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# Chapter 7 - CHAPTER 1

By Melissa Green,2014-05-07 17:10
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Chapter 7 - CHAPTER 1

CHAPTER 7

Introduction to Risk, Return, and

The Opportunity Cost of Capital

1. Recall from Chapter 3 that:

(1 + r) = (1 + r) ? (1 + inflation rate) nominalreal

Therefore:

r = [(1 + r)/(1 + inflation rate)] 1 realnominal

a. The real return on the stock market in each year was:

1999: 20.4%

2000: -13.8%

2001: -12.4%

2002: -22.8%

2003: 29.1%

b. From the results for Part (a), the average real return was: 0.10%

c. The risk premium for each year was:

1999: 18.9%

2000: -16.8%

2001: -14.8%

2002: -22.6%

2003: 30.6%

d. From the results for Part (c), the average risk premium was: 0.94%

e. The standard deviation (?) of the risk premium is calculated as follows:

??12222??? σ??(0.189?(?0.0094))?(?0.168 ?(?0.0094))?(?0.148?(?0.0094))??5?1??

22?(?0.226?(?0.0094))?(0.306?(?0.0094))]?0.057530

σ?0.057530?0.2399?23.99%

2. Internet exercise; answers will vary.

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3. a. A long-term United States government bond is always absolutely safe in

terms of the dollars received. However, the price of the bond fluctuates as

interest rates change and the rate at which coupon payments received

can be invested also changes as interest rates change. And, of course,

the payments are all in nominal dollars, so inflation risk must also be

considered.

b. It is true that stocks offer higher long-run rates of return than do bonds, but

it is also true that stocks have a higher standard deviation of return. So,

which investment is preferable depends on the amount of risk one is

willing to tolerate. This is a complicated issue and depends on numerous

factors, one of which is the investment time horizon. If the investor has a

short time horizon, then stocks are generally not preferred.

c. Unfortunately, 10 years is not generally considered a sufficient amount of

time for estimating average rates of return. Thus, using a 10-year average

4. In the context of a well-diversified portfolio, the only risk characteristic of a single

security that matters is the security’s contribution to the overall portfolio risk. This

contribution is measured by beta. Lonesome Gulch is the safer investment for a

diversified investor because its beta (+0.10) is lower than the beta of

Amalgamated Copper (+0.66). For a diversified investor, the standard deviations

are irrelevant.

5. The risk to Hippique shareholders depends on the market risk, or beta, of the

investment in the black stallion. The information given in the problem suggests

that the horse has very high unique risk, but we have no information regarding

the horse’s market risk. So, the best estimate is that this horse has a market risk

about equal to that of other racehorses, and thus this investment is not a

particularly risky one for Hippique shareholders.

= 0.60 ? = 0.10 6. xII

x = 0.40 ? = 0.20 JJ

a. ρ?1IJ

22222σ?[xσ?xσ?2(xxρσσ)] pIIJJIJIJIJ

2222?[(0.60)(0.10)?(0.40)(0.20)?2(0.60)(0.40)(1)(0.10)(0.20)]?0.0196

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b. ρ?0.50IJ

22222σ?[xσ?xσ?2(xxρσσ)] pIIJJIJIJIJ

2222?[(0.60)(0.10)?(0.40)(0.20)?2(0.60)(0.40)(0.50)(0.10)(0.20)]?0.0148

c. ρ?0ij 22222σ[xσxσ2(xxρσσ)]??? pIIJJIJIJIJ

2222?[(0.60)(0.10)?(0.40)(0.20)?2(0.60)(0.40)(0)(0.10)(0.20)]?0.0100

7. a. Refer to Figure 7.11 in the text. With 100 securities, the box is 100 by 100.

