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Fundamentals of Coporate Finance Ch9~11 Ross

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Fundamentals of Coporate Finance Ch9~11 Ross

CHAPTER 9

    RISK AND RETURN: LESSONS FROM

    MARKET HISTORY

Answers to Concepts Review and Critical Thinking Questions

    1. They all wish they had! Since they didn’t, it must have been the case that the stellar performance was

    not foreseeable, at least not by most.

    2. As in the previous question, it’s easy to see after the fact that the investment was terrible, but it

    probably wasn’t so easy ahead of time.

    3. No, stocks are riskier. Some investors are highly risk averse, and the extra possible return doesn’t

    attract them relative to the extra risk.

    4. Unlike gambling, the stock market is a positive sum game; everybody can win. Also, speculators

    provide liquidity to markets and thus help to promote efficiency.

    5. T-bill rates were highest in the early eighties. This was during a period of high inflation and is

    consistent with the Fisher effect.

    6. Before the fact, for most assets, the risk premium will be positive; investors demand compensation

    over and above the risk-free return to invest their money in the risky asset. After the fact, the

    observed risk premium can be negative if the asset’s nominal return is unexpectedly low, the

    risk-free return is unexpectedly high, or if some combination of these two events occurs.

7. Yes, the stock prices are currently the same. Below is a diagram that depicts the stocks’ price

    movements. Two years ago, each stock had the same price, P. Over the first year, General 0

    Materials’ stock price increased by 10 percent, or (1.1) ( P. Standard Fixtures’ stock price declined 0

    by 10 percent, or (0.9) ( P. Over the second year, General Materials’ stock price decreased by 10 0

    percent, or (0.9)(1.1) ( P, while Standard Fixtures’ stock price increased by 10 percent, or (1.1)(0.9) 0

    ( P. Today, each of the stocks is worth 99 percent of its original value. 0

     2 years ago 1 year ago Today

    General Materials P(1.1)P (1.1)(0.9)P = (0.99)P0 000

    Standard Fixtures P(0.9)P (0.9)(1.1)P= (0.99)P0 00 0

    8. The stock prices are not the same. The return quoted for each stock is the arithmetic return, not the

    geometric return. The geometric return tells you the wealth increase from the beginning of the period

    to the end of the period, assuming the asset had the same return each year. As such, it is a better

    measure of ending wealth. To see this, assuming each stock had a beginning price of $100 per share,

    the ending price for each stock would be:

     Lake Minerals ending price = $100(1.10)(1.10) = $121.00

     Small Town Furniture ending price = $100(1.25)(.95) = $118.75

     CHAPTER 9 B- 2

     Whenever there is any variance in returns, the asset with the larger variance will always have the

    greater difference between the arithmetic and geometric return.

    9. To calculate an arithmetic return, you simply sum the returns and divide by the number of returns.

    As such, arithmetic returns do not account for the effects of compounding. Geometric returns do

    account for the effects of compounding. As an investor, the more important return of an asset is the

    geometric return.

    10. Risk premiums are about the same whether or not we account for inflation. The reason is that risk

    premiums are the difference between two returns, so inflation essentially nets out. Returns, risk

    premiums, and volatility would all be lower than we estimated because aftertax returns are smaller

    than pretax returns.

Solutions to Questions and Problems

    NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem.

     Basic

    1. The return of any asset is the increase in price, plus any dividends or cash flows, all divided by the

    initial price. The return of this stock is:

     R = [($94 83) + 1.40] / $83

     R = .1494 or 14.94%

    2. The dividend yield is the dividend divided by price at the beginning of the period, so:

     Dividend yield = $1.40 / $83

     Dividend yield = .0169 or 1.69%

     And the capital gains yield is the increase in price divided by the initial price, so:

     Capital gains yield = ($94 83) / $83

     Capital gains yield = .1325 or 13.25%

3. Using the equation for total return, we find:

     R = [($76 83) + 1.40] / $83

     R = .0675 or 6.75%

     And the dividend yield and capital gains yield are:

     Dividend yield = $1.40 / $83

     Dividend yield = .0169 or 1.69%

     CHAPTER 9 B- 3

     Capital gains yield = ($76 83) / $83

     Capital gains yield = .0843 or 8.43%

     Here’s a question for you: Can the dividend yield ever be negative? No, that would mean you were

    paying the company for the privilege of owning the stock. It has happened on bonds. Remember the

    Buffett bond’s we discussed in the bond chapter.

