Chapter_11_Solutions

Chapter 11

Systematic Risk and the Equity Risk Premium

Note: All problems in this chapter are available in MyFinanceLab. An asterisk (*) indicates problems

with a higher level of difficulty.

1. Plan: The expected return on any portfolio is the weighted average of the expected returns of the securities in the portfolio. Therefore we will compute the weighted average return on this portfolio. Execute:[](70%)(20%)(30%)(15%)18.5%p E R =+= Evaluate: The expected return on this portfolio is 18.5%.

2. Plan: Perform the calculations to answer the questions in the problem.

Execute:

a. Let i n be the number of share in stock i , then

200,0000.5

4,000

25

200,0000.2562580

200,0000.25

25,0002G M V n n n ×=

=×==×=

=

The new value of the portfolio is

30603$232,500

G M p n n nv =++=

b. Return 232,500

1200,00016.25%

=

?=

126 Berk/DeMarzo/arford ? Fundamentals of Corporate Finance

c. The portfolio weight are the fraction of value invested in each stock

30

GoldFinger: 51.61%232,50060

Moosehead: 16.13%232,5003

Venture: 32.26%232,500

G M V n n n ×=

×=

×=

Evaluate:

a. The new value of the portfolio is $232,500.

b. The return on the portfolio was 16.25%.

c. If you don’t buy or sell shares after the price change, your new portfolio weights are GoldFinger 51.61%, Moosehead 16.13%, and Venture 32.26%. 3. Both calculations of expected return of a portfolio give the same answer. 4. If the price of one stock goes up, the other stock price always goes up as well. 5. Plan: Download the data and make the calculations required in the problem.

Execute:

a. Correlation between Dell and Starbucks (using Excel’s “correl” function): 0.191685

b. Monthly standard deviation for: Dell: 14.56% Starbucks: 11.85%

c. Monthly variance and standard deviation of a portfolio containing 30% Dell and 70% Starbucks:

22222(0.3)(0.1456)(0.7)(0.1185)2(0.7)(0.3)(0.1456)(0.1185)(0.1917)0.0101790.100910.09%

σσ=++===

Evaluate: The correlation between the returns of Dell and Starbucks is 0.191685, which is low positive correlation. The monthly standard deviations of the returns of Dell are 14.56%, and Starbuck is 11.85%. The monthly standard deviation of a portfolio of these two stocks is 10.09, which is lower than either of the standard deviations of the stocks that make up the portfolio.

6. Plan: Calculate the expected return and volatility of Stock A and Stock B.

Realized Returns

Year Stock A

Stock B

1998 ?10% 21% 1999 20% 30% 2000 5% 7% 2001 ?5% ?3% 2002 2% ?8% 2003

9%

25%

Chapter 11 Systematic Risk and the Equity Risk Premium 127

Execute:

10205529

6

3.5%

2130738256

12%A B R R ?++?++=

=++??+=

=

2222

22(0.10.035)(0.20.08)1Variance of (0.050.035)(0.050.035)5(0.020.035)(0.090.035)

0.01123

Volatility of ()10.60%

A A A SD R ??+?+???

?=?+??+?????+???

====

2222

22(0.210.12)(0.30.12)1Variance of (0.070.12)(0.030.12)5(0.080.12)(0.250.12)

0.02448

Volatility of ()15.65%

B B B SD R ?+?+???

?=?+??+??????+???

====

Evaluate: The return on Stock A is 3.5% with a volatility of 10.60%. The return on Stock B is

12% with a volatility of 15.65%.

7. Plan: Calculate the volatility of a portfolio that is 70% invested in Stock A and 30% invested in Stock B.

Execute:

0.105110.51%

σ===

Evaluate: The volatility of a portfolio of 70% invested in Stock A and 30% in Stock B is 10.51%.

8. Plan: Calculate the average monthly return and volatility for the stock of KO and XOM.

Date KO XOM 19900131 –10.84% ?6.00%19900228 2.36% 1.28%19900330 6.60% –1.86%19900430 2.01% –1.90%19900531 18.36% 7.40%19900629 –1.22% ?0.26%19900731 2.25% 8.36%

Continued

128 Berk/DeMarzo/arford ? Fundamentals of Corporate Finance

Date KO XOM 19900831 –6.89% –2.46%

19900928 –6.04% –2.00%19901031 13.61% 0.00%19901130 3.51% 4.68%19901231 0.54% 2.22%

Execute: The mean for KO is 2.02%; the mean for XOM is 0.79%.

The standard deviation (i.e., volatility) for KO is 8.24%; the standard deviation for XOM is 4.25%.

Evaluate: KO has a higher mean return (2.02%) than XOM (0.79%). But KO has more

volatility (8.24%) than XOM (4.25%). This is consistent with Finance Theory—higher risk is associated with higher average return.

9. All three methods result have the same result: The standard deviation (i.e., volatility) is 5.90%. 10. Plan: Use Equations (11.3) and (11.9) to compute the expected return and volatility of the

indicated portfolio.

Execute: In this case, the portfolio weights are x j = x w = 0.50. From Eq. (11.3),

[][][]

0.50(7%)0.50(10%)8.5%

P j j w w E R x E R x E R =+=+=

We can use Eq. (11.9)

()14.1%

P SD R = Evaluate: The portfolio would have an expected return of 8.5% and a standard deviation of return of 14.1%.

11. Plan: You must estimate the expected return and volatility of each portfolio created by adding

Stock A or Stock B. You will select that portfolio that gives you the greatest return or the least volatility.

Execute: The expected return of the portfolio will be the same (17.4%) if you pick A or B, since both A and B have the same expected return. Therefore, the choice of A or B depends on how risky the portfolio becomes when you add A or B.