The variance terms are the diagonal terms, and thus there are 100

variance terms. The rest are the covariance terms. Because the box has

(100 times 100) terms altogether, the number of covariance terms is:

2 100 100 = 9,900

Half of these terms (i.e., 4,950) are different.

b. Once again, it is easiest to think of this in terms of Figure 7.11. With 50

stocks, all with the same standard deviation (0.30), the same weight in the

portfolio (0.02), and all pairs having the same correlation coefficient (0.40),

the portfolio variance is:

222222 σ = 50(0.02)(0.30) + [(50) 50](0.02)(0.40)(0.30) =0.03708

σ = 0.193 = 19.3%

c. For a fully diversified portfolio, portfolio variance equals the average

covariance:

2 σ = (0.30)(0.30)(0.40) = 0.036

σ = 0.190 = 19.0%

8. Internet exercise; answers will vary.

9. Internet exercise; answers will vary depending on time period.

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10. The table below uses the format of Table 7.11 in the text in order to calculate the

portfolio variance. The portfolio variance is the sum of all the entries in the matrix.

Portfolio variance equals: 0.0598355

Alcan BP Deutsche KLM LVMH Nestle Sony Alcan 0.0018613 0.0005745 0.0012915 0.0018138 0.0015790 0.0002484 0.0010539 BP 0.0005745 0.0011657 0.0004274 0.0007709 0.0004507 0.0002268 0.0003244 Deutsche 0.0012915 0.0004274 0.0029625 0.0015256 0.0015675 0.0001928 0.0014404 KLM 0.0018138 0.0007709 0.0015256 0.0060617 0.0022890 0.0005517 0.0010038 LVMH 0.0015790 0.0004507 0.0015675 0.0022890 0.0036000 0.0000266 0.0020357 Nestle 0.0002484 0.0002268 0.0001928 0.0005517 0.0000266 0.0004903 0.0001503 Sony 0.0010539 0.0003244 0.0014404 0.0010038 0.0020357 0.0001503 0.0046046

11. Internet exercise; answers will vary depending on time period.

12. “Safest” means lowest risk; in a portfolio context, this means lowest variance of

return. Half of the portfolio is invested in Deutsche Bank stock, and half of the

portfolio must be invested in one of the other securities listed. Thus, we calculate

the portfolio variance for six different portfolios to see which is the lowest. The

safest attainable portfolio is comprised of Deutsche Bank and Nestle.

Stocks Portfolio Variance

Deutsche & Alcan 0.090733

Deutsche & BP 0.061042

Deutsche & KLM 0.147923

Deutsche & LVMH 0.118795

Deutsche & Nestle 0.047021

Deutsche & Sony 0.127987

13. Internet exercise; answers will vary depending on time period.

14. a. In general, we expect a stock’s price to change by an amount equal to

(beta ? change in the market). Beta equal to 0.30 implies that, if the DAX

suddenly increases by 5 percent, then the expected change in the stock’s

price is 1.5 percent. If the DAX falls by 5 percent, then the expected

change is +1.5 percent.

b. “Safest” implies lowest risk. Assuming the well-diversified portfolio is

invested in typical securities, the portfolio beta is approximately one. The

largest reduction in beta is achieved by investing the ?30,000 in a stock

with a negative beta. Therefore, invest in the stock with β = 0.30.

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+ ?(r r) c. r = rfmf

For the stock with ? = 0.30:

0.064 = 0.04 + 0.30(r r) (r r) = 0.08 ?mfmf

For the stock with ? = 0.30:

r = 0.04 + 0.30(0.08) = 0.016 = 1.6%

15. Internet exercise; answers will vary depending on time period.

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Challenge Questions

1. a. In general:

22222 Portfolio variance = ? = x? + x? + 2xx??? P1122121212

Thus:

22222 ? = (0.5)(0.530)+(0.5)(0.475)+2(0.5)(0.5)(0.72)(0.530)(0.475) P

2 ? = 0.21726 P

Standard deviation = ? = 0.466 = 46.6% P

b. We can think of this in terms of Figure 7.11 in the text, with three

securities. One of these securities, T-bills, has zero risk and, hence, zero

standard deviation. Thus:

22222 ? = (1/3)(0.530)+(1/3)(0.475)+2(1/3)(1/3)(0.72)(0.530)(0.475) P

2 ? = 0.09656 P

Standard deviation = ? = 0.311 = 31.1% P

Another way to think of this portfolio is that it is comprised of one-third

T-Bills and two-thirds a portfolio which is half Dell and half Microsoft.