    4. The total pound return is the change in price plus the coupon payment, so:

     Total pound return = ?1,074 1,120 + 90

     Total pound return = ?44

     The total percentage return of the bond is:

     R = [(?1,074 1,120) + 90] / ?1,120

     R = .0393 or 3.93%

    Notice here that we could have simply used the total pound return of ?44 in the numerator of this

    equation.

     Using the Fisher equation, the real return was:

     (1 + R) = (1 + r)(1 + h)

     r = (1.0393 / 1.030) 1

     r = .0090 or 0.90%

5. The nominal return is the stated return, which is 12.40 percent. Using the Fisher equation, the real

    return was:

     (1 + R) = (1 + r)(1 + h)

     r = (1.1240)/(1.031) 1

     r = .0902 or 9.02%

    6. Using the Fisher equation, the real returns for government and corporate bonds were:

     (1 + R) = (1 + r)(1 + h)

     r = 1.058/1.031 1 G

     r = .0262 or 2.62% G

     r= 1.062/1.031 1 C

     r = .0301 or 3.01% C

     CHAPTER 9 B- 4 7. The average return is the sum of the returns, divided by the number of returns. The average return for each

    stock was:

    N??.11.06.08.28.13!,XxN .1000 or 10.00% ??i51??i

    N??.36.07.21.12.43!, YyN .1620 or 16.20%??i51??i

     We calculate the variance of each stock as:

    N??22;;;;sxxN1??Xii1??

    1222222;;;;;;;;;;??s.11.100.06.100.08.100.28.100.13.100.016850 X51

    1222222;;;;;;;;;;??s.36.162.07.162.21.162.12.162.43.162.061670Y51

     The standard deviation is the square root of the variance, so the standard deviation of each stock is:

     1/2 s = (.016850) X

     s = .1298 or 12.98% X

     1/2 s = (.061670) Y

     s = .2483 or 24.83% Y

8. We will calculate the sum of the returns for each asset and the observed risk premium first. Doing so,

    we get:

     Year Large co. stock return T-bill return Risk premium

     1973 14.69% 7.29% 21.98%

     1974 26.47 7.99 34.46

     1975 37.23 5.87 31.36

     1976 23.93 5.07 18.86

     1977 7.16 5.45 12.61

     1978 6.57 7.64 1.07

     19.41 39.31 19.90

     a. The average return for large company stocks over this period was:

     Large company stock average return = 19.41% /6

     Large company stock average return = 3.24%

     CHAPTER 9 B- 5

     And the average return for T-bills over this period was:

     T-bills average return = 39.31% / 6

     T-bills average return = 6.55%

     b. Using the equation for variance, we find the variance for large company stocks over this period

    was:

     2222 Variance = 1/5[(.1469 .0324) + (.2647 .0324) + (.3723 .0324) + (.2393 .0324) + 22 (.0716 .0324)+ (.0657 .0324)]

     Variance = 0.058136

     And the standard deviation for large company stocks over this period was:

     1/2 Standard deviation = (0.058136)

     Standard deviation = 0.2411 or 24.11%

     Using the equation for variance, we find the variance for T-bills over this period was:

     2222 Variance = 1/5[(.0729 .0655) + (.0799 .0655) + (.0587 .0655) + (.0507 .0655) + 22 (.0545 .0655) + (.0764 .0655)]

     Variance = 0.000153

     And the standard deviation for T-bills over this period was:

     1/2 Standard deviation = (0.000153)

