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# ch7 Risk and Rates of Return (solutions_nss_nc_8)

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ch7 Risk and Rates of Return (solutions_nss_nc_8)

Chapter 8

Risk and Rates of Return

Learning Objectives

After reading this chapter, students should be able to:

; Define risk and calculate the expected rate of return, standard deviation, and coefficient of variation for

a probability distribution.

; Specify how risk aversion influences required rates of return.

; Graph diversifiable risk and market risk; explain which of these is relevant to a well-diversified investor. ; State the basic proposition of the Capital Asset Pricing Model (CAPM) and explain how and why a

portfolio’s risk may be reduced.

; Explain the significance of a stock’s beta coefficient, and use the Security Market Line to calculate a

stock’s required rate of return.

; List changes in the market or within a firm that would cause the required rate of return on a firm’s

stock to change.

; Identify concerns about beta and the CAPM.

; Explain the implications of risk and return for corporate managers and investors.

Chapter 8: Risk and Rates of Return Learning Objectives 179

Lecture Suggestions

Risk analysis is an important topic, but it is difficult to teach at the introductory level. We just try to give students an intuitive overview of how risk can be defined and measured, and leave a technical treatment to advanced courses. Our primary goals are to be sure students understand (1) that investment risk is the uncertainty about returns on an asset, (2) the concept of portfolio risk, and (3) the effects of risk on required rates of return.

What we cover, and the way we cover it, can be seen by scanning the slides and Integrated Case solution for Chapter 8, which appears at the end of this chapter solution. For other suggestions about the lecture, please see the ―Lecture Suggestions‖ in Chapter 2, where we describe how we conduct our classes.

DAYS ON CHAPTER: 3 OF 58 DAYS (50-minute periods)

180 Lecture Suggestions Chapter 8: Risk and Rates of Return

8-1 a. No, it is not riskless. The portfolio would be free of default risk and liquidity risk, but inflation

could erode the portfolio’s purchasing power. If the actual inflation rate is greater than that

expected, interest rates in general will rise to incorporate a larger inflation premium (IP) and

as we saw in Chapter 7the value of the portfolio would decline.

b. No, you would be subject to reinvestment rate risk. You might expect to ―roll over‖ the

Treasury bills at a constant (or even increasing) rate of interest, but if interest rates fall, your

investment income will decrease.

c. A U.S. government-backed bond that provided interest with constant purchasing power (that is,

an indexed bond) would be close to riskless. The U.S. Treasury currently issues indexed bonds.

8-2 a. The probability distribution for complete certainty is a vertical line.

b. The probability distribution for total uncertainty is the X-axis from - to +.

8-3 a. The expected return on a life insurance policy is calculated just as for a common stock. Each

outcome is multiplied by its probability of occurrence, and then these products are summed.

For example, suppose a 1-year term policy pays \$10,000 at death, and the probability of the

policyholder’s death in that year is 2%. Then, there is a 98% probability of zero return and a

2% probability of \$10,000:

Expected return = 0.98(\$0) + 0.02(\$10,000) = \$200.

This expected return could be compared to the premium paid. Generally, the premium will

be larger because of sales and administrative costs, and insurance company profits, indicating a

negative expected rate of return on the investment in the policy.

b. There is a perfect negative correlation between the returns on the life insurance policy and the

returns on the policyholder’s human capital. In fact, these events (death and future lifetime

earnings capacity) are mutually exclusive. The prices of goods and services must cover their

costs. Costs include labor, materials, and capital. Capital costs to a borrower include a return

to the saver who supplied the capital, plus a mark-up (called a ―spread‖) for the financial

intermediary that brings the saver and the borrower together. The more efficient the financial

system, the lower the costs of intermediation, the lower the costs to the borrower, and, hence,

the lower the prices of goods and services to consumers.

c. People are generally risk averse. Therefore, they are willing to pay a premium to decrease the

uncertainty of their future cash flows. A life insurance policy guarantees an income (the face

value of the policy) to the policyholder’s beneficiaries when the policyholder’s future earnings

capacity drops to zero.

8-4 Yes, if the portfolio’s beta is equal to zero. In practice, however, it may be impossible to find

individual stocks that have a nonpositive beta. In this case it would also be impossible to have a stock portfolio with a zero beta. Even if such a portfolio could be constructed, investors would probably be better off just purchasing Treasury bills, or other zero beta investments.

