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a. 1
b. 2
c. 3
d. 4
8. The variable (A) in the utility formula represents the:
a. investorâ€™s return requirement.
b. investorâ€™s aversion to risk.
c. certainty equivalent rate of the portfolio.
d. preference for one unit of return per four units of risk.
Use the following expectations on Stocks X and Y to answer questions 9 through 12 (round to
the nearest percent).
Bodieâˆ’Kaneâˆ’Marcus: II. Portfolio Theory 5. Risk and Return: Past Â© The McGrawâˆ’Hill
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Bear Market Normal Market Bull Market
Probability 0.2 0.5 0.3
Stock X 20% 18% 50%
Stock Y 15% 20% 10%

9. What are the expected returns for Stocks X and Y?

Stock X Stock Y
a. 18% 5%
b. 18% 12%
c. 20% 11%
d. 20% 10%

10. What are the standard deviations of returns on Stocks X and Y?

Stock X Stock Y
a. 15% 26%
b. 20% 4%
c. 24% 13%
d. 28% 8%

11. Assume that of your \$10,000 portfolio, you invest \$9,000 in Stock X and \$1,000 in
Stock Y. What is the expected return on your portfolio?
a. 18%
b. 19%
c. 20%
d. 23%
12. Probabilities for three states of the economy, and probabilities for the returns on a
particular stock in each state are shown in the table below.

Probability of
Stock Performance
Probability of Stock in Given
State of Economy Economic State Performance Economic State
Good .3 Good .6
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Neutral .3
Poor .1
Neutral .5 Good .4
Neutral .3
Poor .3
Poor .2 Good .2
Neutral .3
Poor .5
Bodieâˆ’Kaneâˆ’Marcus: II. Portfolio Theory 5. Risk and Return: Past Â© The McGrawâˆ’Hill
Essentials of Investments, and Prologue Companies, 2003
Fifth Edition

162 Part TWO Portfolio Theory

The probability that the economy will be neutral and the stock will experience poor
performance is
a. .06 c. .50
b. .15 d. .80
13. An analyst estimates that a stock has the following probabilities of return depending on
the state of the economy:

State of Economy Probability Return
Good .1 15%
Normal .6 13
Poor .3 7

The expected return of the stock is:
a. 7.8%
b. 11.4%
c. 11.7%
d. 13.0%
14. XYZ stock price and dividend history are as follows:

Year Beginning-of-Year Price Dividend Paid at Year-End
1999 \$100 \$4
2000 \$110 \$4
2001 \$ 90 \$4
2002 \$ 95 \$4

An investor buys three shares of XYZ at the beginning of 1999 buys another two shares
at the beginning of 2000, sells one share at the beginning of 2001, and sells all four
remaining shares at the beginning of 2002.
a. What are the arithmetic and geometric average time-weighted rates of return for the
investor?
b. What is the dollar-weighted rate of return. Hint: Carefully prepare a chart of cash
flows for the four dates corresponding to the turns of the year for January 1, 1999, to
January 1, 2002. If your calculator cannot calculate internal rate of return, you will
have to use trial and error.
15. a. Suppose you forecast that the standard deviation of the market return will be 20% in
the coming year. If the measure of risk aversion in equation 5.6 is A 4, what would
be a reasonable guess for the expected market risk premium?
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b. What value of A is consistent with a risk premium of 9%?
c. What will happen to the risk premium if investors become more risk tolerant?
annual HPR on the S&P 500 stock portfolio if the current risk-free interest rate is 5%?
17. What has been the historical average real rate of return on stocks, Treasury bonds, and
Treasury notes?
18. Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be
either \$50,000 or \$150,000, with equal probabilities of 0.5. The alternative riskless
investment in T-bills pays 5%.
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a. If you require a risk premium of 10%, how much will you be willing to pay for the
portfolio?
b. Suppose the portfolio can be purchased for the amount you found in (a). What will
the expected rate of return on the portfolio be?
c. Now suppose you require a risk premium of 15%. What is the price you will be
willing to pay now?
between the required risk premium on a portfolio and the price at which the portfolio
will sell?
For problems 19â€“23, assume that you manage a risky portfolio with an expected rate of re-
turn of 17% and a standard deviation of 27%. The T-bill rate is 7%.
19. a. Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill
money market fund. What is the expected return and standard deviation of your
clientâ€™s portfolio?
b. Suppose your risky portfolio includes the following investments in the given
proportions:

Stock A 27%
Stock B 33%
Stock C 40%

What are the investment proportions of your clientâ€™s overall portfolio, including the
position in T-bills?
c. What is the reward-to-variability ratio (S) of your risky portfolio and your clientâ€™s
overall portfolio?
d. Draw the CAL of your portfolio on an expected return/standard deviation diagram.
What is the slope of the CAL? Show the position of your client on your fundâ€™s CAL.
20. Suppose the same client in problem 19 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?
b. What are your clientâ€™s investment proportions in your three stocks and the T-bill
fund?
c. What is the standard deviation of the rate of return on your clientâ€™s portfolio?
21. Suppose the same client in problem 19 prefers to invest in your portfolio a proportion
(y) that maximizes the expected return on the overall portfolio subject to the constraint
that the overall portfolioâ€™s standard deviation will not exceed 20%.
a. What is the investment proportion, y?
b. What is the expected rate of return on the overall portfolio?
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22. You estimate that a passive portfolio invested to mimic the S&P 500 stock index yields
an expected rate of return of 13% with a standard deviation of 25%. Draw the CML and
your fundâ€™s CAL on an expected return/standard deviation diagram.
a. What is the slope of the CML?
b. Characterize in one short paragraph the advantage of your fund over the passive
fund.
23. Your client (see problem 19) wonders whether to switch the 70% that is invested in your
fund to the passive portfolio.
Bodieâˆ’Kaneâˆ’Marcus: II. Portfolio Theory 5. Risk and Return: Past Â© The McGrawâˆ’Hill
Essentials of Investments, and Prologue Companies, 2003
Fifth Edition

164 Part TWO Portfolio Theory

b. Show your client the maximum fee you could charge (as a percent of the
investment in your fund deducted at the end of the year) that would still leave him
at least as well off investing in your fund as in the passive one. (Hint: The fee
will lower the slope of your clientâ€™s CAL by reducing the expected return net of
the fee.)
24. What do you think would happen to the expected return on stocks if investors perceived
an increase in the volatility of stocks?
25. The change from a straight to a kinked capital allocation line is a result of the:
a. Reward-to-variability ratio increasing.
b. Borrowing rate exceeding the lending rate.
c. Investorâ€™s risk tolerance decreasing.
d. Increase in the portfolio proportion of the risk-free asset.
26. You manage an equity fund with an expected risk premium of 10% and an expected
standard deviation of 14%. The rate on Treasury bills is 6%. Your client chooses to
invest \$60,000 of her portfolio in your equity fund and \$40,000 in a T-bill money
market fund. What is the expected return and standard deviation of return on your
clientâ€™s portfolio?

Expected Return Standard Deviation of Return
a. 8.4% 8.4%
b. 8.4 14.0
c. 12.0 8.4
d. 12.0 14.0

27. What is the reward-to-variability ratio for the equity fund in problem 26?
a. .71
b. 1.00
c. 1.19
d. 1.91
For problems 28â€“30, download Table 5.3: Rates of return, 1926â€“2001, from www.mhhe.com/
blkm.
28. Calculate the same subperiod means and standard deviations for small stocks as Table
5.5 of the text provides for large stocks.
a. Do small stocks provide better reward-to-variability ratios than large stocks?
b. Do small stocks show a similar declining trend in standard deviation as Table 5.5
documents for large stocks?
29. Convert the nominal returns on both large and small stocks to real rates. Reproduce
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Table 5.5 using real rates instead of excess returns. Compare the results to those of
Table 5.5.
30. Repeat problem 29 for small stocks and compare with the results for nominal rates.
Bodieâˆ’Kaneâˆ’Marcus: II. Portfolio Theory 5. Risk and Return: Past Â© The McGrawâˆ’Hill
Essentials of Investments, and Prologue Companies, 2003
Fifth Edition

165
5 Risk and Return: Past and Prologue

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