Because the risk of T-bills is zero, the portfolio standard deviation is two-

thirds of the standard deviation computed in Part (a) above:

Standard deviation = (2/3)(0.466) = 0.311 = 31.1%

c. With 50 percent margin, the investor invests twice as much money in the

portfolio as he had to begin with. Thus, the risk is twice that found in Part

(a) when the investor is investing only his own money:

Standard deviation = 2 ? 46.6% = 93.2%

d. With 100 stocks, the portfolio is well diversified, and hence the portfolio

standard deviation depends almost entirely on the average covariance of

the securities in the portfolio (measured by beta) and on the standard

deviation of the market portfolio. Thus, for a portfolio made up of 100

stocks, each with beta = 1.77, the portfolio standard deviation is

approximately: (1.77 ? 15%) = 26.55%. For stocks like Microsoft, it is:

(1.70 ? 15%) = 25.50%.

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2. For a two-security portfolio, the formula for portfolio risk is:

2222? + x? + 2xx??? Portfolio variance = x112212?1212

If security one is Treasury bills and security two is the market portfolio, then ? is 1

zero, ? is 20 percent. Therefore: 2

2222 Portfolio variance = x? = x(0.20) 222

Standard deviation = 0.20x 2

Portfolio expected return = x(0.06) + x(0.06 + 0.85) 12

Portfolio expected return = 0.06x + 0.145x 12

Expected Standard Portfolio X X 12Return Deviation

1 1.0 0.0 0.060 0.000

2 0.8 0.2 0.077 0.040

3 0.6 0.4 0.094 0.080

4 0.4 0.6 0.111 0.120

5 0.2 0.8 0.128 0.160

6 0.0 1.0 0.145 0.200

Portfolio Return & RiskPortfolio Return & Risk

0.2

?

0.15?

?

?0.1

?

? Expected Return0.05

0

00.050.10.150.20.25 Standard Deviation

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3. Internet exercise; answers will vary.

4. The matrix below displays the variance for each of the seven stocks along the

diagonal and each of the covariances in the off-diagonal cells:

Alcan BP Deutsche KLM LVMH Nestle Sony Alcan 0.0912040 0.0281494 0.0632841 0.0888786 0.0773724 0.0121706 0.0516420 BP 0.0281494 0.0571210 0.0209436 0.0377740 0.0220836 0.0111135 0.0158935 Deutsche 0.0632841 0.0209436 0.1451610 0.0747522 0.0768096 0.0094488 0.0705803 KLM 0.0888786 0.0377740 0.0747522 0.2970250 0.1121610 0.0270320 0.0491863 LVMH 0.0773724 0.0220836 0.0768096 0.1121610 0.1764000 0.0013020 0.0997500 Nestle 0.0121706 0.0111135 0.0094488 0.0270320 0.0013020 0.0240250 0.0073625 Sony 0.0516420 0.0158935 0.0705803 0.0491863 0.0997500 0.0073625 0.2256250

The covariance of Alcan with the market portfolio (σ) is the mean of the Alcan, Market

seven respective covariances between Alcan and each of the seven stocks in the

portfolio. (The covariance of Alcan with itself is the variance of Alcan.) Therefore,

σ is equal to the average of the seven covariances in the first row or, Alcan, Market

equivalently, the average of the seven covariances in the first column. Beta for

Alcan is equal to the covariance divided by the market variance (see Practice

Question 10). The covariances and betas are displayed in the table below:

Covariance Beta

Alcan 0.0589573 0.9853

BP 0.0275826 0.4610

Deutsche 0.0658542 1.1006

KLM 0.0981156 1.6398

LVMH 0.0808398 1.3510

Nestle 0.0132078 0.2207

Sony 0.0742914 1.2416

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