Chapter 8: Risk and Rates of Return Integrated Case 181

8-5 Security A is less risky if held in a diversified portfolio because of its negative correlation with other

> and CV > CV. stocks. In a single-asset portfolio, Security A would be more risky because ABAB

8-6 No. For a stock to have a negative beta, its returns would have to logically be expected to go up in

the future when other stocks’ returns were falling. Just because in one year the stock’s return

increases when the market declined doesn’t mean the stock has a negative beta. A stock in a given

year may move counter to the overall market, even though the stock’s beta is positive.

8-7 The risk premium on a high-beta stock would increase more than that on a low-beta stock.

RP = Risk Premium for Stock j = (r r)b. jMRFj

If risk aversion increases, the slope of the SML will increase, and so will the market risk premium th(r r). The product (r r)b is the risk premium of the j stock. If b is low (say, 0.5), then MRFMRFjj

the product will be small; RP will increase by only half the increase in RP. However, if b is large jMj

(say, 2.0), then its risk premium will rise by twice the increase in RP. M

8-8 According to the Security Market Line (SML) equation, an increase in beta will increase a company’s

expected return by an amount equal to the market risk premium times the change in beta. For

example, assume that the risk-free rate is 6%, and the market risk premium is 5%. If the

company’s beta doubles from 0.8 to 1.6 its expected return increases from 10% to 14%. Therefore,

in general, a company’s expected return will not double when its beta doubles.

8-9 a. A decrease in risk aversion will decrease the return an investor will require on stocks. Thus,

prices on stocks will increase because the cost of equity will decline.

b. With a decline in risk aversion, the risk premium will decline as compared to the historical

difference between returns on stocks and bonds.

c. The implication of using the SML equation with historical risk premiums (which would be higher

than the ―current‖ risk premium) is that the CAPM estimated required return would actually be

higher than what would be reflected if the more current risk premium were used.

182 Integrated Case Chapter 8: Risk and Rates of Return

Solutions to End-of-Chapter Problems

ˆ7-1 = (0.1)(-50%) + (0.2)(-5%) + (0.4)(16%) + (0.2)(25%) + (0.1)(60%) r

= 11.40%.

2222 = (-50% 11.40%)(0.1) + (-5% 11.40%)(0.2) + (16% 11.40%)(0.4) 22 + (25% 11.40%)(0.2) + (60% 11.40%)(0.1) 2 = 712.44; = 26.69%.

26.69%CV = = 2.34. 11.40%

7-2 Investment Beta

\$35,000 0.8

40,000 1.4

Total \$75,000

b = (\$35,000/\$75,000)(0.8) + (\$40,000/\$75,000)(1.4) = 1.12. p

7-3 r = 8%; r = 12%; b = 0.8; r = ? RFM

r = r + (r r)b RFMRF

= 8% + (12% 8%)0.8

= 11.2%.

7-4 r = 5.5%; RP = 6.5%; r = ? RFMM

r = 5.5% + (6.5%)1 = 12%. M

r when b = 1.2 = ?

r = 5.5% + 6.5%(1.2) = 13.3%.

7-5 a. r = 11%; r = 7%; RP = 4%. RFM

r = r + (r r)b RFMRF

11% = 7% + 4%b

4% = 4%b

b = 1.

Chapter 8: Risk and Rates of Return Integrated Case 183

= 7%; RP = 6%; b = 1. b. rRFM

r = r + (r r)b RFMRF

= 7% + (6%)1

= 13%.

N

ˆ7-6 a. . rPriii1

ˆ = 0.1(-35%) + 0.2(0%) + 0.4(20%) + 0.2(25%) + 0.1(45%) rY

= 14% versus 12% for X.

N2ˆb. = . (r;r)Piii1

2222 = (-10% 12%)(0.1) + (2% 12%)(0.2) + (12% 12%)(0.4) σX22 + (20% 12%)(0.2) + (38% 12%)(0.1) = 148.8%.

= 12.20% versus 20.35% for Y. X

ˆCV = / = 12.20%/12% = 1.02, while rXXX

CV = 20.35%/14% = 1.45. Y

If Stock Y is less highly correlated with the market than X, then it might have a lower beta than

Stock X, and hence be less risky in a portfolio sense.

