.
State of Economy
Probability
Boom
.25
Normal Growth
.45
Recession
.30
The expected variance of these returns is
a. 66.6.
b. 35.5.
c. 29.4.
d. 40.5.
e. none of the above
HPR
25%
15%
9%
13. b
[.25 (25 - 15.7)2 + .45 (15 - 15.7)2 + .30 (9 - 15.74)2] = 35.47
14. Which of the following statements regarding risk-averse investors is true?
a. They only accept risky investments that offer risk premiums over the risk-free
b.
c.
d.
e.
rate.
They accept investments that are fair games.
They only care about rate of return.
They are willing to accept lower returns and high risk.
A and B.
14. a
Risk-averse investors only accept risky investments that offer risk premiums over
the risk-free rate.
15. An investor invests 60 percent of his wealth in a risky asset with an expected rate of
return of 0.14 and a variance of 0.32 and 40 percent in a T-bill that pays 3 percent. His
portfolio's expected return and standard deviation are __________ and __________,
respectively.
a. 0.096; 0.339
b. 0.087; 0.267
c. 0.295; 0.123
d. 0.087; 0.182
e. none of the above
15. a
E(rP) = 0.6(14%) + 0.4(3%) = 9.6%; sP = 0.6(0.32)1/2 = .3394.
16. When a portfolio consists of only a risky asset and a risk-free asset, increasing the fraction
of the overall portfolio invested in the risky asset will
a.
b.
c.
d.
e.
increase the expected return on the portfolio.
increase the standard deviation of the portfolio.
decrease the standard deviation of the portfolio.
A and B are true.
A and C are true.
16. d
When a portfolio consists of only a risky asset and a risk-free asset, increasing the
fraction of the overall portfolio invested in the risky asset will increase the expected
return and the standard deviation of the portfolio
17. Olivia is a risk-averse investor. Alex is a less risk-averse investor than Olivia. Therefore,
a.
b.
c.
d.
e.
for the
for the
for the
for the
cannot
same risk, Alex requires a higher rate of return than Olivia.
same return, Alex tolerates higher risk than Olivia.
same risk, Olivia requires a lower rate of return than Alex.
same return, Olivia tolerates higher risk than Alex.
be determined.
17. b
The more risk averse the investor, the less risk that is tolerated, given a rate of
return
18. Given the capital allocation line, an investor's optimal portfolio is the portfolio that
a.
b.
c.
d.
e.
maximizes her expected utility.
maximizes her risk.
minimizes both her risk and return.
maximizes her expected profit.
none of the above
18. a
Given the capital allocation line, an investor's optimal portfolio is the portfolio that
maximizes her expected utility
19. If a T-bill pays 5 percent (standard deviation of T-bill assumes to be equal 0), which of the
following investments would not be chosen by a risk-averse investor?
a. An asset that pays 10 percent with a probability of 0.60 or 2 percent with a
probability of 0.40.
b. An asset that pays 10 percent with a probability of 0.40 or 2 percent with a
probability of 0.60.
c. An asset that pays 10 percent with a probability of 0.30 or 3.75 percent with a
probability of 0.70.
d. An asset that pays 10 percent with a probability of 0.20 or 3.75 percent with a
probability of 0.80.
e. neither A nor B would be chosen
19. d
If you multiply the probability of each outcome by the return if that outcome occurs,
you will have the expected return for each of the responses. Since the T-bill return
5% without risk, a risk-averse investor would never choose any risky investment
that returned 5% or less. Answer D would never be chosen
Using the following expectations on Stocks X and Y to answer questions 20 through 22
Probability
Stock X
Stock Y
Bear Market
0.2
-20%
-15%
Normal Market
0.5
18%
20%
Bull Market
0.3
50%
10%
20. What are the expected returns for X and Y
E(rX) = [0.2 × (–20%)] + [0.5 × 18%] + [0.3 × 50%)] = 20%
E(rY) = [0.2 × (–15%)] + [0.5 × 20%] + [0.3 × 10%)] = 10%
21. What are standard deviation of returns on X and Y
σX2 = [0.2 × (–20 – 20)2] + [0.5 × (18 – 20)2] + [0.3 × (50 – 20)2] = 592
σX = 24.33%
σY = [0.2 × (–15 – 10)2] + [0.5 × (20 – 10)2] + [0.3 × (10 – 10)2] = 175
σY = 13.23%
22. Assume that of your $10,000 portfolio, you invest $9000 in stock X and $1000 in stock Y.
What is the expected return on your portfolio?
E(r) = (0.9 × 20%) + (0.1 × 10%) = 19%
Using the following information to answer question 23-26
Assume you manage a risky portfolio with an expected rate of return of 17% and a standard
deviation of 27%. The T-bill rate (risk-free rate) is 7%
23. Your client chooses to invest 70% of a portfolio in your fund (risky portfolio) and 30% in a Tbill money market fund. What is expected return and standard deviation of your client’s complete
portfolio.
