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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

Answers to Practice Questions

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

    is likely to be misleading.

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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