\$400,000\$600,000\$1,000,000\$2,000,0007-8 Portfolio beta = (1.50) + (-0.50) + (1.25) + (0.75) \$4,000,000\$4,000,000\$4,000,000\$4,000,000

b = (0.1)(1.5) + (0.15)(-0.50) + (0.25)(1.25) + (0.5)(0.75) p

= 0.15 0.075 + 0.3125 + 0.375 = 0.7625.

r = r + (r r)(b) = 6% + (14% 6%)(0.7625) = 12.1%. pRFMRFp

Alternative solution: First, calculate the return for each stock using the CAPM equation

[r + (r r)b], and then calculate the weighted average of these returns. RFMRF

r = 6% and (r r) = 8%. RFMRF

Stock Investment Beta r = r + (r r)b Weight RFMRF

A \$ 400,000 1.50 18% 0.10

B 600,000 (0.50) 2 0.15

C 1,000,000 1.25 16 0.25

D 2,000,000 0.75 12 0.50

Total \$4,000,000 1.00

r = 18%(0.10) + 2%(0.15) + 16%(0.25) + 12%(0.50) = 12.1%. p

184 Integrated Case Chapter 8: Risk and Rates of Return

7-9 In equilibrium:

ˆ = = 12.5%. rrJJ

r = r + (r r)b JRFMRF

12.5% = 4.5% + (10.5% 4.5%)b

b = 1.33.

7-10 We know that b = 1.50, b = 0.75, r = 13%, r = 7%. RSMRF

r = r + (r r)b = 7% + (13% 7%)b. iRFMRFii

r = 7% + 6%(1.50) = 16.0% R

r = 7% + 6%(0.75) = 11.5 S

4.5%

7-11 An index fund will have a beta of 1.0. If r is 12.0% (given in the problem) and the risk-free rate is M

5%, you can calculate the market risk premium (RP) calculated as r r as follows: MMRF

r = r + (RP)b RFM

12.0% = 5% + (RP)1.0 M

7.0% = RP. M

Now, you can use the RP, the r, and the two stocks’ betas to calculate their required returns. MRF

r = r + (RP)b BRFM

= 5% + (7.0%)1.45

= 5% + 10.15%

= 15.15%.

Farley:

r = r + (RP)b FRFM

= 5% + (7.0%)0.85

= 5% + 5.95%

= 10.95%.

The difference in their required returns is:

15.15% 10.95% = 4.2%.

7-12 r = r* + IP = 2.5% + 3.5% = 6%. RF

r = 6% + (6.5%)1.7 = 17.05%. s

Chapter 8: Risk and Rates of Return Integrated Case 185

= r + (r r)b = 9% + (14% 9%)1.3 = 15.5%. 7-13 a. riRFMRFi

b. 1. r increases to 10%: RF

r increases by 1 percentage point, from 14% to 15%. M

r = r + (r r)b = 10% + (15% 10%)1.3 = 16.5%. iRFMRFi

2. r decreases to 8%: RF

r decreases by 1%, from 14% to 13%. M

r = r + (r r)b = 8% + (13% 8%)1.3 = 14.5%. iRFMRFi

c. 1. r increases to 16%: M

r = r + (r r)b = 9% + (16% 9%)1.3 = 18.1%. iRFMRFi

2. r decreases to 13%: M

r = r + (r r)b = 9% + (13% 9%)1.3 = 14.2%. iRFMRFi

7-15 Using Stock X (or any stock):

9% = r + (r r)b RFMRFX

9% = 5.5% + (r r)0.8 MRF

(r r) = 4.375%. MRF

\$142,500\$7,5007-16 Old portfolio beta = (b) + (1.00) \$150,000\$150,000

1.12 = 0.95b + 0.05

1.07 = 0.95b

1.1263 = b.

New portfolio beta = 0.95(1.1263) + 0.05(1.75) = 1.1575 1.16.

Alternative solutions:

1. Old portfolio beta = 1.12 = (0.05)b + (0.05)b + ... + (0.05)b 1220

1.12 = (0.05) (b)i

= 1.12/0.05 = 22.4. bi

New portfolio beta = (22.4 1.0 + 1.75)(0.05) = 1.1575 1.16.