For

A:

0.2548

25.48%

σ===

Chapter 11 Systematic Risk and the Equity Risk Premium 129

For

B:

0.2659

26.59%

σ===

Evaluate: Since the portfolio is less risky when A is added, you should add A to the portfolio. 12. Plan: Compute the total market value of the total portfolio and the weighted percent that each

individual stock would be in the market portfolio.

Execute: Total value of the market 10102012835014520$1.314 billion.=×+×+×+×+×= Stock Portfolio Weight A 1010

7.61%1314

×=B 2012

18.26%1314

×=C 83

1.83%1314

×= D 50

3.81%1314

= E

4520

68.49%1314

×=

Evaluate: The market portfolio would have a value of $1.314 billion. Stock A would be 7.61% of the market portfolio, Stock B would be 18.26%, Stock C would be 1.83%, Stock D would be 3.81%, and Stock E would be 68.49%.

13. Plan: Compute the total market value of the total portfolio and the weighted percent that each

individual stock would be in the market portfolio.

Execute: Total value of all four stocks =×+×+×+×=13 1.0022 1.254330510$1,380.5 billion. Stock Portfolio Weight Golden Seas 13 1.00

0.942%

1380.5

×=Jacobs and Jacobs 22 1.25

1.992%

1380.5

×=MAG 4330

93.444%

1380.5

×=PDJB

510

3.622%

1380.5

×

=

130 Berk/DeMarzo/arford ? Fundamentals of Corporate Finance

Evaluate: The market portfolio would have a value of $1.380.5 billion. Golden Seas would be

0.942% of the market portfolio, Jacobs and Jacobs would be 1.992%, MAG would be 93.444%,

and PDJB would be 3.622%.

14.Nothing needs to be done. The portfolio is still value-weighted.

15.Plan: Compute the excess returns of Apple and Proctor and Gamble.

Execute:

a. The best guess to Apple’s return today is the product of the market return and Apple’s beta.

Apple’s return=?×=?

2% 1.4 2.8%.

b. P&G’s return=?×=?

2% 0.5 1%.

Evaluate: Apple’s excess return is –2.8% and P&G’s is –1.0%.

16. Plan: Go to the MyFinanceLab Web site and access the Excel spreadsheet. Use the slope function

to estimate the slope coefficient of the data, which is our estimate of beta.

Execute: Using Excel’s slope function, the beta of Nike’s stock is 0.5679.

Evaluate: The estimate of beta for Nike is 0.5679.

17. Plan: Go to the MyFinanceLab Web site and access the Excel spreadsheet. Use the slope function

to estimate the slope coefficient of the data, which is our estimate of beta.

Execute:

a. Solving for Microsoft’s beta using the slope function in Excel:

1.4110

1987–1991:

0.8544

1992–1996:

1997–2001:

1.8229

1.0402

2002–2006:

Evaluate:

b. It decreased in the early 1990s as Microsoft established itself as the dominant operating

software company, but increased during the Internet bubble in the late 1990s (when tech

stocks were soaring). It has since decreased.

18. Plan: Compute the expected return for Johnson and Johnson.

Execute:Expected Return4%0.32(0.100.04) 5.92%

=+×?=

Evaluate: The expected return for Johnson and Johnson is 5.92%.

19.The sign of the risk premium for a negative beta stock is negative. This is because the negative

beta stock acts as “recession insurance,” and thus investors are willing to pay for this insurance

in the form of a lower return than the risk-free rate.

Chapter 11 Systematic Risk and the Equity Risk Premium 131

20. Plan: Compute the expected returns of Intel and Boeing as well as the portfolio beta. Then

compute the expected return of the portfolio.

Execute:

a. Intel’s Expected Return =+×?=4% 1.6(0.100.04)13.6%.

b. Boeing’s Expected Return =+×?=4% 1.0(0.100.04)10%.

c. Portfolio beta =+= (60%)(1.6)(40%)(1.0) 1.36.

d. Portfolio’s Expected Return 4% 1.36(0.100.04)12.16%=+×?=

or Portfolio’s Expected Return =+=(60%)(13.6%)(40%)(10%)12.16%.

Evaluate: Intel’s expected return is 13.6%, Boeing’s expected return is 10%, the portfolio beta is 1.36, and the expected return of the portfolio is 12.16%. *21. Plan: Compute the necessary beta.

Execute: Return on the stock =

1117100

18%100

+?=

For 18% to be the expected return on the stock, solve for beta:

Expected Return 18% 4.5%beta (6%)

18% 4.5%

beta 6%2.25

==+×?=

= Evaluate: A beta of 2.25 would be consistent with an 18% return on the stock.

*22. Plan: Compute what the expected return for a stock with a beta of 1.2 should be. You should

buy the stock if the expected return is 11% or less.

Execute: Expected Return =+×=5% 1.2(6%)12.2%.

Evaluate: No, you should not buy the stock. You should expect a return of 12.2% for taking

on an investment with a beta of 1.2. But since this stock only returns 11%, it does not fully compensate you for the risk of the stock, and you can find other investments that will return 11% with less risk.

23. Plan: Compute the expected return for Starbucks.

Execute: Expected Return =+×=5%0.6(5.5%)8.3%.

Evaluate: Starbucks should produce a return of 8.3% to compensate its equity investors for the

riskiness of their investment.

132 Berk/DeMarzo/arford ? Fundamentals of Corporate Finance

24. Plan: Compute the expected return and the realized return for Apple.

Execute:

Expected Return 4.5% 1.4(6%)

12.9%198.08 84.84Realized Return

84.84133.47%

=+×=?

==

Evaluate: Apple’s managers greatly exceeded the required return of investors, as given by the CAPM.