     Standard deviation = 0.0124 or 1.24%

     c. The average observed risk premium over this period was:

     Average observed risk premium = 19.90% / 6

     Average observed risk premium = 3.32%

     The variance of the observed risk premium was:

     222 Variance = 1/5[(.2198 .0332) + (.3446 .0332) + (.3136 .0332) + 222 (.1886 .0332) + (.1261 .0332) + (.0107 .0332)]

     Variance = 0.062078

     And the standard deviation of the observed risk premium was:

     1/2 Standard deviation = (0.06278)

     Standard deviation = 0.2492 or 24.92%

9. a. To find the average return, we sum all the returns and divide by the number of returns, so:

     Arithmetic average return = (2.16 +.21 + .04 + .16 + .19)/5

     Arithmetic average return = .5520 or 55.20%

     CHAPTER 9 B- 6

     b. Using the equation to calculate variance, we find:

     2222 Variance = 1/4[(2.16 .552) + (.21 .552) + (.04 .552) + (.16 .552) + 2 (.19 .552)]

     Variance = 0.081237

     So, the standard deviation is:

     1/2 Standard deviation = (0.81237)

     Standard deviation = 0.9013 or 90.13%

10. a. To calculate the average real return, we can use the average return of the asset and the average

    inflation rate in the Fisher equation. Doing so, we find:

     (1 + R) = (1 + r)(1 + h)

     = (1.5520/1.042) 1 r

     = .4894 or 48.94% r

     b. The average risk premium is simply the average return of the asset, minus the average risk-free

    rate, so, the average risk premium for this asset would be:

     RP R Rf

     RP = .5520 .0510

     RP = .5010 or 50.10%

11. We can find the average real risk-free rate using the Fisher equation. The average real risk-free rate

    was:

     (1 + R) = (1 + r)(1 + h)

     = (1.051/1.042) 1 rf

     = .0086 or 0.86% rf

     And to calculate the average real risk premium, we can subtract the average risk-free rate from the

    average real return. So, the average real risk premium was:

     = 4.41% 0.86% rp rrf

     = 3.55% rp

12. Apply the five-year holding-period return formula to calculate the total return of the stock over the

    five-year period, we find:

    5-year holding-period return = [(1 + R)(1 + R)(1 +R)(1 +R)(1 +R)] 1 12345

     5-year holding-period return = [(1 .0491)(1 + .2167)(1 + .3257)(1 + .0619)(1 + .3185)] 1

     5-year holding-period return = 1.1475 or 114.75%

     CHAPTER 9 B- 7

    13. To find the return on the zero coupon bond, we first need to find the price of the bond today. Since

    one year has elapsed, the bond now has 19 years to maturity, so the price today is:

     19 P = ?1,000/1.10 1

     P = ?163.51 1

     There are no intermediate cash flows on a zero coupon bond, so the return is the capital gains, or:

     R = (?163.51 152.37) / ?152.37

     R = .0731 or 7.31%

14. The return of any asset is the increase in price, plus any dividends or cash flows, all divided by the

    initial price. This preferred stock paid a dividend of ?5, so the return for the year was:

     R = (?80.27 – 84.12 + 5.00) / ?84.12

     R = .0137 or 1.37%

15. The return of any asset is the increase in price, plus any dividends or cash flows, all divided by the

    initial price. This stock paid no dividend, so the return was:

     R = (Au$42.02 38.65) / Au$38.65

     R = .0872 or 8.72%

     This is the return for three months, so the APR is:

     APR = 4(8.72%)

     APR = 34.88%

     And the EAR is:

     4 EAR = (1 + .0872) 1

     EAR = .3971 or 39.71%

    16. To find the real return each year, we will use the Fisher equation, which is:

     1 + R = (1 + r)(1 + h)

     Using this relationship for each year, we find:

     T-bills Inflation Real Return

     1926

    0.0330 (0.0112) 0.0447

     1927

    0.0315 (0.0226) 0.0554

     1928

    0.0405 (0.0116) 0.0527

     1929

    0.0447 0.0058 0.0387

     1930

    0.0227 (0.0640) 0.0926

     CHAPTER 9 B- 8

     1931

    0.0115 (0.0932) 0.1155

     1932

    0.0088 (0.1027) 0.1243

     CHAPTER 9 B- 9

     So, the average real return was:

     Average = (.0447 + .0554 + .0527 + .0387 + .0926 + .1155 + .1243) / 7

     Average = .0748 or 7.48%

     Notice the real return was higher than the nominal return during this period because of deflation, or

    negative inflation.

    17. Looking at the long-term corporate bond return history in Figure 9.2, we see that the mean return

    was 6.2 percent, with a standard deviation of 8.6 percent. The range of returns you would expect to

    see 68 percent of the time is the mean plus or minus 1 standard deviation, or:

     R? ? 1~ = 6.2% ? 8.6% = 2.40% to 14.80%

     The range of returns you would expect to see 95 percent of the time is the mean plus or minus 2

    standard deviations, or:

     R? ? 2~ = 6.2% ? 2(8.6%) = 11.00% to 23.40%

    18. Looking at the large-company stock return history in Figure 9.2, we see that the mean return was

    12.4 percent, with a standard deviation of 20.3 percent. The range of returns you would expect to see

    68 percent of the time is the mean plus or minus 1 standard deviation, or:

     R? ? 1~ = 12.4% ? 20.3% = 7.90% to 32.70%

     The range of returns you would expect to see 95 percent of the time is the mean plus or minus 2

    standard deviations, or:

     R? ? 2~ = 12.4% ? 2(20.3%) = 28.20% to 53.00%

19. To find the best forecast, we apply Blume’s formula as follows:

    30 - 55 - 1 R(5) = × 10.7% + × 12.8% = 12.51% 2929

    10 - 130 - 10 R(10) = × 10.7% + × 12.8% = 12.15% 2929

    30 - 2020 - 1 R(20) = × 10.7% + × 12.8% = 11.42% 2929

     CHAPTER 9 B- 10

    20. The best forecast for a one year return is the arithmetic average, which is 12.4 percent. The

    geometric average, found in Table 9.3 is 10.4 percent. To find the best forecast for other periods, we

    apply Blume’s formula as follows:

    5 - 180 - 5 R(5) = × 10.4% + × 12.4% = 12.30% 80 - 180 - 1

    80 - 2020 - 1 R(20) = × 10.4% + × 12.4% = 11.92% 80 - 180 - 1

    30 - 180 - 30 R(30) = × 10.4% + × 12.4% = 11.67% 80 - 180 - 1

     Intermediate

    21. Here we know the average stock return, and four of the five returns used to compute the average

    return. We can work the average return equation backward to find the missing return. The average

    return is calculated as:

     .55 = .08 .13 .07 + .22 + R

     R = .45 or 45%

     The missing return has to be 45 percent. Now we can use the equation for the variance to find:

     22222 Variance = 1/4[(.08 .11) + (.13 .11) + (.07 .11) + (.22 .11) + (.45 .11)]

     Variance = 0.054650

     And the standard deviation is:

     1/2 Standard deviation = (0.054650)

     Standard deviation = 0.2338 or 23.38%

    22. The arithmetic average return is the sum of the known returns divided by the number of returns, so:

     Arithmetic average return = (.21 + .14 + .23 .08 + .09 .14) / 6

     Arithmetic average return = .075 or 7.5%

     Using the equation for the geometric return, we find:

     1/T Geometric average return = [(1 + R) × (1 + R) × … × (1 + R)] 1 12T(1/6) Geometric average return = [(1 + .21)(1 + .14)(1 + .23)(1 .08)(1 + .09)(1 .14)] 1

     Geometric average return = .0655 or 6.55%

     Remember, the geometric average return will always be less than the arithmetic average return if the

    returns have any variation.

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