E(rc) = (0.3 × 7%) + (0.7 × 17%) = 14% per year
σc = 0.7 × 27% = 18.9% per year
24. Suppose your risky portfolio includes the following investments in the given proportions:
(K)
What are the investment proportions of each security in your client’s overall portfolio,
including the position in T-bills?
Investment
Proportions
Security
T-Bills
Stock A
Stock B
Stock C
0.7 × 27% =
0.7 × 33% =
0.7 × 40% =
30.0%
18.9%
23.1%
28.0%
25. What is the reward-to-variability ratio of your risky portfolio and your client’s overall
complete portfolio.
Your Reward-to-variability ratio = S =
17 − 7
= 0.3704
27
Client's Reward-to-variability ratio =
14 − 7
= 0.3704
18.9
26. Suppose your client decides to invest in your risky portfolio a proportion (y) of his total
investment budget so that his overall portfolio will have an expected rate of return of 15%.
a. What is the proportion y?
E(rc) = (1 – y)rf + y E(rP) = (1-y)7 + 17y = 7 + 10y
If the expected rate of return for the portfolio is 15%, then, solving for y:
15 = 7 + 10y ⇒ y =
15 − 7
= 0.8
10
Therefore, in order to achieve an expected rate of return of 15%, the client
must invest 80% of total funds in the risky portfolio and 20% in T-bills.
b. What are your client’s investment proportions in your three stocks and the T-bill fund?
Investment
Proportions
Security
T-Bills
Stock A
Stock B
0.8 × 27% =
0.8 × 33% =
20.0%
21.6%
26.4%
Stock C
c.
0.8 × 40% =
32.0%
What is the standard deviation of your client’s portfolio?
σP = 0.8 × 27% = 21.6% per year
27. A portfolio of nondividend-paying stocks earned geometric mean return of 5 percent between
January 1, 2001 and December 31, 2007. The arithmetic mean return for the same period was 6
percent. If the market value of the portfolio at the beginning of 2001 was $100,000, what was the
market value of the portfolio at the end of 2007?
Value(12/31/2007) = Value(1/1/2001) × (1 + g)7
= $100,000 × (1.05)7 = $140,710.04
Using the following information to answer question 28-30
Consider historical data showing that the average annual return on the S&P 500 portfolio over the
past 80 years have averaged roughly 8.5% more than the T-bill return and that the S&P 500
standard deviation has been about 20% per year. Assume these values are representative of
investors' expectations for future performance and that the current T-bill rate is 5%
28. Calculate the expected return and variance of portfolios invested in T-bills and the S&P 500
index with weights as follows
W(bills)
W(index)
0
0.2
0.4
0.6
0.8
1.0
1.0
0.8
0.6
0.4
0.2
0
E(Rp)
σ 2 Portfolio
The portfolio expected return and variance are computed as follows:
(1)
WBills
0.0
0.2
0.4
0.6
(2)
rBills
5%
5%
5%
5%
(3)
WIndex
1.0
0.8
0.6
0.4
(4)
rIndex
13.5%
13.5%
13.5%
13.5%
rPortfolio
(1)×(2)+(3)×(4)
13.5% = 0.135
11.8% = 0.118
10.1% = 0.101
8.4% = 0.084
σPortfolio
(3) × 20%
20% = 0.20
16% = 0.16
12% = 0.12
8% = 0.08
σ 2 Portfolio
0.0400
0.0256
0.0144
0.0064
0.8
1.0
5%
5%
0.2
0.0
13.5%
13.5%
6.7% = 0.067
5.0% = 0.050
4% = 0.04
0% = 0.00
0.0016
0.0000
29. Calculate the utility levels for each of the portfolio of problem 28 for an investor with
A = 3. Assume the utility function is U = E(r) – 0.5 × Aσ 2
What do you conclude (i.e., which portfolio gives the investor highest utility)
WBills
0.0
0.2
0.4
0.6
0.8
1.0
WIndex
1.0
0.8
0.6
0.4
0.2
0.0
rPortfolio
0.135
0.118
0.101
0.084
0.067
0.050
σPortfolio
0.20
0.16
0.12
0.08
0.04
0.00
σ2Portfolio
0.0400
0.0256
0.0144
0.0064
0.0016
0.0000
U(A = 3)
0.0750
0.0796
0.0794
0.0744
0.0646
0.0500
U(A = 5)
0.0350
0.0540
0.0650
0.0680
0.0630
0.0500
The column labeled U(A = 3) implies that investors with A = 3 prefer a portfolio
that is invested 80% in the market index and 20% in T-bills to any of the other
portfolios in the table.
30. Repeat the problem 29 for an investor with A = 5. What do you conclude (i.e., which
portfolio gives the investor highest utility)?
The column labeled U(A = 5) in the table above is computed from:
U = E(r) – 0.5Aσ 2 = E(r) – 2.5σ 2
The more risk averse investors prefer the portfolio that is invested 40% in the
market index, rather than the 80% market weight preferred by investors with A = 3.