index model, 194 portfolio, 185

PROBLEM 1. A three-asset portfolio has the following characteristics:

SETS

Expected Standard

Asset Return Deviation Weight

X 15% 22% 0.50

Y 10 8 0.40

Z 6 3 0.10

What is the expected return on this three-asset portfolio?

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2. An investor is considering adding another investment to a portfolio. To achieve the

maximum diversification benefits, the investor should add, if possible, an investment

that has which of the following correlation coefficients with the other investments in the

portfolio?

a. 1.0

b. 0.5

c. 0.0

d. 1.0

3. Consistent with capital market theory, systematic risk:

i. Refers to the variability in all risky assets caused by macroeconomic and other

aggregate market-related variables.

ii. Is measured by the coefficient of variation of returns on the market portfolio.

iii. Refers to nondiversifiable risk.

a. i only

b. ii only

c. i and iii only

d. ii and iii only

4. Suppose that the returns on the stock fund presented in Spreadsheet 6.1 were 14%,

13%, and 30% in the three scenarios.

a. Would you expect the mean return and variance of the stock fund to be more than,

less than, or equal to the values computed in Spreadsheet 6.2? Why?

b. Calculate the new values of mean return and variance for the stock fund using a

format similar to Spreadsheet 6.2. Confirm your intuition from part (a).

c. Calculate the new value of the covariance between the stock and bond funds using a

format similar to Spreadsheet 6.4. Explain intuitively why covariance has increased.

5. Use the rate of return data for the stock and bond funds presented in Spreadsheet 6.1,

but now assume that the probability of each scenario is: Recession: 0.4; Normal: 0.2;

Boom: 0.4.

a. Would you expect the mean return and variance of the stock fund to be more than,

less than, or equal to the values computed in Spreadsheet 6.2? Why?

b. Calculate the new values of mean return and variance for the stock fund using a

format similar to Spreadsheet 6.2. Confirm your intuition from part (a).

c. Calculate the new value of the covariance between the stock and bond funds using

a format similar to Spreadsheet 6.4. Explain intuitively why the absolute value of

the covariance has increased.

The following data apply to problems 6“10.

A pension fund manager is considering three mutual funds. The first is a stock fund,

the second is a long-term government and corporate bond fund, and the third is a T-bill

money market fund that yields a sure rate of 5.5%. The probability distributions of the

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risky funds are:

Expected Return Standard Deviation

Stock fund (S) 15% 32%

Bond fund (B) 9 23

The correlation between the fund returns is 0.15.

6. Tabulate and draw the investment opportunity set of the two risky funds.

Use investment proportions for the stock fund of 0 to 100% in increments of 20%.

What expected return and standard deviation does your graph show for the minimum

variance portfolio?

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202 Part TWO Portfolio Theory

7. Draw a tangent from the risk-free rate to the opportunity set. What does your graph

show for the expected return and standard deviation of the optimal risky portfolio?

8. What is the reward-to-variability ratio of the best feasible CAL?

9. Suppose now that your portfolio must yield an expected return of 12% and be efficient,

that is, on the best feasible CAL.

a. What is the standard deviation of your portfolio?

b. What is the proportion invested in the T-bill fund and each of the two risky funds?

10. If you were to use only the two risky funds and still require an expected return of 12%,

what would be the investment proportions of your portfolio? Compare its standard

deviation to that of the optimal portfolio in the previous problem. What do you

conclude?

11. Stocks offer an expected rate of return of 10% with a standard deviation of 20% and

gold offers an expected return of 5% with a standard deviation of 25%.

a. In light of the apparent inferiority of gold to stocks with respect to both mean return

and volatility, would anyone hold gold? If so, demonstrate graphically why one

would do so.

b. How would you answer (a) if the correlation coefficient between gold and stocks

were 1.0? Draw a graph illustrating why one would or would not hold gold. Could

these expected returns, standard deviations, and correlation represent an equilibrium

for the security market?

12. Suppose that many stocks are traded in the market and that it is possible to borrow at

the risk-free rate, rf . The characteristics of two of the stocks are as follows:

Stock Expected Return Standard Deviation

A 8% 40%

B 13 60

Correlation 1

Could the equilibrium rf be greater than 10%? (Hint: Can a particular stock portfolio be

substituted for the risk-free asset?)

13. Assume expected returns and standard deviations for all securities, as well as the risk-

free rate for lending and borrowing, are known. Will investors arrive at the same

optimal risky portfolio? Explain.