2. excluding the stock with the beta equal to 1.0 is 22.4 1.0 = 21.4, so the beta of the bi

portfolio excluding this stock is b = 21.4/19 = 1.1263. The beta of the new portfolio is:

1.1263(0.95) + 1.75(0.05) = 1.1575 1.16.

186 Integrated Case Chapter 8: Risk and Rates of Return

= 1.8; b = 0.6. No changes occur. 7-17 bHRILRI

r = 6%. Decreases by 1.5% to 4.5%. RF

r = 13%. Falls to 10.5%. M

Now SML: r = r + (r r)b. iRFMRFi

r = 4.5% + (10.5% 4.5%)1.8 = 4.5% + 6%(1.8) = 15.3% HRI

r = 4.5% + (10.5% 4.5%)0.6 = 4.5% + 6%(0.6) = 8.1% LRI

Difference 7.2%

7-18 Step 1: Determine the market risk premium from the CAPM:

0.12 = 0.0525 + (r r)1.25 MRF

(r r) = 0.054. MRF

Step 2: Calculate the beta of the new portfolio:

(\$500,000/\$5,500,000)(0.75) + (\$5,000,000/\$5,500,000)(1.25) = 1.2045.

Step 3: Calculate the required return on the new portfolio:

5.25% + (5.4%)(1.2045) = 11.75%.

7-19 After additional investments are made, for the entire fund to have an expected return of 13%, the portfolio

must have a beta of 1.5455 as shown below:

13% = 4.5% + (5.5%)b

b = 1.5455.

Since the fund’s beta is a weighted average of the betas of all the individual investments, we can

calculate the required beta on the additional investment as follows:

(\$20,000,000)(1.5)\$5,000,000X 1.5455 = + \$25,000,000\$25,000,000

1.5455 = 1.2 + 0.2X

0.3455 = 0.2X

X = 1.7275.

7-20 a. (\$1 million)(0.5) + (\$0)(0.5) = \$0.5 million.

b. You would probably take the sure \$0.5 million.

c. Risk averter.

d. 1. (\$1.15 million)(0.5) + (\$0)(0.5) = \$575,000, or an expected profit of \$75,000.

2. \$75,000/\$500,000 = 15%.

3. This depends on the individual’s degree of risk aversion.

Chapter 8: Risk and Rates of Return Integrated Case 187

4. Again, this depends on the individual.

5. The situation would be unchanged if the stocks’ returns were perfectly positively correlated.

Otherwise, the stock portfolio would have the same expected return as the single stock

(15%) but a lower standard deviation. If the correlation coefficient between each pair of

for stocks is stocks was a negative one, the portfolio would be virtually riskless. Since

generally in the range of +0.35, investing in a portfolio of stocks would definitely be an

improvement over investing in the single stock.

ˆ7-21 = 10%; b = 0.9; = 35%. rXXX

ˆ = 12.5%; b = 1.2; = 25%. rYYY

r = 6%; RP = 5%. RFM

a. CV = 35%/10% = 3.5. CV = 25%/12.5% = 2.0. XY

b. For diversified investors the relevant risk is measured by beta. Therefore, the stock with the

higher beta is more risky. Stock Y has the higher beta so it is more risky than Stock X.

c. r = 6% + 5%(0.9) X

= 10.5%.

r = 6% + 5%(1.2) Y

= 12%.

ˆd. r = 10.5%; = 10%. rXX

ˆr = 12%; = 12.5%. rYY

Stock Y would be most attractive to a diversified investor since its expected return of 12.5% is

greater than its required return of 12%.

e. b = (\$7,500/\$10,000)0.9 + (\$2,500/\$10,000)1.2 p

= 0.6750 + 0.30

= 0.9750.

r = 6% + 5%(0.975) p

= 10.875%.

f. If RP increases from 5% to 6%, the stock with the highest beta will have the largest increase M

in its required return. Therefore, Stock Y will have the greatest increase.

Check:

r = 6% + 6%(0.9) X

= 11.4%. Increase 10.5% to 11.4%.

r = 6% + 6%(1.2) Y

= 13.2%. Increase 12% to 13.2%.

188 Integrated Case Chapter 8: Risk and Rates of Return

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