14. Your assistant gives you the following diagram as the efficient frontier of the group of

stocks you asked him to analyze. The diagram looks a bit odd, but your assistant insists

he got the diagram from his analysis. Would you trust him? Is it possible to get such a

diagram?

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B

A

Expected return

Standard deviation

15. What is the relationship of the portfolio standard deviation to the weighted average

of the standard deviations of the component assets?

16. A project has a 0.7 chance of doubling your investment in a year and a 0.3 chance of

halving your investment in a year. What is the standard deviation of the rate of return on

this investment?

17. Investors expect the market rate of return this year to be 10%. The expected rate of

return on a stock with a beta of 1.2 is currently 12%. If the market return this year turns

out to be 8%, how would you revise your expectation of the rate of return on the stock?

18. The following figure shows plots of monthly rates of return and the stock market for

two stocks.

a. Which stock is riskiest to an investor currently holding her portfolio in a diversified

portfolio of common stock?

b. Which stock is riskiest to an undiversified investor who puts all of his funds in only

one of these stocks?

rA “ rf rB “ rf

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rM “ rf rM “ rf

Bodie’Kane’Marcus: II. Portfolio Theory 6. Efficient Diversification © The McGraw’Hill

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204 Part TWO Portfolio Theory

19. Here are rates of return for six months for Generic Risk, Inc. What is Generic™s beta?

(Hint: Find the answer by plotting the scatter diagram.)

Month Market Return Generic Return

1 0% 2%

2 0 0

3 1 0

4 1 2

5 1 4

6 1 2

The following data apply to problems 20“22:

Hennessy & Associates manages a $30 million equity portfolio for the multimanager

Wilstead Pension Fund. Jason Jones, financial vice president of Wilstead, noted that Hen-

nessy had rather consistently achieved the best record among the Wilstead™s six equity

managers. Performance of the Hennessy portfolio had been clearly superior to that of the

S&P 500 in four of the past five years. In the one less favorable year, the shortfall was

trivial.

Hennessy is a “bottom-up” manager. The firm largely avoids any attempt to “time the

market.” It also focuses on selection of individual stocks, rather than the weighting of

favored industries.

There is no apparent conformity of style among the six equity managers. The five

managers, other than Hennessy, manage portfolios aggregating $250 million, made up of

more than 150 individual issues.

Jones is convinced that Hennessy is able to apply superior skill to stock selection, but

the favorable results are limited by the high degree of diversification in the portfolio.

Over the years, the portfolio generally held 40“50 stocks, with about 2% to 3% of total

funds committed to each issue. The reason Hennessy seemed to do well most years was

because the firm was able to identify each year 10 or 12 issues that registered particularly

large gains.

Based on this overview, Jones outlined the following plan to the Wilstead pension

committee:

Let™s tell Hennessy to limit the portfolio to no more than 20 stocks. Hennessy will double the

commitments to the stocks that it really favors and eliminate the remainder. Except for this

one new restriction, Hennessy should be free to manage the portfolio exactly as before.

All the members of the pension committee generally supported Jones™s proposal, be-

cause all agreed that Hennessy had seemed to demonstrate superior skill in selecting

stocks. Yet, the proposal was a considerable departure from previous practice, and several

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committee members raised questions.

20. Answer the following:

a. Will the limitation of 20 stocks likely increase or decrease the risk of the portfolio?

Explain.

b. Is there any way Hennessy could reduce the number of issues from 40 to 20 without

significantly affecting risk? Explain.

21. One committee member was particularly enthusiastic concerning Jones™s proposal.

He suggested that Hennessy™s performance might benefit further from reduction in the

number of issues to 10. If the reduction to 20 could be expected to be advantageous,

explain why reduction to 10 might be less likely to be advantageous. (Assume that

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Fifth Edition

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6 Efficient Diversification

Wilstead will evaluate the Hennessy portfolio independently of the other portfolios in

the fund.)

22. Another committee member suggested that, rather than evaluate each managed portfolio

independently of other portfolios, it might be better to consider the effects of a change

in the Hennessy portfolio on the total fund. Explain how this broader point of view

could affect the committee decision to limit the holdings in the Hennessy portfolio

to either 10 or 20 issues.

23. What percent of the variance of stock ABC in Example 6.4 is systematic (market) risk?

STANDARD & POOR™S

1. Go to www.mhhe.com/edumarketinsight. Use data from Market Insight to

calculate the beta of Apple Computer (AAPL). Start by copying the daily price

changes of Apple and the S&P 500 (Daily Adjusted Prices Report) into Excel,

calculating the daily rate of return for each of the series, and then calculating a

regression with Apple™s return as the dependent variable and the S&P 500 return